Forecasting New York State Electricity Prices

using Time-Series Models

Senior Thesis

Presented toThe Faculty of the School of Arts and Sciences

Brandeis University

Undergraduate Program in Economics

George Hall, Advisor

In partial fulfillment of the requirements for the degree ofBachelor of Arts

by

Jacob N. SilverMay 2016

Copyright byJacob N. Silver

Forecasting New York State Day-Ahead Electricity

Spot Prices using Time-Series Models

Jacob N. Silver ’16

May 12, 2016

Abstract

This thesis examines the forecasting proficiency of several linear and non-linear timeseries models on the day-ahead hourly electricity spot prices of the New York statemarket, the New York Independent System Operator (NYISO). The study considersMarkov Chains, Linear Regressions, VAR, AR, ARMA, and ARIMA models. Theanalysis shows that the ARIMA models consistently forecast with the greatest accuracyacross each of the NYISO’s geographically diverse zones and in each season studied.ARIMA models handle large price spikes during peak periods with the greatest accuracy,consistently forecast evening hours with the highest demand and volatility, and maintainaccuracy during mid-week days with historically volatile prices. The analysis also showsthat exogenous explanatory variables increase forecast accuracy substantially. Thestrongest models for the NYISO, and markets that function similarly to the NYISO,should incorporate forecasted values of future price and load. ARIMA models withthese exogenous variables consistently have MAPEs of 4% compared to past studieswith MAPEs of 6-10%. Translated over a two week period, this results in roughly $3.46million in savings.

Senior Thesis

Brandeis University Department of Economics

Primary Advisor: George Hall

Secondary Advisor: Daniel Tortorice

Third Reader: Winston Bowman

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1 IntroductionElectricity price forecasting is a product of the mass liberalization of electricity markets

in Western Europe and North America during the 1990s. Deregulated markets transformedthe entire electric power sector by separating generator from distributor and creating elec-tricity trading through regional wholesale markets. The liberalization of electricity createdmany economic questions, one of which is how to forecast wholesale electricity spot pricesaccurately. To begin to answer this question, this paper will examine price-forecasting tech-niques in the state of New York.

This research shows that ARIMA models perform the most accurate forecasts across allseasons and every geographic zone in New York. ARIMA models with exogenous variablesthat include forecasted future values of price and load are the most accurate across peakpricing periods as well as periods with the greatest demand at the level of daily and hourlyaverages. The paper reports 4% Mean Absolute Percentage Errors (MAPEs)1 for ARIMAmodels compared to past studies 6-10% MAPEs which translates to $3.46 million in savingsover a two week period in 20142. These findings apply not only to New York State, but toareas with similar climate and electricity sectors like New England, Quebec, Ontario, theNorthern Midwest, Scandinavian Europe and parts of the northern United Kingdom.

Accurate price forecasts in the New York market, and similar deregulated markets, arecrucial to increase efficiency and expand our generation capabilities (Weron 2014). Electricitymarkets generally trade at hourly levels. Hourly electricity price forecasts send signals togenerators alerting them of profitable and efficient points to produce power and enter themarket (Shahidehpour, Yamin, and Li 2002).

A pervasive question in the 21st century is the viability of renewable power. Renewablepower is highly capital intensive and requires the price of electricity to be at a certain level inorder to maintain economic viability. With increased accuracy in price forecasts, renewablegenerators will know exactly when they can economically operate. Outside of renewablegeneration, the electricity sector can achieve gains in overall efficiency from increasinglyaccurate electricity price forecasts. Oil and coal plants are necessary to meet peak demandacross the Northeast; however, they are very expensive and inefficient to turn on. Accurateprice forecasts can send these plants signals to increase efficiency in generation scheduling.

Demand Response is an efficiency and reliability tool that has grown in popularity sincethe turn of the millennium. It is directly linked to accurate electricity price forecasts. De-

1Mean Absolute Percentage Errors, which measure the percentage deviation of forecasts from the actualprice.

2Over a two week period approximately $173 billion of electricity is traded in the NYISO.

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mand Response helps relieve the system of demand at crucial points in the day. Largeindustrial users rely on Demand Response to make economic decisions about electricity useand generation. Accurate hourly spot price forecasts are essential to make these economicdecisions.

With increased technological innovation, Demand Response can be used to increase mar-ket place efficiency and thereby avoid price volatility and extreme shocks seen throughoutelectricity markets. This will also lower demand and put fewer constraints on both genera-tors and distributors. Accurate hourly electricity price forecasts are crucial for any form ofDemand Response as well as efficient economic generation.

There have been several overviews of electricity price forecasting, most focused on Eu-rope’s liberalized markets. There are many techniques used across multiple disciplines toaccurately forecast these prices. This paper explores the validity and robustness of thestatistical-regression based forecasts used across Europe and Canada to predict the day-ahead hourly electricity prices in the state of New York. New York is an ideal market toanalyze because it has liberalized and deregulated over the last 15 years, leaving ample loadand price data. New York’s electricity-market is divided into 11 regions spread throughoutthe state with contrasting geographic and population attributes, allowing a variety of modelsto be tested against a diverse set of backgrounds.

I was motivated to undertake this project after reading an overview of electricity priceforecasting methods (Weron 2014). There are few publicly available studies on the North-east generally or New York specifically. As the Northeast’s electricity sector continues toderegulate and modernize, price forecasts will grow increasingly significant. I was deeplyinterested in exploring the leading variables to determine the wholesale price of electricity.There is a void in the study of forecasts in the Northeast; this paper attempts to begin tofill that gap as the region pushes for a more modern and innovative grid.

There are three types of electricity price forecasts: short-term, medium-term and long-term. Short-term forecasts are used to establish strategies in the market, including generatoroperation. Medium-term forecasts are used by demand serving entities to determine costeffective power supply strategies. Long-term forecasts are used to make long-run business andpolicy decisions including sites of generators, types of generators and transmission capacityexpansions. Accurate electricity price forecasts are essential for all three, with large economicramifications for entire regions of the country.

This paper builds upon the past research to test which predictive models perform thebest in the New York hourly day-ahead electricity market, a previously unstudied state withcomplex weather patterns and regulatory structures. These forecasts can influence electricpower and energy policy including renewable resource integration and demand side response,

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as well as individual market players’ actions.

2 NYISO and the New York ElectricitySector

In this section, I provide a general overview of New York’s electricity sector. The sectiondescribes the New York Independent System Operator’s (NYISO) role as the administratorof the wholesale market and provides an overview of NYISO’s service territory. A history ofNYISO and New York’s electricity sector deregulation can be found in Section A. NYISOAppendix.

2.1 General Overview

New York’s electricity sector is one of many deregulated and competitive electricitymarkets in the country. A deregulated market generally has competitive supply, whereElectric Load Serving Entities (LSEs) purchase electricity in a heavily regulated market froma group of competitive generators and then distribute the electricity to end use customersacross the LSEs service territory. A noncompetitive electricity sector usually consists ofvertically integrated utilities that provide generation, transmission and distribution servicesacross their service territory. The integrated utilities are responsible for both wholesale andretail electricity3. With competition, LSEs, such as utilities, are obligated to transmit anddistribute retail electricity, but generation (wholesale electricity) is separated and traded in acompetitive market or secured by LSEs through bilateral contracts. Deficiencies in meetingload (scarcities) are penalized with serious regulatory sanctions. Over the period studied,the NYISO has not seen a scarcity or electricity shortage (NYISO 2014a).

New York’s electricity sector joined many states across the country and deregulatedin 1999. A deregulated electricity sector requires additional participants to manage thecompetitive wholesale market. In the United States and Canada these actors are referred toas Independent Service Operators or ISOs4. ISOs administer regions’, or states’, wholesaleelectricity markets and transmission services to ensure adequate system reliability. To ensuresystem reliability, ISOs manage generator dispatch. To simplify, ISOs facilitate wholesale

3As electricity moves from generation its voltage changes, wholesale electricity is electricity flowingthrough transmission lines at high voltage levels. Retail electricity is electricity that has had its voltagestepped down through transformers and travels through distribution lines to customers.

4In some regions ISO’s are referred to as Regional Transmission Operators.

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energy transactions between generators and LSEs such as distribution utilities. Competitiveelectricity markets exist throughout the world; each has an entity that functions like an ISO(Weron 2014).

2.2 NYISO’s Objective and Roles

NYSIO has a broad and complicated role in New York’s electric power sector. NYISOperforms a majority of the former New York Power Pool’s (NYPP) functions5. Primarily,NYSIO administers the competitive wholesale electricity markets. NYISO divides the stateinto 11 regional zones that function as 11 separate markets within the greater NYISO terri-tory. It is important to note the geographic and population disparities between the NYISOregions. The regions range from Long Island and New York City to rural upstate New York(See Figure 1 below). In 2014 the NYISO market oversaw the transaction of 37,978 MWand 11,056 miles of transmission (NYISO 2014b).

Figure 1: NYISO Zones

Source: NYISO. 2015. “New York Control Area Load Zones.”5The NYPP coordinated the sale of wholesale electric power and performed essential reliability functions.

To do this it dispatched generating units according to schedules provided by the utilities, balanced electricsystem supply and demand in real time, maintained transmission line voltage, managed operating reserves,and monitored contingencies that required rapid response to assure system reliability. A full description ofthe NYPP can be found in Section A. NYISO Appendix

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Broadly, NYISO is charged with preserving system reliability by using price signals toattract investments in the grid as they are needed. NYISO must preserve system reliabilitywith the cheapest available resources. They place a value on energy that reflects systemconditions and enables the system to shift the risk of investment in the grid from consumersto investors (NYISO 2014b). Prior to NYISO, the risk of grid investment was placed solely onthe consumer through electric rate increases for the vertically integrated utilities to recovertheir stranded capital in generation assets. With generator and distributer detached fromone another, the investor bears the risk.

This paper primarily focuses on day-ahead energy markets. To preserve New York’s elec-tric system reliably, NYISO has three mechanisms to ensure load is always met: bilateralcontracts, day-ahead and real-time markets. Bilateral transactions are settled outside of theNYISO market; the power producer and the LSE set their quantities and prices indepen-dently, accounting for approximately 40% of New York’s electricity. The remaining 60% issettled in the market. Both the day-ahead and real-time markets function as two settlementprocess auctions where the price of energy is based on the market and grid conditions at spe-cific times. In the market, 94% of energy is traded day-ahead and the reaming 6% real-time(NYISO 2014b).

NYISO must secure the schedule for a set of generators to be available for dispatch ineach hour of the next day and must ensure that a set of LSEs is scheduled to buy a certainamount of load at the day-ahead price; this is the day-ahead market. Essentially, NYISOis responsible for ensuring the market clears for each hour, real-time and day-ahead. Tomaintain the schedule, NYISO establishes Locational Based Marginal Prices (LBMPs) foreach of the 11 NYISO zones. LBMPs are the marginal cost of energy added to natural energylosses due to transmission and finally subtracting marginal cost of congestion6.

LBMP = MCE +MCL�MCC (1)

LBMPs are computed, then forward contracts are established for generation and loadaccordingly to maintain the schedule. Differences in generation levels and load consump-tion from the day-ahead market (first settlement process) must be resolved in the secondsettlement (real-time market) or by the power producers. The New York Public ServiceCommission (NYPSC), New York State Reliability Council (NYSRC), and the Federal En-ergy Regulatory Commission (FERC) subject the NYISO and LSEs to heavy regulatoryscrutiny to ensure system reliability and avoid scarcity.

It is easiest to understand the day-ahead electricity market from the perspective of a6The Marginal Cost of Energy is constant across each NYISO zone while the transmissions losses and

congestion costs are independent to each zone.

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LSE. The LSE purchases electricity in the market corresponding to its distribution terri-tory. In this market, generators compete through LBMPs to trade electricity to the LSE.Once the LSE purchases the electricity from the generators, it is dispatched throughout theLSEs service territory and distributed to the LSE’s residential, commercial, and industrialcustomers. The cost is not passed directly through due to the heavily regulated nature ofelectricity rates.

In addition to the energy market, NYISO administers a capacity market (ICAP). ICAPensures affordable, reliable electricity during peak demand. LSEs purchase capacity fromtransmission operators to fulfill their load obligations during peak periods. ICAP functionssimilarly to the energy market in that load can be met through bilateral contracts and auctionbased markets, which account for 45% of capacity. There are three separate auctions: a six-month seasonal auction, monthly auctions and monthly spot market auctions. Also similarto energy markets, ICAP prices are established by load and market conditions, often heavilyinfluenced by seasonal weather conditions (NYISO 2014b).

The ancillary service market also allows LSEs to trade capacity outside of peaking periods.This market supports the transmission of energy from generator to LSEs prior to retaildistribution. The market functions hourly, the products are regulated reserve and operatingreserve transmission (spinning) capacity. This market sends signals to generators to controltheir output due to sudden moment-to-moment changes. In this way, the ancillary servicemarket helps account for energy imbalances in the transmission system. The market sendsmessages to generators to shutdown or run. This allows the NYISO to play an active rolein demand response programs, which can help remove demand from the grid and lowerwholesale prices for all consumers.

NYSRC and NYPSC heavily regulate the NYISO markets, both energy and capacity, toensure constant reliability and avoid energy shortages. Certain suppliers are deemed piv-otal suppliers, which means they must constantly have a pre-determined amount of capacityavailable at auction during all times. NYISO also operates with a price floor to preventundercutting by any buyer regardless of market share. NYISO markets trade at hourly in-tervals, but commitment and energy flows must be reported to the NYISO every five minutesto allow the NYISO to maintain system balance and dispatch generators if market conditionspoint toward a coming shortage or scarcity. NYSRC, NYPSC and the FERC oversee eachof these actions to ensure the reliability of the electric system and avoid blackouts.

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2.3 NYISO’s Service Territory

The maps below highlight two important characteristics of NYISO. First, the NYISO isan incredibly diverse territory spanning the entire state of New York (See Figure 2 below).New York varies significantly in both geography and population density, therefore the pricesin the zones will change accordingly. New York City’s prices will differ greatly from regionssuch as North and West due to differences in population density and weather. Statisticalforecasts that are accurate in New York City and Long Island could behave entirely differentin other Zones of the state.

Figure 2: NYISO Zones and Counties

Source: NYISO. 2015. “New York Control Area Load Zones.”

Second, it is important to note NYISO’s proximity to the IESO of Ontario, ISONE, PJMISO and Quebec7 (See Figure 3 below). These entities, all in close proximity to NYISO,trade electricity with NYISO and have an impact on NYISO’s LBMPs. NYISO’s neighborsabundance of electricity result in New York operating as a net importer of power largely dueto New York’s natural geographic and load constraints.

7Quebec’s electricity is generated and distributed by Hydro Quebec a state owned utility.

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Figure 3: RTOs/ISOs of North America

Source: Walsh, Linda L. 2015. “FERC 101 for Environmental Lawyers.” In American Bar Association,edited by Hunton & Williams. Washington DC.

3 Literature ReviewIn this section, I provide a survey of the major trends in wholesale electricity spot price

forecasting literature and specify the econometric tests I will run based on the methodologyput forward in tests performed on similar markets. Wholesale electricity markets are aproduct of worldwide electricity deregulation that began in the United Kingdom and NordicEurope during the late 1980s and early 1990s, because of this, comprehensive electricity priceforecasting studies only began in the early 2000s. Electricity price forecasting is still in itsearly stages and there are only a handful of accepted statistical or regression based models.

Initially, econometric forecasting was focused on electricity demand known as “load”.However, in recent years there has been a push toward price forecasting. These forecaststake on a variety of components from various econometric and financial models. Electricityprice forecasting extends across many disciplines, this paper only addresses the statisticalregression based forecasting techniques currently used. Rafal Weron has produced a compre-

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hensive overview of electricity price forecasting methods in the 21st century (Weron 2014).Summarized below is an overview of the statistical regression based electricity price fore-casts sampled in this paper. Section D. Literature Appendix provides an overview of modelsexcluded from the analysis, but reviewed for the study and considered when determiningmodels to incorporate.

3.1 Issues and Considerations with Electricity Price Forecasting

The electricity price forecasting literature highlights several unique characterizations ofelectricity markets that are essential for all forecasts. First, there is no effective mechanismto store electricity in bulk so the market must constantly clear (Weron 2014). Transmissioncongestion can prevent free exchange among control areas (Shahidehpour, Yamin, and Li2002). Electricity prices and load follow many patterns. First, price experiences large sea-sonal and calendar effects as well as intra-week and intra-day trends. Power system stabilitynecessitates constant balancing of supply and demand, which means that the governmentregulator plays a critical role in the process. Price volatility is far greater than all othercommodities and financial assets (Weron 2014). Electricity spot prices are often nonlinear,adding another layer of complexity (Bunn 2000). Negative prices are permissible and occurwhen demand is very low for two reasons. First, Some regulatory structures, including NewYork, require select generators to run at all times to maintain system reliability. Second, asmentioned above, there is no efficient storage mechanism for electricity so generation mustbe consumed (Weron 2014).

Derek Bunn of the London Business School provides an overview of statistical basedelectricity price forecasting (Bunn and Karakatsani 2003). He explains that the statisticalissues stem from the instantaneous nature of electricity where production and consumptionmust be perfectly synchronized (Bunn and Karakatsani 2003). There is very little short-term elasticity of demand to price. Bunn notes that price exhibits great complexity to meetdemand, which it must due to the heavily regulated nature of the industry and electricity’sprominent roles in the economy. Bunn stresses that electricity prices have long term meanreversion, multi-scale seasonality, calendar effects, and erratic extreme behavior with fastreverting spikes, excessive volatility, as well as technological effects. Not all generators areuniform, which leads to varying degrees of efficiency. Electricity markets are also uniquebecause generators can go off the grid due to technical problems, which will have an impacton the price of power (Bunn and Karakatsani 2003).

Several other studies highlight that the mean and variance of electricity prices are non-constant (Aggarwal, Saini, and Kumar 2009). In most markets, generation is oligopolistic and

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market equilibrium is influenced by both demand and supply side uncertainties (Aggarwal,Saini, and Kumar 2009). These unique attributes combine and make forecasting wholesaleelectricity different from many other commodities or assets.

3.1.1 Price Volatility

As mentioned above price volatility is a large issue with electricity price forecasting.Volatility is defined as the measure of change in the price of electricity over a given periodof time (Shahidehpour, Yamin, and Li 2002). Price is more volatile than load. The loadcurve is homogenous with a high degree of cyclicality. The price curve is highly volatile,non-homogeneous with limited cyclicality.

Price is volatile but not random (Shahidehpour, Yamin, and Li 2002). Transmissioncongestion leads to price spikes, but generally price reverts back to the mean, highlightingthe mean-reverting quality of electricity prices. Volatility can also be a result of fuel pricefluctuations, load uncertainty, variations in large generator output, generation uncertainties(outages), behavior of market participants, and market manipulation.

3.1.2 Significant Indicators of Wholesale Electricity Prices

The literature points toward several significant indicators used in electricity price fore-casting. The first important variable is time: year, month, day and hour (Shahidehpour,Yamin, and Li 2002). Time, including special days and holidays, has a large impact on pricemovements. Next, transmission capacity and transmission reserve, both historical and fore-casted, are important because they predict congestion, which heavily influences price spikes.Historical price is the most significant variable in all of the regression-based forecasting mod-els (Crespo Cuaresma et al. 2004), (Weron 2014), and (Zareipour 2012). Load fluctuationsalso move price. Most studies incorporate historical and forecasted load (Shahidehpour,Yamin, and Li 2002) (Karakatsani and Bunn 2008). Finally, electricity prices move withthe price of marginal fuel. Over the time horizon of this paper the marginal fuel has transi-tioned from nuclear power to natural gas in New York. In the Spanish market the significantfluctuating generation source used in many forecasts is wind energy (Weron 2014) while inthe Nord Pool and other Northern European countries it is hydroelectric power reserves andgeneration (Koopman, Ooms, and Carnero 2007).

3.1.3 The Importance of Electricity Price Forecasts

The volatile nature of electricity combined with the economic and societal consequencesof electricity market failure makes electricity price forecast accuracy crucial despite the inher-

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ent difficulties mentioned above. When electricity markets malfunction and prices surge therecan be widespread electricity blackouts. While rolling blackouts currently affect countrieswith non-modernized or centralized grids such as India, there have been several large-scaleblackouts in the United States since deregulation. California in 2000-2001 and the greaterSouthwest in the summer of 1996 experienced massive price surges that resulted in a seriesof rolling power outages. In both of these case studies demand peaked due to cooling on hotdays, which resulted in price spikes with supply and demand imbalances (Lambert 2006).

These two cases highlight the intrinsic obstacles associated with forecasting electricityprices and load. However, had there been more accurate electricity price forecasts in bothof these markets, the grid administrators could have pushed for more bilateral contracts,alleviating some of the imbalances in the market. New York’s geographic location leaves itsusceptible to winter peaking and price surges. Electricity price forecasting is an importantcomponent of generator scheduling through which the system can avoid market failures andblackouts from peaking demand and surging prices.

3.2 Linear Regression and AR Models

Despite many alternatives, linear regressions remain amongst the most popular electric-ity price forecasting techniques to date (Weron 2014). Jesus Crespo Cuaresma analyzes theLeipzig Power Exchange (LPX) with autoregressive, autoregressive-moving average and un-observed component models between June 16, 2000 and October 15, 2001 (Crespo Cuaresmaet al. 2004). LPX functions very similarly to the NYISO, which makes this study ideal forcomparison.

LPX is a day-ahead market with bids for buying and selling put forward at each hourfor next 24 hours, the same as NYISO. Weekday prices are higher than weekend prices. Themarket experiences shocks that lead to a high frequency of price spikes (Crespo Cuaresmaet al. 2004). Supply shocks occur when large power plants have unplanned outages con-trasting demand shocks which occur during heat waves in the summer and cold snaps in thewinter. Market mechanism failures such as capacity constraints and deviations from perfectcompetition due to oligopolistic buyers and sellers also lead to price spikes. New York Statefollows a similar pattern.

Crespo Cuaresma makes several important assumptions. First, the probability of a pricespike is assumed constant at each hour of the day. Prices higher than the V quintile arenormally distributed and centered around the empirical mean of electricity spot prices. Next,seasonal patterns are assumed to remain constant throughout time and are modeled asdummies while irregularities are assumed to be white noise. Cyclical trends in the data

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are assumed to follow sine-cosine waves with constant frequency. Finally, the probability ofjumps and all hour specific shocks are assumed to be uncorrelated with each other (CrespoCuaresma et al. 2004). Taking these assumptions Crespo Cuaresma tests the regressionmodels listed above.

AR(1):pt = ↵ + �pt�1 + nt ⇠ NID(0, �2

n) (2)

AR(1) Time-Varying Intercept

pt = ↵t + �pt�1 + nt ⇠ NID(0, �2n) (3)

↵t = ↵0t+24X

i=1

↵1,iI +7X

j=1

↵2,jI +12X

k=1

↵3,kI (4)

Linear Regression:pj,t = Xj,t�j + "j,t N(0, �2

j ) (5)

In equations 2 and 3, pt is the spot-price of electricity and the error term nt is assumed tobe white noise. In each of the linear regression run by Crespo Cuaresma the regressions canbe unaggregated and run for each specific hour. The Time-Varying Intercept AR process(equation 3) allows the intercept to adjust at hourly, daily and monthly levels which is anecessary limitation of the linear regression models. in equation 4 I is t hour i, t day j, tmonth k.

Equation 5 comes from a study conducted by Nektaria Karakatsani and Derek Bunn(Karakatsani and Bunn 2008). Here pj,t is the spot price for day t in period j so that 48 isthe max for j because Karakatsani and Bunn test on two day intervals. Xjt is a 16x1 vectorof exogenous regressors, T is the sample size, �j, is a 16x1 vector of coefficients, "j,t is ani.i.d error term. More specifically than above:

Xj,t = (1, Pj(t�1), Pj(t�7),MPt�1, Demand.Linjt, Demand.Quadjt,

DemandCurvjt,Marginjt,Marginj(t�1), Scarcityjt, DemandV oljt,

T rend, Seasonalityjt (6)

Karakatsani and Bunn test two variants of equation 5. The two equations are variationsof Xjt that include different components of he 16x1 vector (Karakatsani and Bunn 2008).

Pt = �0 + �1Pt�1 + �2Demand.Lint + �3Margint + �4PriceV olt + "t (7)

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Pt = �0 + �1Pt�1 + �2Demand.Lint + �3Demand.Quadt+

�4Margint + �5Margint�1 + �6DemandCurvt + "t

(8)

Karakatsani and Bunn tested the simple linear regressions and found that the linear mod-els were still fundamentally valid. The MAPE for the linear regressions generally approached8%. The regression was tested between June of 2001 and January of 2002 in the EnglishSpot Market. However, time-varying parameters and moving averages tests often yieldedMAPEs between 4-6.5% (Karakatsani and Bunn 2008).

The results of Crespo Cuaresma were tested for accuracy using a Diebold-Mariano Test(DM). They show that the separable crossed autoregressive-moving average models for eachhour was the most accurate forecast. To conclude, the study notes that more researchneeds to be done on both linear and nonlinear models. Although linear regressions oftenover simply electricity price forecasts, they are a common starting place for many moresophisticated models.

3.2.1 VAR

A final variant of the AR model referenced in the literature is the Vector AutoregressiveModel (VAR). This model has been put forward as a way to help handle the relationshipbetween price and load (Weron 2014). The goal of these models is to attempt to handlethe correlation between hours of both price and load for the highly correlated hours of theday (Weron 2014). The central assumption the VAR models is that both load and pricemove together (Weron 2014). While in theory this seems logical, it has already been notedthat load is cyclical while price is highly volatile (Shahidehpour, Yamin, and Li 2002). Thisis the fundamental flaw with VAR model for electricity price forecasting (Weron 2014),by assuming that load and price should move together the model does not allow price torespond to greater shocks. Nonetheless, it is valuable to study the VAR as it may helpincrease forecasting accuracy during normal days. The VAR is defined below.

pt = ⌫ + b1pt−1 + ...+ �kpt−k + ut

lt = ⌫ + g1lt−1 + ...+ �klt−k + ut

(9)

In Equation 9, t = time, v is a constant vector for the intercept and �t and �t, t = 1, ..., k

are kxk matrices, and ut is a multivariate white noise process. matrices. pt is a matrix ofprice and lt a matrix of load such that:

Xt =pt

lt

(10)

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Xt = v+b1pt−1 + ...+ �kpt−k + g1lt−1 + ...+ �klt−k + ut (11)

In this way, load and price are allowed to move together. However, as mentioned abovethis model does not handle the extreme volatility prevalent in electricity price forecastingdue to the cyclical nature of load and the volatile nature of electricity prices. These modelsgenerally perform stronger than the basic AR model and the linear regression but not asstrong as the models that are more adept at handling shocks (Weron 2014).

3.3 Markov Regime Models

Karakatsani and Bunn analyze regressions with first-order Markov Regime-Switching.Karakatsani and Bunn and Weron argue that Markov Regime-Switching allows for consec-utive spikes naturally and dynamically due to their versatile nature (Bunn and Karakatsani2008)(Weron 2014). They also allow for temporary dependence and mean reversion.

Markov Chain:

Pt = ↵Rt + (1−�Rt)Pt−1 + �Rt|Pt−1|�Rt"t, "t ⇠ N(0, 1) (12)

In Equation 12, Pt denotes the spot price and Rt is a Regime Switching process. Thesemodels, like the Linear Regression models, are not expected to be the most accurate hourlyforecasts, they are meant to pick up the characteristics of electricity spot prices (Weron 2014).Markov Regime Switching process generally pick up on peak prices period but struggle duringnormal days (Bunn and Karakatsani 2008)(Weron 2014).

3.4 Non-Linear Regressions

3.4.1 ARMAAntonio Conejo contrasts Crespo Cuaresma by arguing that linear regressions cannot

be used due to the presence of serial correlation in the errors (Conejo et al. 2005). Thefirst non-linear model mentions is the Auto Regressive Moving Average Model (ARMA).The ARMA process is generally the first non-linear model tested (Crespo Cuaresma et al.2004). The ARMA model specified in Crespo Cuaresma follows the same logic as equations2 through 4 (Crespo Cuaresma et al. 2004). The difference is the incorporation of a movingaverage which allows for increased accuracy during peak periods (Weron 2014). The modelspecified by Crespo Cuaresma, Weron and Silvano Bordignon is shown below (Bordignon etal. 2013)(Crespo Cuaresma et al. 2004)(Weron 2014).

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pt = ↵t+pX

i=1

Fipt�i + "t+qX

j=1

✓j"t�j " ⇠ NID(0, �2") (13)

Here, the specifications follow equations 2 through 4. "t is a white-noise process with aconstant variance. This process performs well with electricity prices mean revertive prop-erties. ARMA models generally perform better than AR models (Crespo Cuaresma et al.2004).

3.4.2 ARIMA

The Auto Regressive Integrative Moving Average Regression (ARIMA) is often timesreferred to as the “back bone of all electricity price forecasts” (Weron 2014). In an ARIMAmodel, historical demand does not significantly improve predictions, therefore only priceis included as a power sector variable (Conejo et al. 2005). Historical price is lagged asday � 1. Explicative variables, such as temperature, may be included but for simplicity heargues that prices from the past hour already incorporate explicative variables (Conejo et al.2005). Backshift operators can be used to adjust for price volatility and seasonality (Conejoet al. 2005).

The first step of the ARIMA is to make the time series stationary (constant mean andvariance) (Conejo et al. 2005), accomplished with a logarithmic transformation (Conejo etal. 2005). Differentiation can be hourly, daily, or weekly depending on the data (Conejo etal. 2005). Next polynomial parameters are estimated through plot observation and physicalknowledge of historical data (Conejo et al. 2005). Next the model hypothesis is validatedand the prediction is executed. Conejo tested the ARIMA over the PJM Interconnection ofthe Mid-Atlantic8, United States between February 18 and 24, May 20 and 26, August 19and 25, and November 18 to 24 all for 2002. The MAPEs were approximately 6-15% (Conejoet al. 2005).

ARIMA:

F(B)ph = c+ ✓(B)"h (14)

F(B) = 1�FX

k=1

FkBk (15)

8This is the independent system operator that administer the electricity markets of Delaware, Illinois, In-diana, Kentucky, Maryland, Michigan, New Jersey, North Carolina, Ohio, Pennsylvania, Tennessee, Virginia,West Virginia and the District of Columbia

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F(B)ph = 1� F1ph�1 � �2ph�2 � ...� �nph�n (16)

✓(B) = 1�✓X

l=1

✓lBl (17)

✓(B)"h = 1� ✓1"h�1 � ✓2"h�2 � ...� ✓n"h�n (18)

In equations 14 through 18, ph is the price at the specified hour, c is the constant intercept�(B) and ✓(B) are the backshift operators in polynomial functions. Many studies onlyinclude past price in the ARIMA specifications (Conejo et al. 2005). ARIMA models relateactual price to past price through �(B) and actual errors to past errors through ✓(B). Ifthe model is properly specified "h is white noise.

ARIMA model’s have also been used to forecast the wholesale spot prices in the Span-ish electricity market over 2002 and the PJM Interconnection over 2006 (Nogales et al.2002)(Tan et al. 2010). The findings of this study highlight that the basic ARIMA model hassome of the lowest MAPEs out of all non-linear regression models (Nogales et al. 2002)(Tanet al. 2010). ARIMA models can also utilize GARCH specifications. An overview of theseGARCH specifications as they are applied to ARIMA models can be found in Section D.Literature Appendix.

4 DataTo accurately forecast New York State’s wholesale electricity hourly spot prices, I needed

data from the electric power sector, macro-economy and climate. I compiled data from NY-ISO, the Energy Information Agency, and the National Oceanic & Atmospheric Administra-tion into a unique data set for each of NYISO’s eleven regions. Section B. Data Appendixfeatures supplemental description of data including in depth energy and fuel information,correlation data, macroeconomic data and a description of the data assembly process.

4.1 NYISO Power Sector Data Set

The most important data used in this analysis came from NYISO’s electronic Market &Operation Database. Due to the heavily regulated nature of the electricity sector, key dataon price and load are publicly available. There are two price variables: day-ahead locationalbased marginal price (da-lbmp), and real- time locational based marginal price (rt-lbmp),

18

both in dollars per megawatt hour ($/MWh) . The da-lbmp is the clearing price for the day-ahead market while the rt-lbmp is the clearing price for the real-time market. The NYISOprovides da-lbmps and rt-lbmps across each of its eleven regions and for all of its electricityimports and exports at hourly levels. The study uses NYISO’s hourly data from January1, 2005 to December 31, 2014, representing 87,368 observations. Table 1 contains summarystatistics on the two most important variables, da-lbmp and da-load.

Table 1: DA-LBMP and Load Summary Statistics(1) (2) (3) (4) (5)

VARIABLES N mean sd min max

da-lbmp 964,018 55.73 33.34 1.940 999

energy-bid-load 962,578 812.0 689.2 24 5,211

Electricity demand data (load) can also be acquired through NYISO’s electronic Market& Operation Database. As with price, load data is available at hourly increments for theday-ahead (energy-bid-load) and real-time markets (integrated-load). NYISO also providesan hourly load forecast for each hour (da-load). All load data is reported in megawatt hours(MWhs). A description of the summary statistics for NYISO load can be found below.

4.1.1 Summary Statistics of Power Sector Data Set

There are some apparent trends in the data that coincide with the existing literature.Price is highly volatile and reverts to its mean throughout the day, week, month and year.Load is seasonally cyclical. During the winter there are various price spikes. However, priceis highly mean-revertive. Across all zones, early 2014 had the highest price spike due toincreased congestion in the grid. Please see Figure 4 on the following page and Figure 5 onpage 21 for an overview of da-lbmp and da-load over the time period.

19

Figure 4: Day-Ahead LBMP 2005-2015

20

Figure 5: Day-Ahead Load 2005-2015

Figures 4 and 5 highlight the cyclical nature of load as well as the extreme volatility andmean reversion apparent in da-lbmp. Figure 6 on the next page gives an overview of da-lbmpand da-load means throughout the time period. There is no clear trend. It is important tonote that in 2014 there were large price spikes, but the yearly averages do not reflect this,again highlighting electricity prices mean reversion.

21

Figure 6: Day-Ahead LBMP and Load Yearly Means

The NYISO power sector data follows the usual trends in electricity prices establishedin the literature review. There are peak periods more often throughout January, February,July and August due to New York’s extreme weather, increased load and congestion, in partcaused by a diversion of natural gas9 to heating from electricity generation. Major spikes incongestion occur at the same time as major price spikes. Please see Figure 7 on the followingpage and Figure 8 on page 24 for more detail.

9Natural gas has grown to approximately 40% of electricity production in New York over the time periodanalyzed

22

Figure 7: DA-LBMP and DA-Load Monthly Means

23

Figure 8: DA-LBMP and Congestion

Price also varies greatly during the week, falling off on the weekends, see Table 9. Duringthe day electricity prices are highest between the hours of 15 and 18, rapidly declining until3 of the next day, then gradually increasing from hours 9 through 15; this can be seen inTable 10. See B. Data Appendix for detailed data tables.

24

Figure 9: DA-LBMP and Load Weekly Trends

Figure 10: DA-LBMP and Load Hourly Trends

25

LBMPs vary greatly across the NYISO Zone. The mean da-lbmp across the NYISO Zonewas $55.73/MWh, ranging from $47/MWh in the North and Genesee Zones to $72/MWh forLong Island. New York City’s mean over the period was $64.57. The da-lbmp’s also variedgreatly over the course of the sample, starting significantly higher in 2005 and falling slightlyuntil today. Load behaves cyclically across the period as mentioned above, exhibiting strongseasonality and calendar effects. The greatest load is seen in the New York City and LongIsland zones, while the smallest is in the North and Millwood Zones. See Table 2 for moredetail.

Table 2: Zonal Mean and Standard Deviations: Day-Ahead Price and LoadZone da-lbmp ($/MWh) da-load (MWh) Zone da-lbmp ($/MWh) da-load (MWh)

CAPITL 57.57 929.14 MHK VL 50.71 498.06

(SD) (33.03) (302.26) (27.20) (158.50)

CENTRL 49.07 1076.65 MILLWD 59.99 213.46

(25.88) (308.98) (35.15) (80.18)

DUNWOD 60.16 228.64 N.Y.C. 64.58 2287.28

(35.24) (66.58) (38.83) (593.17)

GENESE 47.16 568.47 NORTH 47.29 124.60

(24.95) (158.61) (26.35) (50.24)

HUD VL 59.25 682.52 WEST 45.20 873.31

(33.85) (199.69) (23.53) (210.26)

LONGIL 72.07 1466.85 NYISO 55.73 811.95

(44.70) (721.64) (33.34) (689.20)

4.2 Energy Data

There are several energy inputs that are not reported directly through the NYISO.Data on New York’s various generation sources (oil, coal, natural gas, nuclear, hydro, andrenewables) is publicly available through the Energy Information Agency (EIA) at monthlylevels in MWhs. Equally important to the share of each generation source is the price ofrelevant fuels. The EIA publishes monthly natural gas prices for each state and end use indollars per thousand cubic square feet of natural gas ($/MCF). This study uses electricitygeneration end use prices. The EIA data was assumed constant across each month it wasreported. Generators do not purchase fuel on daily or even weekly contracts; thereforemonthly fuel prices are more accurate for the price generators pay for fuel.

The coal and oil and price data come from CRSP. Both oil and coal price data aremonthly indices. Coal price data are the Dow Jones Industrial Average Coal Index, dollarsper British Thermal Unites ($/BTUs). The oil price data are the Cushing Oklahoma WTISpot Price, dollars per barrel ($/barrel). All energy input data is assumed constant across

26

the NYISO region. For analysis of the energy input data see Section B. Data Appendix.

4.3 Weather data

The study utilizes weather data from the National Oceanic & Atmospheric Adminis-tration. NOAA publishes Heating Degree Day and Cooling Degree Day data daily acrosstheir own regions of New York, which closely align with the eleven NYISO Regions. HDDsand CDDs combine precipitation, humidity, wind speed and temperature to give the actualdeviation from 65ºF. HDDs measure any temperature below 65ºF and CDDs measure anytemperature above 65ºF. For example if it were 35ºF the HDD would be 30 while the CDDwould be 0. If it were 75ºF the HDD would be 0 and the CDD would be 10. HDD added toCDD create Degree Days, which succinctly combine all components of weather that impactelectricity demand and supply. Degree days were applied to each hour of the day throughoutthe data set. Below, Figure 11 plots degree days with da-lbmp and Figure 12 plots degreedays with da-load.

Figure 11: DA-LBMP and Degree Day

Price generally moves with degree day, however, price spikes are far more severe than thedegree day spikes in the winter. Degree days and price follow a similar pattern as degreedays and load shown below. Contrastingly, load reaches its peak in the summer because ofcooling, in New York Degree Days reach their peaks in the winter from severe weather.

27

Figure 12: DA-Load and Degree Day

For discussion of New York State fuel price and usage please see Section B. Data Appendix.Energy movement as an input has limited effect on electricity price forecasts. The only rele-vant variable is the marginal fossil fuel used at the time. Analysis of the marginal fossil fuelcan be found in Section B. Data Appendix.

5 Econometric SpecificationsIn this section I provide an explanation for the the exogenous variables chosen for the

models specified in Section 3 as well as how the sample and forecast periods was chosen for2014.

5.1 Sample Selection

Sample period selection occurs similarly throughout the literature. A forecast periodconsists of one to two weeks for each season of the calendar year (winter, spring, summer, andfall). Each forecast period has a sample period of several months that ends the hour beforethe forecast period begins (Conejo et al. 2005) (Crespo Cuaresma et al. 2004) (Nogales etal. 2002) (Zareipour 2012).

The winter period in the Northern Hemisphere tests the models ability to hit peak periods.

28

The two week forecast period generally falls in January or February. For each winter forecastperiod there is a month to two month sample period. The spring forecast period occurs duringApril with a sample period beginning in January. The spring period is the transition outof extreme volatility, but the period generally includes several peak period. The summerforecast period takes place during August. In a geographic area like New York cooling doesnot generate large peak periods and has the least volatility. The sample period begins inJune for the summer forecast. The fall forecast period is between periods of low and highvolatility, generally in mid-November. Therefore, the sample period begins in January togive the model information on both peak and non-peak periods.

The NYISO overview selected sample and forecast periods as follows with each forecastperiod beginning at hour 0 of the first day and ending at hour 23 of the last day. Eachsample period begins at hour 0 and ends at hour 23 of the day prior to the first day of theforecast period. The winter forecast period begins on January 6, 2014 and runs throughJanuary 19, 2014 with the winter sample period beginning November 1, 2013. The springforecast period runs from April 21, 2014 to May 4, 2014. The spring sample period beginsJanuary 1, 2014. The summer forecast period begins August 11, 2014 and ends August 24,2014. The sample period begins on June 1, 2014. Finally the fall forecast period runs fromNovember 10, 2014 to November 23, 2014 with the sample period beginning on January 1,2014.

5.2 Exogenous Variables

Each model specified in Section 3 tested three sets of exogenous variables. One set cre-ated independently, the “JNS” exogenous variables, combines the analysis from the literatureto create a new and independent set of variables. The other two models are established in(Zareipour 2012) “Z”, and (Bunn and Karakatsani 2008) “KB” respectively.

The JNS variables take into consideration factors such as the weather, neighboring regionimport and export price, load and price fluctuation and volatility, as well lagged and currentprice and load. The variables selected and created are:

• da� lbmp, da� lbmpt�1, da� lbmpt�24, da� lbmpt�48: price is lagged for an hour, oneday, two days and one week because of a consensus in the literature that these periodhave the greatest relationship to the forecasted period.

• energy � bid � load: load is generally agreed up to be one of the most importantexplanatory variables for electricity prices. However, lagged load is excluded to avoidoverfit and allow for load volatility and fluctuation to pick up the important effects ofpast load.

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• mprice : the average price for the past 24 hours from the forecasted period.

• weekly � price � volatility: Weekly price volatility is the standard deviation of theda � lbmp for the last week divided by the mean of the da � lbmp for the same week(Bunn and Karakatsani 2008).

• daily � load � volatility: Same specification as weekly � price � volatility but withload and not da� lbmp

• dd: Degree days, measuring the distance from 65F, which is dd = 0. Degree days helppick up peak periods in the data and develop patterns for peak days.

• marginal � fuel � price: The price of the largest fuel generator across the NYISOregion at any given hour. This is natural gas for 2014.

• neighbor�price: The average da-lbmp of neighboring markets that import and exportelectricity with NYISO.

The KB variables focus on lagged values of price and load with a focus on their volatility andfluctuations. However, it does not include and forecasted values, weather, or other exogenousvariables. These variables are:

• da� lbmp, da� lbmpt�1, da� lbmpt�24, da� lbmpt�48, da� lbmp168

• energy� bid� load, energy� bid� load

2, energy� bid� load.curv: Quadratic demand

is included to help handle the natural curvature in load throughout the day (Bunn andKarakatsani 2008). energy � bid � load.curv captures aspects of inter-day the interdemand curve. It is the second time-difference of energy � bid� load.

• weekly � price� volatility

• weekly � load� volatility

• loadfluct: Load fluctuation is the difference between forecasted load for the hour andactual load. NYISO publishes hourly day ahead forecasts for load. loadfluct is thedifference between this and observed load.

• marginal � fuel � price

The Z variables focus largely on forecasted values of price and load offering an alternativeset of exogenous variables. These variables are:

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• da� lbmp, f3� da� lbmp: f3� da� lbmp is the price forecasted for the next 3 hours,which attempts to pick up an ascension toward peak periods.

• energy � bid� load, f3� energy � bid� load, da� load :f3� energy � bid� load isthe same as f3 � da � lbmp for load. da � load is the previous days forecasted valuefor the load at the current hour.

• loadfluct

• demandsl: The first time difference of energy � bid� load. (Zareipour 2012)

• hq � da� lbmp, npx� da� lbmp, oh� da� lbmp, pjm� da� lbmp: The da� lbmp

0s

of all of the NYISO’s neighboring regions

For all three of these exogenous variable sets, log specifications were applied to the LinearRegression, AR, ARMA, and ARIMA Models as consistent with the literature (Weron 2014).

6 ResultsIn this section, I provide an overview of the results from the battery of statistical tests

run on the NYISO. The models tested include, Linear Regression, Markov Chain RegimeSwitching, AR and VAR Regression, ARMA, and ARIMA. Each model, except for theLinear Regression model, was tested with and without exogenous variables. The results ofthese tests for each of the NYISO Zones are analyzed below. ARIMA and ARMA modelsperformed with the greatest accuracy over the NYISO territory. These models, coupled withthe Z exogenous variables containing forecasted future values of price and load, consistentlyoutperformed the other sets of exogenous variables across each sample, during peak pricingperiods and hours with historically high and volatile demand, as well as each day of theweek. The literatures comparative accuracy measure can be found in Section D. LiteratureAppendix.

To assess the best forecasting techniques the analysis looked at forecast accuracy fromseveral levels and compared the results to existing findings. First the analysis finds whichmodel performs the best generally over the two-week period. Then the analysis moves on todetermine the most accurate set of exogenous variables over the general period. After themost accurate model and exogenous variables are selected the analysis breaks down accuracyduring the peak period and then at a daily and hourly levels. These three measures ofaccuracy are crucial to the electric power sector and generator scheduling. Therefore, it is

31

necessary to compare how the selected models perform at each interval to assess which modelforecasts with the most accuracy during critical times for the market and grid.

6.1 General Overview

The analysis tested each model across all of NYISO’s zones during each sample andforecast period. The Linear Regression and Markov Chain tests provide a baseline for forecastaccuracy and highlight the robustness of the ARIMA and ARMA models results. Thecomplex models have a greater degree of accuracy across the full period and hit the peakprice periods more consistently than the baseline tests. Table 3 provides an overview ofeach test’s Mean Absolute Percentage Error (MAPE) and Mean Absolute Error (MAE).

(Zareipour 2012) and (Crespo Cuaresma et al. 2004) define MAE as: 1n

s+nX

t=s+1

ˆ(pt�pt) and

MAPE as: 1n

s+nX

t=s+1

(p̂t�pt)pt

. Here, pt is the actual observed price and p̂t is the forecasted price

for the period t. n is 336, the number of hours over the two week forecast period. Beloware the results of the MAPEs and MAEs for each test run presented as an average of all theforecast periods. MAPE’s and MAEs for each season and zone can be found in Section A.NYISO Appendix.

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Table 3: Baseline Tests: 2014 Sample Period AccuracyModel MAPE MAE Model MAPE MAE

Markov Chain KB 8.76% 4.56 ARMAX KB 7.07% 3.43

Markov Chain JNS 10.07% 5.11 ARMAX Z 2.31% 1.25

Markov Chain Switching KB 8.71% 4.64 ARMA with Logs 7.92% 0.08

Markov Chain Switching JNS 9.49% 4.94 ARMAX with Logs JNS 2.91% 0.03

Linear Regression JNS 7.98% 4.06 ARMAX with Logs KB 4.57% 0.05

Linear Regression KB 12.23% 5.74 ARMAX with Logs Z 1.91% 0.02

Linear Regression Z 6.23% 3.33 ARIMA 8.39% 4.42

Linear Regression with Logs JNS 5.92% 0.06 ARIMAX JNS 3.86% 2.06

Linear Regression with Logs KB 8.91% 0.09 ARIMAX KB 6.41% 3.17

Linear Regression with Logs Z 6.32% 0.06 ARIMAX Z 2.15% 1.17

AR 5.57% 2.88 ARIMA with Logs 8.06% 0.08

ARX JNS 4.89% 2.59 ARIMAX with Logs JNS 2.90% 0.03

ARX KB 6.51% 3.20 ARIMAX with Logs KB 4.50% 0.05

ARX Z 6.51% 3.20 ARIMAX with Logs Z 1.86% 0.02

AR with Logs 4.78% 0.05 ARFIMA 8.48% 4.36

ARX with Logs JNS 3.85% 0.04 ARFIMAX JNS 3.83% 2.06

ARX with Logs KB 4.81% 0.05 VARX JNS 7.12% 3.55

ARX with Logs Z 2.91% 0.03 VARX KB 9.17% 4.25

ARMA 8.71% 4.48 VARX Z 5.76% 2.91

ARMAX JNS 3.83% 2.07

The ARIMA and ARMA models consistently outperform the other models across allsample periods and in each of the NYISO’s zones. They are robust to GARCH and logspecifications. Change occurs in relative accuracy, each model performs consistently withone another in each season and zone. However, the absolute accuracy depends on the regionand season. ARIMA and ARMA are the most accurate models and can be thought of astarget models for the study. This is consistent with the literature which generally reportsARIMA and ARMA models performing with MAPEs between 6% and 10% depending onthe season.

Generally zones with a balance of rural and urban areas perform with greater veracitythan solely urban and densely populated or rural and sparsely populated zones. In thewinter, the northern and rural zones have the lowest accuracy due to the severity of theweather. In the summer, the urban zones with high population density perform with thelowest accuracy. This highlights the dichotomy between heating and cooling periods. Zonessuch as the Capital Zone around Albany and the Hudson Zone in between Albany and NewYork City contain both rural and urban centers. They consistently perform with the highestaccuracy10. In part, this may be due to the increased local balancing available in the zones

10For full zonal analysis please see Tables 18 through 24 in Section A. NYISO Appendix

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that combine both rural and urban populations. These areas have a larger base load thanpurely rural areas acting as a stabilizing force but are less prone to spikes caused by highdemand in urban areas.

Exogenous variables in forecasts across the NYISO greatly increase forecast accuracy. TheARMAX and ARIMAX models generally perform better than the models without exogenousvariables. This is a break from some past studies (Contreras et al. 2003)(Nogales et al.2002)(Weron 2014), but consistent with others (Zareipour 2012)(Crespo Cuaresma et al.2004).

Table 4 takes the average of the MAPEs for the ARMA and ARIMA models to determinewhich of the high performing models performed with greater accuracy. This furthers thefocus of the analysis on the most accurate model. The analysis shows that the ARIMA isthe strongest performing model across the sample period. The MAPEs highlighted belowfor the best performing models are more accurate than the baseline tests shown above.

Table 4: NYISO Average Exogenous Model MAPE and MAEModel MAPE MAE

ARMAX 4.40% 2.25

ARIMAX 4.14% 2.14

These results not only perform better than the baseline tests run on the NYISO butthey are also consistent with the literature, often slightly outperforming previous studies’accuracy. In analysis considering ARIMA and ARMA models, MAPEs fluctuate between 6%and 10% depending on the season and exogenous variables (Bordignon et al. 2013)(Bunnand Karakatsani 2008)(Contreras et al. 2003)(Tan et al. 2010)(Weron 2014)(Zareipour2012)(Tan et al. 2010).

The seasonal performance of each model is important for market participants; a consistentmodel must be able to accurately predict prices in volatile and stable seasons. Table 5 breaksdown the ARIMAX and ARMAX models seasonally. This enables the analysis to focus onhow separate models perform in different seasons and in this way determines the strongestperforming model seasonally.

Table 5: NYISO Average Exogenous Model MAPE and MAE SeasonallySeason Winter Spring Summer Fall

Model MAPE MAE MAPE MAE MAPE MAE MAPE MAE

ARMAX 7.23% 4.89 5.09% 1.86 1.48% 0.38 3.81% 1.86

ARIMAX 6.46% 4.53 4.77% 1.76 1.47% 0.38 3.85% 1.88

The result is not as clear as above; the ARIMAX models perform the best out of the

34

selected models outside of the fall forecast period. However, the difference in MAPE duringthe fall is approximately .05%. The ARIMAX model picks up on the daily, weekly andseasonal trends more accurately than the ARMAX model.

The analysis of these time series models highlights that in the NYISO, which may beindicative of the Upper Midwest (MISO) and New England (ISO-NE), ARIMA models arethe strongest performing autoregressive models. The ARIMA’s integration of trend andshock persistence makes it ideal for electricity price forecasting in regions with seasonalpeaking. The model is stationary and in some ways functions like a random walk which isessential to consider the instantaneous nature of the market and the extreme and randompeak price spikes. As shocks move price up and down the ARIMA model is able to moveits optimal forecasts up and down to a greater degree than any other model, even its basespecification, the ARMA.

6.3 Exogenous Variable Analysis

The NYISO analysis demonstrates that exogenous variables are significant when fore-casting electricity prices. Past studies have been ambivalent on the role of exogenous vari-ables in electricity price forecasting (Crespo Cuaresma et al. 2004)(Werond 2014). Thisanalysis considers three separate groups of exogenous variables. One set, the KB variables,incorporates price and load volatility in the forecast; the Z variables include forecasted fu-ture values in the forecast, and JNS variables combine volatility and outside informationincluding weather, fuel price, and electricity import price. This analysis leads to several keyfindings. Table 6 breaks down the MAPEs for each exogenous test referenced in Table 3.

Table 6: Full Tests Exogenous Variable MAPEsVariables Winter Spring Summer Fall 2014

JNS 6.47% 4.13% 1.99% 3.97% 4.14%

KB 8.64% 5.98% 3.49% 5.35% 5.87%

Z 4.15% 2.20% 1.68% 3.24% 2.82%

The JNS and Z variables consistently outperform the KB variables. During the summerand the winter periods JNS and Z are closer in accuracy due to low volatility, however Z isthe most accurate group of exogenous variables. The above analysis incorporates the non-target baseline models identified above. Incorporating volatility as in the KB models, evenduring the most volatile winter period, does not improve the overall accuracy like futureforecasted variables do.

The average MAPE of all the models with Z variables still outperforms the baseline

35

results put forward by Bordignon. Bordignon finds a baseline MAPE of approximately 4%(Bordignon et al. 2013). The JNS model is largely consistent with this baseline accuracywhile the Z variables generally outperform the baseline.

Table 7 explores the different exogenous variables for the ARIMAX and ARMAX modelsidentified above as the target models for the NYISO.

Table 7: Selected Tests Exogenous Variable MAPEsWinter Spring Summer Fall

Model MAPE MAE MAPE MAE MAPE MAE MAPE MAE

ARMAX JNS 6.21% 4.437 4.62% 1.75 0.59% 0.158 3.90% 1.923

ARMAX KB 11.58% 7.446 8.45% 2.98 3.21% 0.826 5.05% 2.470

ARMAX Z 3.91% 2.800 2.19% 0.85 0.64% 0.167 2.48% 1.191

ARIMAX JNS 5.87% 4.267 4.82% 1.81 0.60% 0.159 4.14% 2.014

ARIMAX KB 9.53% 6.617 7.93% 2.81 3.20% 0.824 4.99% 2.431

ARIMAX Z 4.00% 2.700 1.56% 0.65 0.62% 0.163 2.42% 1.183

Across each forecasting period the exogenous variables perform differently. Nonetheless,the JNS and Z models outperform the KB models in each season. The 2014 NYISO summerperiod contains abnormally low volatility, which has not been matched in previous studies.However, the other periods contain standard electricity market volatility. The Z variableswith future values consistently outperform Bordignon’s 4% MAPE baseline accuracy andConejo’s 6-10% MAPE ARIMA accuracy (Bordignon et al. 2013) (Conejo et al. 2005).In part this may be because Conejo excludes exogenous variables that are not necessary.However, the Z variables also beat the 9% MAPE reported as a baseline for ARIMAXmodels11 (Contreras et al. 2003).

Figures 13 through 18 compare the forecasted ARIMAX models with JNS and Z exoge-nous variables to the observed price. Displayed below are the winter and summer forecastperiods, winter displaying extreme volatility and summer a stable trend. The Capital, North,and New York City (NYC) Zones were selected because each represents a component of NewYork’s geography and population density. The North Zone covers the northeast of the Statealong the Canadian and Vermont border. It is rural with low population density. The Cap-ital Zone is centered around Albany with both rural and urban geographies, high and lowpopulation densities. The population dense NYC Zone covers the entirety of New York City

11Part the large discrepancy in MAPEs between this analysis and the literature is due to the differentmarket structures and varying levels of market reliability in the commonly analyzed markets of Spain andNorthern Europe. These markets incorporate higher shares of renewable intermittent generation than NewYork, which adds volatility to the market as their intermittent resources continuously enter and exit themarket. NYISO administers a large share of natural gas and nuclear generation, which puts relativelyconstant load into the market despite the severe winter weather conditions in New York.

36

and several of the surrounding suburban areas. Graphs for the three zones in the fall andspring periods can be found in the Section C. Results Appendix.

Figure 13: Winter 2014 Target Models Results North Zone

Notes:

1. Observed Da-Lbmp is the price recorded during the forecast period

2. JNS Forecasted Da-Lbmp is the price predicted by the ARIMA model with the JNS ExogenousVariables: da� lbmp, da� lbmpt�1, da� lbmpt�24, da� lbmpt�48,energy�bid� load, mprice, weekly�price� volatility, daily � load� volatility, dd,marginal � fuel � price, neighbor � price

3. Z Forecasted Da-Lbmp is the price predicted by the ARIMA model with the Z Exogenous Variables:da�lbmp, f3�da�lbmp,energy�bid�load, f3�energy�bid�load, da�load, loadfluct, demandsl, hq�da� lbmp, npx� da� lbmp, oh� da� lbmp, pjm� da� lbmp

37

Figure 14: Winter 2014 Target Models Results NYC Zone

Figure 15: Winter 2014 Target Models Results Capital Zone

38

Figure 16: Summer 2014 Target Models Results North Zone

Figure 17: Summer 2014 Target Models Results NYC Zone

39

Figure 18: Summer 2014 Target Models Results Capital Zone

The figures above highlight several significant findings for exogenous and explanatoryvariable selection. Future variables for price and load appear to be some of the strongestindicators for future price in both the stable, predictable summer and the highly volatilewinter. Models that contain forecasted or future values of price load such as the Z modelsreferenced above consistently hit the peak period. The JNS model, which considers outsideinformation such as weather and fuel price, is better than the KB model, which considersonly past values of price and load with volatility. JNS is in line with the Z models forecastedvalues during periods with limited volatility, but does not compare to the Z set of explanatoryvariables in periods of extreme volatility. The JNS weather consideration is more explanatorythan KB’s sole focus on volatility, but not as valuable as forecasted power sector inputs. Thesummer period has the least volatility out of all the periods selected. Each model performsvery well in this period. The stable summer period greatly contrasts the volatile winterperiod.

Thus far the analysis has shown two important findings. First, the most accurate modelis the ARIMA model. Second, exogenous variables greatly improve forecast accuracy andexogenous variables that include forecasted future values of both price and load are generallythe most accurate exogenous variables across the two week sample period. These results are

40

robust with GARCH and log specifications12.

6.3 Peak Analysis

The most significant component of electricity price forecasting is the accuracy withwhich models can hit peak periods. Here, the peak period is defined mathematically as anyprice a full standard deviation from the mean. Table 8 breaks down the occurrence of peakhours across the winter and fall forecasting periods. The spring and summer forecastingperiods did not contain any peak periods.

Table 8: Peak Hours in the 2014 SampleZone Fall Winter Zone Fall Winter

Capital 3 102 Millwood 10 96

Central 4 57 Mohawk 4 58

Dunwoodie 10 95 North 3 45

Genesee 4 55 NYC 8 94

Hudson 3 96 West 3 60

Long Island 3 101 NYISO 55 859

The ARIMA and ARMA models identified above as the target models perform thestrongest during peak periods. However, a complete breakdown of peak period performanceas well as zonal peak period analysis can be found in Section C. Results Appendix. Table 9highlights the target models peak period performance.

Table 9: Select Model Peak Period AccuracyWinter Fall Winter Fall

Model MAPE MAE MAPE MAE Model MAPE MAE MAPE MAE

ARMA 13.71% 21.67 20.74% 19.268 ARIMA 13.93% 21.86 21.81% 20.271

ARMAX JNS 6.16% 9.51 5.49% 5.251 ARIMAX JNS 6.04% 9.31 5.69% 5.444

ARMAX KB 10.55% 15.91 8.96% 8.445 ARIMAX KB 10.67% 15.84 8.96% 8.432

ARMAX Z 4.03% 6.00 4.65% 4.243 ARIMAX Z 3.40% 5.16 4.87% 4.445

Table 9 demonstrates empirically what the Figures 13 and 16 displayed graphically. Theexogenous variables tested in the Z models containing forecasted prices and loads hit thepeak period prices with the greatest accuracy. In each of the selected models the Z speci-fications outperform both the JNS and the KB specifications. The ARIMA model with Zspecifications is the most accurate model analyzed for both peak and general periods. Forpeak price forecasting the ARIMA and ARMA maintain their robustness with GARCH and

12See Section C.6 ARIMA and ARMA with Log Specifications.

41

log specifications that existed in general forecasting (See Section C.6 ARIMA and ARMAwith Log Specifications.).

During peak periods overall accuracy declines as is to be expected because of the volatileand sometimes random nature of peaks. Nonetheless, the ARIMA model with Z exogenousvariables still outperform Bordignon’s baseline accuracy of 4% MAPE as well as the consensusARIMA accuracy of 6-10% MAPE. In many ways accurate forecasts during peak periods aremore significant to market participants than general forecast accuracy. The ARIMA modelwith Z exogenous variables maintains its accuracy during these periods better than the othermodels tested.

6.4 Hourly Analysis

Certain hours of the day are prone to electricity price spikes and volatility due toincreased demand. In New York, these spikes do not always constitute peak periods, buthours during the late afternoon, generally from hour 16-19, are prone to high demand. For amodel to be successful it must accurately hit these hours in the same way that it is essentialto hit peak periods. From the prior analysis, we can focus the study on the ARMAX andARIMAX models with the JNS and Z exogenous variables. The models with the greatestaccuracy over the entire day and the hours of afternoon with the highest demand, hours16-20, are the ARIMAX and ARMAX with Z and JNS exogenous variables13. These hoursare the hours that market participants are the most concerned with. Figure 19 shows howthese models perform over each hour14.

13These results are also robust with regard to log and GARCH specifications.14Section C. Results Appendix contains deeper analysis of hourly MAPEs across the period.

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Figure 19: 2014 NYISO Hourly Average MAPEs

The ARIMAX with Z exogenous variables performs the strongest during the day andduring the period in the late afternoon to evening that has the highest demand where fore-casts are crucial for generator scheduling. However, during periods of the day with lowerdemand such as the early morning, hours 23-03, accuracy generally declines. During theseperiods ARMAX Z exogenous variables outperform the ARIMAX model. The errors at thispoint are greater overall than during the day but the ARMAX with Z exogenous variablesperforms slightly better. Broken down at the hourly level, the Z exogenous variables out-performs the JNS variables. The JNS variables follow the same pattern as the Z variableswhere the ARIMAX is better during the day but at night the ARMAX is stronger. TheARIMAX with Z exogenous variables performs with the highest accuracy at the point of theday that is the most important to system reliability.

6.5 Daily Analysis

Daily mean accuracy for forecasts is in line with hourly forecast accuracy. Hourlyanalysis is important for the market to clear, but daily forecasts can be more importantfor generator scheduling. Weekdays generally have greater demand, making the forecasts forthese days more important. The analysis was limited to the ARIMA and ARMA models withJNS and Z exogenous variables. Days with higher demand are better forecasted by different

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models than days with low demand. Table 10 provides an overview of these models’ dailyperformance. The results at the daily level were robust with GARCH and log specifications.

Table 10: 2014 NYISO Daily Average MAPEsDay ARIMAX JNS ARIMAX Z ARMAX JNS ARMAX Z All Test Average

1 4.62% 3.26% 4.44% 2.92% 3.58%

2 4.12% 2.57% 3.76% 2.22% 3.35%

3 3.46% 2.50% 3.15% 1.97% 3.04%

4 2.60% 2.19% 2.92% 2.05% 2.78%

5 3.88% 3.34% 3.86% 3.36% 3.26%

6 5.43% 4.28% 5.41% 4.49% 4.37%

7 6.12% 4.18% 6.20% 4.28% 4.92%

Notes: Day 1 corresponds to Monday and day 7 corresponds Sunday

The strongest performing models here differ slightly than the results from the full sample,peak, and hourly accuracy. The ARMA model outperforms the ARIMA models duringthe week when demand is generally higher. However, on the weekend the ARIMA modelsaccuracy overtakes the ARMA model. In general, the average accuracy is greater in themiddle of the week when demand is highest. This is the period that market participantscare the most about because it is the period of highest demand. At the daily level the Zexogenous variables containing forecasted future values of price and load remains the mostaccurate set of variables.

In the NYISO, ARIMA and ARMA models produce the most accurate and robustforecasts. They consistently outperform past studies from other markets. Z exogenousvariables that include future forecasted variables greatly increase the overall accuracy ofthese models across the full two-week period of each season, during peak periods, and onhours and days with the highest demand.

7 ConclusionThe analysis of the NYISO shows that in electricity sectors that function with a similar

market and climate as New York the strongest model to accurately forecast day-ahead hourlyprices is the ARIMA with exogenous variables containing forecasted future values of priceand load. These models combined with the selected exogenous variables generally have aMAPE of 4% or lower, consistent with and often times more accurate than previous studieson different markets that report MAPEs ranging from 6-10% for ARIMA models.

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The use of forecasted future variables is not prevalent in the literature, however, acrosseach of the seasons and all of NYISO’s zones the future values in the exogenous variablesproduced the forecasts with the greatest accuracy. Fluctuations in forecast accuracy arelargely due to seasonal factors and population densities. Each model and set of exogenousvariables tested performs similarly relative to the other models studied. The ARIMA modeland future value exogenous variables routinely perform the best throughout.

The results from the study of NYISO will be useful for accurately forecasting electricityprices in comparable liberalized markets that exist in regions with similar climates andgeographies including: New England, the northern Midwest of the United States, Quebec,Ontario, Scandinavian Europe, and the northern United Kingdom. Each of these regionsoperates a similar market with a similar geography and climate to New York. Models thatfunction like the ARIMA, with the ability to incorporate and process shocks, are crucialfor electricity price forecasting given the inherent volatility and randomness to electricitymarkets.

Improved electricity price forecasts are crucial as more regions deregulate and liberal-ize their wholesale electricity supply. These forecasts will improve the electricity sector inthree ways. First, they can increase market efficiency through smarter generator scheduling.Second, accurate price forecasts make Demand Response programs more viable. Finally,stronger forecasts allow for a greater incorporation of renewable energy while improving theefficiency of our legacy fossil fuel plants. Broadly, more accurate electricity price forecastscan increase the overall performance of both the grid and the market while providing pivotalinformation for policy makers at a critical point in the history of United States electricitypolicy.

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Appendix InformationA. NYISO Appendix

1. History of NYISO2. NYISO Today

B. Data Appendix

1. Energy Data Movement2. Correlations3. Macroeconomic Data4. Electricity Data

C. Results Appendix

1. Seasonal2. Zonal3. Exogenous4. Peak5. Hourly6. ARIMA and ARMA Log Specifications

D. Literature Review Appendix

1. Jump Diffusion Models2. Similar Day and Exponential Smoothing3. Dynamic Regression and Transfer Functions4. Literature Accuracy Comparison

E. Glossary

46

A. NYISO AppendixThis appendix provides additional and supporting detail for the NYISO overview sec-

tion. Presented is history of the NYISO and the institutions involved in wholesale electricityprior to the NYISO.

A.1 The History of NYISOPrior to the 1999 deregulation, the majority of electricity in New York was gener-

ated, transmitted, and distributed by several vertically integrated, investor owned utilities(IOUs)15 and municipalities. The New York Power Authority (NYPA) was the largest gov-ernment managed utility in the state. It was responsible for the transmission and generationof the municipality’s electricity. The chief regulator was the New York Public Service Com-mission (NYPSC). The IOUs and the NYPA largely owned generation assets, however, therewere several independent power producers (IPPs) whom mostly owned single, small gener-ating facilities (Tierney 2010).

The final participant, but the most important for understanding the role of NYISO, wasthe New York Power Pool (NYPP). The NYPP coordinated the sale of wholesale electricpower and performed essential reliability functions. The NYPP was created in response tothe Blackout of 1965. The NYPP dispatched generating units according to schedules pro-vided by the utilities, balanced electric system supply and demand in real time, maintainedtransmission line voltage, managed operating reserves, and monitored contingencies that re-quired rapid response to assure system reliability. The NYPP even facilitated short termelectricity trading among the utilities (Tierney 2010). These functions would all be assumedby NYISO upon its inception.

Prior to deregulation, New York had some of the highest electric rates in the country. By1997, New York’s electric rates were 140-160% of US average (Tierney 2010). Large industrialcustomers had already begun to push for relief. They began to enter into contracts with LSEsto pay lower rates, which in turn put upward pressure on residential and commercial customerrates. This led the NYPA and the State of New York to begin looking for solutions. Duringthe late 1990s the United States electricity policy was transitioning toward competition andderegulation. The precedent had been set in the United Kingdom starting in 1990 (Thomas2004). At the very same time, during the late 1990s a majority of the country createdcompetitive and deregulated electricity markets16. State regulators across the country, andespecially in New York, hoped competitive markets and deregulation would lower the ratesfor industrial users without putting added strain on consumer and residential users.

Many regions of the United States were moving toward competitive markets to solveincreases in electric rates because of precedence set by the Federal Energy Regulatory Com-

15Central Hudson Gas & Electric Company (CHG&E), Consolidated Edison Company (ConEd), LongIsland Lighting Company (LILCO), New York State Electric & Gas Co. (NYSEG), Niagara Mohawk PowerCorp (NIMO), Orange & Rockland Co (O&R), and Rochester Gas & Electric (RGE)

16New England established the Independent System Operator New England (ISONE), the Mid-AtlanticStates gave greater power to PJM Interconnection (PJM), the Midwest created Midcontinent IndependentSystem Operator (MISO), California created the California Independent System Operator (CAISO) andTexas established the ERCOT (Electric Reliability Council of Texas).

47

mission (FERC). After the United Kingdom deregulated their electric sector in 1990s, theFERC and the United States Congress began pushing for liberalization in the United Statesenergy sector. In 1992, the FERC and Congress passed the Energy Policy Act, which au-thorized market mechanisms for electric power and natural gas as an attempt to propagatecompetition. Then, in 1996, the FERC issued Order 888, which established formal and legalcompetitive electricity markets. Order 888 required all transmission providers in the UnitedStates, including all IOUs, to provide open access to their transmission systems to affordnon-utility generating companies the opportunity to compete with utility owned power sup-ply. The FERC and NYPSC encouraged distribution utilities and other LSEs to divest fromtheir generating units so that the transmission capacity made available could be used com-petitively (Tierney 2010). This encouraged generation companies of all sizes to enter intothe New York market.

A necessary component of any competitive market is the Independent System Operator.Order 888 encouraged states and groups of states to work together to form ISOs that couldmanage the newly created markets for wholesale electricity17. Order 888 pushed New Yorkto establish the New York Independent System Operator.

In order to establish NYISO and make New York’s electricity sector competitive, manychanges needed to be made. First, the NYPP was dissolved and NYISO assumed its reliabil-ity and dispatch functions, acting as the control area operator. NYISO began to administerthe regional transmission tariffs to price access and use of the transmission system, designedto recover the transmission owner’s costs. NYISO has an independent board of directors,unaffiliated with any market participants in the New York power market. It was establishedas a business, not a regulatory body. The board of directors makes the leading decisions(Tierney 2010).

For the NYISO to be successful, New York’s electricity regulatory structure was forced tochange. To oversee and set system standards, as well as regulate NYISO, the New York StateReliability Council (NYSRC) was created and power supply rules changed in order to allowmarket participants the ability to obtain power through the power exchange or bilateral con-tracts. At a high level, NYISO has three functions. First, maintain the energy market withLocational-Based Marginal Pricing (LBMP), next, maintain the two settlement processesto establish the schedule for dispatching power plants to meet load requirements throughenergy prices in the day-ahead and real time markets, and finally, maintain a centralizedunit commitment and dispatch based on the bids of market participants.

A.2 NYISO TodayThe New York electricity sector has greatly changed since the creation of the NYISO.

Many of the previous IOUs have become fully owned subsidiaries of large multinationalenergy corporations18. IOUs operate the majority of the distribution and transmission lines

17Order 888 had limited impact on the distribution of electricity. Distribution was not deregulated andremained the responsibility of local LSE’s distributing electricity across their service territories.

18CHG&E (doing business as (dba) Fortis)), ConEd, NYSEG (dba Iberdrola USA), NIMO (dba NationalGrid), O&R (dba ConEd), RG&E (dba Iberdrola USA), LILCO was replaced by a joint private-publicpartnership named the Long Island Power Authority (LIPA), administered between the State of New Yorkand the New Jersey Utility Public Service Enterprise Group (PSEG)

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throughout the state, along with several dozen municipal electric providers and the NYPA.However, the majority of electric generation in the state comes from a plethora of IPPS. TheseIPPs range from large multinational generation holding companies to individual power plantowners. Competitive markets and deregulation changed the New York electricity sector fromgeneration to distribution. Electricity prices in New York have continued to rise and remainsome of the highest in the country. However, they have grown at a rate similar to the restof the country (Tierney 2010).

B. Data AppendixThis appendix provides additional and supporting detail for the data used in the analysis

as well as analysis of NYISO data that ended up falling outside of the purview of the studybut may be helpful for further study of the NYISO.

B.1 Energy Data MovementAcross the time period analyzed there were many drastic shifts and movements in gen-

eration mix and fuel prices. Over this time period natural gas overtakes nuclear power asthe marginal generator, shifting from 20% in 2005 to 40% in 2014. Nuclear remains constantat approximately 30% throughout the time studied. Coal and oil independently taper fromapproximately 15% to less than 5%. Renewables grow from less than 1% to close to 5% andhydroelectric power remains constant at 20%. See Figures 20 and 21 below.

Figure 20: New York’s Fossil Generation

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Figure 21: New York’s Non-Fossil Generation

There are large shifts in fuel prices over this period. Between 2005 and 2014 the priceof oil moved from approximately 56$/barrel to 93$/barrel prior to the collapse of the oilmarket in late 2014, early 2015. The price of natural gas falls from 9.4 $/MCF to 5.7$/MCF, explaining the shift in marginal generator. The price of coal falls drastically from107 $/BTU to 60 $/BTU. However, due to increased environmental standards, coals capacityin New York was decreased and the price shift had little impact on electricity prices. Thesame can be said for oil.

DA-LBMP roughly tracks the price of natural gas. However, it was not until 2009 thatnatural gas passed nuclear power as the marginal source of generation in the state. Below,Figure 22 highlights DA-LBMP’s relationship with fuel prices.

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Figure 22: DA-LBMP and Fuel Prices

B.2 CorrelationsDay-Ahead prices in New York across the time period studied show a moderate positive

correlation with Real Time Price, .5719 as well as the price of natural gas, .5857. The Day-Ahead Marginal Cost of Congestion had a strong correlation with the price, as seen in thefigures above. DA MCC had a negative correlation of -0.5707. Please see the table below fordetailed analysis. The results make intuitive sense, the price is roughly correlated with themarginal fuel and congestion. Weather displays limited correlation with price at only .1688.Tables 11 and 12 below provide full correlation detail.

Table 11: DA LBMP Power Sector and Weather CorrelationsVariable Correlation with DA LBMP

Energy Load Bids 0.203

Load Forecasts 0.193

RT LBMP 0.5719

RT Load 0.1745

DA Congestion -0.5707

DA Transmission Losses 0.4756

RT Congestion -0.2573

RT Transmission Losses 0.3951

Degree Days 0.1688

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Table 12: DA LBMP Energy and Macroeconomic CorrelationsVariable DA-LBMP Correlation

Coal Share 0.395

Nat Gas Share -0.2822

Oil Share 0.3806

Renewable Share -0.2598

Hydro Share -0.2124

Wood Share -0.0597

Nuclear Share -0.2044

Other Share -0.4494

Natural Gas Price 0.5857

Coal Price 0.3058

Oil Price -0.0085

Unemployment Rate -0.3057

B.3 Macroeconomic DataMacroeconomic output and unemployment are also thought to have a significant impact

on electricity load and therefore price. New York state regional unemployment and indus-trial output data was acquired through the Federal Reserve Economic Database (FRED)at monthly levels. The unemployment data was city specific with a city corresponding toeach NYISO zone. Unemployment and industrial output were assumed constant across eachmonth throughout the time horizon of the study. The unemployment data was only looselynegatively correlated with the LBMP19.

B.4 Electricity Data

19The unemployment rate had a correlation with da-lbmp of -0.3057.

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Table 13: Hourly Mean and SD: Day-Ahead Price and Loadhour da_lbmp ($/MWh) da_load (MWh) hour da_lbmp ($/MWh) da_load (MWh)

0 44.85 648.77 12 62.23 933.45

(22.86) (528.80) (32.54) (792.05)

1 40.97 621.34 13 62.73 933.93

(20.87) (500.24) (34.64) (799.34)

2 38.54 602.62 14 62.54 934.18

(20.18) (484.08) (36.72) (803.03)

3 37.55 594.46 15 63.34 934.15

(20.00) (475.94) (38.90) (806.97)

4 38.05 602.26 16 67.37 938.01

(20.74) (478.34) (41.09) (805.52)

5 41.81 633.91 17 72.56 948.95

(23.46) (499.07) (44.33) (794.87)

6 49.80 705.10 18 70.06 937.18

(29.26) (551.56) (41.12) (760.33)

7 54.44 777.49 19 67.39 917.43

(32.11) (617.19) (36.66) (732.40)

8 57.01 840.76 20 64.76 897.74

(31.87) (673.65) (34.25) (710.84)

9 59.81 886.21 21 58.36 863.73

(31.84) (727.46) (30.69) (678.41)

10 61.86 914.88 22 51.72 785.94

(32.23) (763.31) (26.54) (630.31)

11 62.45 930.16 23 47.33 703.66

(32.11) (783.66) (23.91) (573.59)

Total 55.73 811.95

(33.34) (689.20)

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Table 14: Daily Mean and SD: Day-Ahead Price and LoadDay of the Week da_lbmp ($/MWh) da_load (MWh)

Mon. 57.56 834.47

(33.67) (713.01)

Tues. 58.48 847.38

(36.66) (722.49)

Wed. 58.07 847.98

(35.97) (723.23)

Thurs. 58.01 845.79

(35.46) (721.66)

Fri. 56.87 833.95

(34.71) (712.20)

Sat. 51.50 747.66

(27.25) (611.26)

Sun. 49.64 726.43

(26.93) (594.73)

Total 55.73 811.95

(33.34) (689.20)

B.5 Data AssemblyThe power sector, energy, and macroeconomic data were all matched on the time variable

provided through with each NYISO dataset. This time variable contains year, month, dayand hour. All monthly-denominated data was assumed constant over the days and hours ofeach month. The final data sets were broken down for each of the NYISO zones and onemaster dataset. Each data set included 87,368 observations for day ahead and real time priceof each zone as well as the price of imports and exports, day ahead load forecast, day aheadload, real time load, generation shares, fuel prices, degree days, unemployment, output aswell as the time kept in the NYISO format and then separated out for year, month, day,hour and weekday.

C. Results AppendixThis appendix provides additional and supporting detail for the results section. Pre-

sented is supporting analysis from the seasonal, zonal, peak, hourly and daily analysis.

C.1 Seasonal

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Table 15: Baseline Tests: 2014 Sample Period MAPEsModel Winter Spring Summer Fall

Markov Chain KB 13.53% 10.33% 4.55% 6.63%

Markov Chain JNS 16.72% 11.00% 4.44% 8.12%

Markov Chain Switching KB 14.47% 8.43% 4.49% 7.45%

Markov Chain Switching JNS 15.52% 9.54% 4.62% 8.29%

Linear Regression JNS 15.44% 7.77% 2.37% 6.31%

Linear Regression KB 19.57% 17.95% 3.99% 7.39%

Linear Regression Z 8.86% 7.06% 1.41% 7.59%

Linear Regression with Logs JNS 5.52% 6.41% 1.68% 6.07%

Linear Regression with Logs KB 7.72% 7.74% 4.79% 8.51%

Linear Regression with Logs Z 6.64% 2.99% 0.82% 6.83%

Table 16: AR Tests: 2014 Sample Period MAPEsModel Winter Spring Summer Fall

AR 9.05% 4.27% 3.67% 5.28%

ARX JNS 8.96% 4.33% 1.63% 4.65%

ARX KB 10.52% 6.88% 3.23% 5.42%

ARX Z 10.52% 6.88% 3.23% 5.42%

AR with Logs 6.87% 3.88% 3.58% 4.80%

ARX with Logs JNS 6.29% 3.66% 1.13% 4.33%

ARX with Logs KB 6.88% 4.13% 3.33% 4.93%

ARX with Logs Z 3.85% 2.67% 0.83% 4.28%

ARMA 9.78% 9.27% 6.24% 9.54%

ARMAX JNS 6.21% 4.62% 0.59% 3.90%

ARMAX KB 11.58% 8.45% 3.21% 5.05%

ARMAX Z 3.91% 2.19% 0.64% 2.48%

ARMA with Logs 9.56% 6.90% 6.09% 9.13%

ARMAX with Logs JNS 5.03% 2.72% 0.43% 3.44%

ARMAX with Logs KB 7.11% 3.56% 3.15% 4.46%

ARMAX with Logs Z 3.37% 1.56% 0.50% 2.21%

ARIMA 11.68% 7.34% 5.72% 8.84%

ARIMAX JNS 5.87% 4.82% 0.60% 4.14%

ARIMAX KB 9.53% 7.93% 3.20% 4.99%

ARIMAX Z 4.00% 1.56% 0.62% 2.42%

ARIMA with Logs 10.26% 7.18% 6.11% 8.67%

ARIMAX with Logs JNS 4.90% 2.69% 0.44% 3.58%

ARIMAX with Logs KB 6.95% 3.40% 3.16% 4.49%

ARIMAX with Logs Z 3.21% 1.46% 0.50% 2.28%

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Table 17: ARFIMA, GARCH, and VAR Tests: 2014 Sample Period MAPEsModel Winter Spring Summer Fall

ARFIMA 11.84% 7.12% 5.71% 9.26%

ARFIMAX JNS 6.08% 4.68% 0.59% 3.96%

AR GARCH 9.94% 7.55% 7.75% 9.51%

ARX GARCH JNS 6.27% 2.63% 0.50% 3.14%

ARX GARCH KB 9.40% 3.83% 3.10% 4.76%

ARX GARCH Z 6.53% 2.92% 2.94% 4.18%

ARMA GARCH 9.68% 6.70% 6.17% 9.22%

ARMAX GARCH JNS 6.06% 2.60% 0.50% 3.10%

ARMAX GARCH KB 9.90% 3.89% 3.13% 4.77%

ARMAX GARCH Z 6.38% 2.80% 2.63% 4.43%

ARIMA GARCH 9.94% 7.29% 5.70% 8.80%

ARIMAX GARCH JNS 5.48% 2.66% 0.55% 3.10%

ARIMAX GARCH KB 8.68% 3.85% 3.10% 4.79%

ARIMAX GARCH Z 5.16% 1.20% 0.58% 2.74%

VAR JNS 13.84% 8.372712049 6.47% 2.518110806

VAR KB 14.84% 8.6971211 10.89% 4.04252048

VAR Z 8.70% 5.915207039 4.64% 1.830522808

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C.2 Zonal - Select Models

Table 18: 2014 Winter Zonal MAPEs Part IModel Capital Central Dunwoodie Genesee Hudson Long Island

ARMA 9.81% 9.67% 9.00% 10.72% 9.03% 9.64%

ARMAX JNS 6.77% 5.37% 4.86% 7.06% 4.76% 7.53%

ARMAX KB 12.14% 12.54% 8.46% 13.85% 8.20% 9.33%

ARMAX Z 4.58% 3.47% 3.10% 4.55% 3.17% 5.67%

ARMA with Logs 9.94% 9.39% 9.06% 9.64% 9.06% 9.82%

ARMAX with Logs JNS 6.37% 4.45% 4.52% 5.21% 4.35% 5.89%

ARMAX with Logs KB 7.59% 6.67% 6.19% 8.10% 6.07% 6.79%

ARMAX with Logs Z 4.15% 2.69% 2.91% 3.47% 2.81% 5.66%

ARIMA 10.68% 11.34% 10.34% 14.62% 10.39% 11.07%

ARIMAX JNS 6.86% 5.23% 4.94% 7.23% 4.83% 6.78%

ARIMAX KB 8.55% 9.49% 7.96% 10.24% 8.11% 8.10%

ARIMAX Z 4.52% 3.40% 3.08% 5.40% 3.15% 6.08%

ARIMA with Logs 9.77% 9.96% 9.37% 11.34% 9.42% 9.92%

ARIMAX with Logs JNS 6.04% 4.44% 4.24% 5.17% 4.14% 5.89%

ARIMAX with Logs KB 7.12% 6.88% 6.12% 8.06% 5.88% 6.65%

ARIMAX with Logs Z 3.95% 2.58% 2.84% 2.95% 2.78% 5.47%

ARX GARCH Z 6.87% 7.33% 5.37% 6.99% 5.36% 7.07%

ARMA GARCH 9.96% 9.97% 8.99% 10.02% 9.02% 10.47%

ARMAX GARCH JNS 6.64% 5.71% 4.95% 6.21% 4.84% 6.35%

ARMAX GARCH KB 8.85% 7.91% 7.45% 9.44% 8.68% 9.65%

ARMAX GARCH Z 6.99% 6.45% 5.53% 6.44% 5.57% 7.25%

ARIMA GARCH 9.86% 11.41% 9.06% 9.47% 9.11% 9.99%

ARIMAX GARCH JNS 6.85% 5.12% 4.63% 5.50% 4.64% 6.50%

ARIMAX GARCH KB 8.24% 8.35% 7.50% 8.30% 7.36% 7.99%

ARIMAX GARCH Z 7.35% 3.67% 4.94% 3.99% 4.83% 6.38%

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Table 19: 2014 Winter Zonal MAPEs Part IIModel Millwood Mohawk Val. North NYC West

ARMA 9.04% 9.88% 11.81% 8.91% 10.06%

ARMAX JNS 4.80% 5.59% 8.71% 4.86% 7.95%

ARMAX KB 8.40% 12.76% 19.70% 8.38% 13.65%

ARMAX Z 3.14% 3.16% 2.46% 3.09% 6.64%

ARMA with Logs 9.10% 9.59% 11.48% 8.95% 9.17%

ARMAX with Logs JNS 4.35% 4.41% 6.89% 4.41% 4.51%

ARMAX with Logs KB 6.13% 7.03% 9.78% 5.97% 7.88%

ARMAX with Logs Z 2.89% 2.45% 2.31% 2.88% 4.88%

ARIMA 10.38% 12.47% 14.47% 10.18% 12.56%

ARIMAX JNS 4.79% 5.31% 8.40% 4.90% 5.27%

ARIMAX KB 8.12% 9.23% 15.46% 8.35% 11.21%

ARIMAX Z 3.10% 3.06% 2.73% 3.04% 6.39%

ARIMA with Logs 9.93% 11.16% 12.75% 9.24% 9.99%

ARIMAX with Logs JNS 4.20% 4.39% 6.75% 4.14% 4.53%

ARIMAX with Logs KB 6.01% 7.10% 9.30% 5.91% 7.46%

ARIMAX with Logs Z 2.86% 2.33% 2.24% 2.77% 4.58%

ARX GARCH Z 5.39% 6.61% 5.36% 5.23% 10.24%

ARMA GARCH 9.04% 10.05% 11.03% 8.91% 9.00%

ARMAX GARCH JNS 4.90% 5.50% 7.96% 4.96% 8.65%

ARMAX GARCH KB 8.42% 8.06% 17.59% 7.38% 15.45%

ARMAX GARCH Z 5.65% 6.06% 5.01% 5.39% 9.87%

ARIMA GARCH 9.11% 9.75% 12.70% 8.96% 9.92%

ARIMAX GARCH JNS 4.71% 5.01% 7.73% 4.54% 5.10%

ARIMAX GARCH KB 7.91% 8.79% 11.80% 7.63% 11.60%

ARIMAX GARCH Z 5.00% 2.32% 1.16% 4.71% 12.39%

58

Table 20: 2014 Spring Zonal MAPEs Part IModel Capital Central Dunwoodie Genesee Hudson Long Island

ARMA 9.27% 9.00% 8.37% 9.45% 8.35% 9.99%

ARMAX JNS 4.62% 2.59% 3.20% 6.66% 2.97% 6.14%

ARMAX KB 8.45% 8.48% 6.14% 38.94% 6.45% 6.87%

ARMAX Z 2.19% 2.70% 1.42% 4.07% 1.45% 5.41%

ARMA with Logs 6.90% 6.25% 6.59% 6.32% 6.52% 7.87%

ARMAX with Logs JNS 2.72% 1.70% 2.27% 2.92% 2.16% 4.05%

ARMAX with Logs KB 3.56% 3.23% 3.26% 4.07% 3.22% 4.30%

ARMAX with Logs Z 1.56% 1.10% 1.26% 1.37% 1.12% 3.57%

ARIMA 7.34% 6.07% 6.12% 6.80% 6.04% 7.73%

ARIMAX JNS 4.82% 2.61% 3.40% 5.82% 2.97% 6.23%

ARIMAX KB 7.93% 9.83% 5.81% 38.80% 6.18% 6.08%

ARIMAX Z 1.56% 2.70% 1.42% 2.99% 1.31% 5.35%

ARIMA with Logs 7.18% 5.99% 6.08% 6.67% 6.02% 7.49%

ARIMAX with Logs JNS 2.69% 1.60% 2.29% 2.02% 2.14% 3.84%

ARIMAX with Logs KB 3.40% 3.15% 3.29% 3.53% 3.09% 4.19%

ARIMAX with Logs Z 1.46% 1.10% 1.20% 1.32% 1.07% 3.39%

ARX GARCH Z 2.92% 2.91% 2.40% 2.58% 2.51% 6.55%

ARMA GARCH 6.70% 5.83% 6.51% 5.77% 6.44% 8.00%

ARMAX GARCH JNS 2.60% 1.73% 1.86% 1.93% 1.81% 3.81%

ARMAX GARCH KB 3.89% 3.44% 3.49% 5.48% 3.54% 4.52%

ARMAX GARCH Z 2.80% 2.72% 2.17% 2.47% 2.25% 6.25%

ARIMA GARCH 7.29% 5.66% 6.13% 6.78% 6.06% 8.49%

ARIMAX GARCH JNS 2.66% 1.79% 1.88% 2.07% 1.80% 3.86%

ARIMAX GARCH KB 3.85% 3.47% 3.50% 4.33% 3.52% 4.47%

ARIMAX GARCH Z 1.20% 1.19% 0.55% 0.94% 0.58% 3.43%

59

Table 21: 2014 Spring Zonal MAPEs Part IIModel Millwood Mohawk Val. North NYC West

ARMA 8.43% 9.41% 10.26% 8.21% 8.18%

ARMAX JNS 2.92% 2.57% 3.57% 4.07% 2.74%

ARMAX KB 5.86% 10.29% 10.54% 6.63% 4.42%

ARMAX Z 1.26% 3.26% 6.03% 3.01% 1.32%

ARMA with Logs 6.62% 6.34% 6.14% 6.56% 5.98%

ARMAX with Logs JNS 2.38% 1.64% 3.26% 2.38% 1.87%

ARMAX with Logs KB 3.35% 3.49% 3.81% 3.44% 3.21%

ARMAX with Logs Z 1.34% 1.21% 2.94% 1.51% 0.97%

ARIMA 6.14% 5.59% 4.69% 7.12% 5.95%

ARIMAX JNS 2.78% 2.68% 3.78% 3.94% 3.21%

ARIMAX KB 5.30% 11.26% 10.27% 6.42% 3.99%

ARIMAX Z 1.16% 3.28% 5.51% 3.21% 1.43%

ARIMA with Logs 6.10% 5.63% 5.51% 6.02% 5.90%

ARIMAX with Logs JNS 2.39% 1.61% 3.48% 2.39% 1.54%

ARIMAX with Logs KB 3.27% 3.30% 3.81% 3.42% 3.25%

ARIMAX with Logs Z 1.38% 1.22% 2.97% 1.41% 1.10%

ARX GARCH Z 2.50% 3.19% 2.49% 2.48% 2.34%

ARMA GARCH 6.53% 5.87% 4.90% 6.46% 5.69%

ARMAX GARCH JNS 2.02% 1.53% 2.74% 2.14% 1.69%

ARMAX GARCH KB 3.58% 3.56% 3.52% 3.54% 3.17%

ARMAX GARCH Z 2.32% 2.94% 2.69% 2.30% 2.24%

ARIMA GARCH 6.15% 5.71% 5.21% 6.05% 5.51%

ARIMAX GARCH JNS 2.04% 1.54% 2.79% 2.16% 1.80%

ARIMAX GARCH KB 3.59% 3.79% 3.80% 3.56% 3.18%

ARIMAX GARCH Z 0.54% 1.38% 1.34% 0.88% 0.86%

60

Table 22: 2014 Summer Zonal MAPEs Part IModel Capital Central Dunwoodie Genesee Hudson Long Island

ARMA 6.35% 6.24% 6.40% 6.09% 6.39% 8.09%

ARMAX JNS 1.03% 0.59% 1.39% 0.83% 1.31% 5.59%

ARMAX KB 3.35% 3.21% 3.66% 3.17% 3.49% 5.52%

ARMAX Z 0.49% 0.64% 0.52% 0.69% 0.47% 4.94%

ARMA with Logs 6.18% 6.09% 6.18% 5.96% 6.18% 7.42%

ARMAX with Logs JNS 0.66% 0.43% 0.69% 0.59% 0.65% 4.48%

ARMAX with Logs KB 3.26% 3.15% 3.32% 3.15% 3.35% 4.48%

ARMAX with Logs Z 0.33% 0.50% 0.33% 0.52% 0.31% 4.58%

ARIMA 5.73% 5.72% 5.80% 5.62% 5.79% 7.68%

ARIMAX JNS 0.90% 0.60% 1.46% 0.84% 1.34% 5.52%

ARIMAX KB 3.37% 3.20% 3.67% 3.18% 3.50% 5.44%

ARIMAX Z 5.69% 0.62% 0.48% 0.70% 0.45% 5.14%

ARIMA with Logs 0.64% 6.11% 5.70% 5.58% 5.69% 7.43%

ARIMAX with Logs JNS 3.27% 0.44% 0.70% 0.66% 0.65% 4.40%

ARIMAX with Logs KB 0.35% 3.16% 3.31% 3.15% 3.36% 4.43%

ARIMAX with Logs Z 0.35% 0.50% 0.34% 0.53% 0.31% 4.36%

ARX GARCH Z 3.31% 2.94% 3.31% 2.83% 3.44% 5.53%

ARMA GARCH 6.36% 6.17% 6.48% 6.03% 6.44% 8.20%

ARMAX GARCH JNS 0.99% 0.50% 0.88% 0.60% 0.68% 4.93%

ARMAX GARCH KB 3.30% 3.13% 3.35% 3.10% 3.41% 5.10%

ARMAX GARCH Z 2.85% 2.63% 2.69% 2.56% 2.84% 5.59%

ARIMA GARCH 5.73% 5.70% 5.73% 5.64% 5.73% 6.70%

ARIMAX GARCH JNS 0.72% 0.55% 1.04% 0.59% 0.67% 5.87%

ARIMAX GARCH KB 3.32% 3.10% 3.35% 3.09% 3.47% 5.93%

ARIMAX GARCH Z 0.44% 0.58% 0.47% 0.53% 0.42% 9.95%

61

Table 23: 2014 Summer Zonal MAPEs Part IIModel Millwood Mohawk Val. North NYC West

ARMA 6.40% 6.23% 6.11% 6.47% 6.67%

ARMAX JNS 1.39% 0.64% 1.03% 1.46% 2.41%

ARMAX KB 3.50% 3.23% 3.29% 3.66% 4.66%

ARMAX Z 0.48% 0.71% 1.01% 0.81% 1.35%

ARMA with Logs 6.19% 6.10% 5.97% 6.27% 6.43%

ARMAX with Logs JNS 0.69% 0.46% 0.81% 0.98% 1.33%

ARMAX with Logs KB 3.21% 3.14% 3.26% 3.34% 4.01%

ARMAX with Logs Z 0.32% 0.62% 0.92% 0.62% 0.93%

ARIMA 5.80% 5.68% 5.60% 5.84% 6.11%

ARIMAX JNS 1.46% 0.65% 1.05% 1.51% 2.33%

ARIMAX KB 3.53% 3.22% 3.30% 3.70% 4.50%

ARIMAX Z 0.46% 0.70% 1.00% 0.77% 1.28%

ARIMA with Logs 5.70% 5.63% 5.55% 5.76% 6.45%

ARIMAX with Logs JNS 0.71% 0.49% 0.83% 0.97% 1.28%

ARIMAX with Logs KB 3.24% 3.15% 3.27% 3.36% 4.05%

ARIMAX with Logs Z 0.33% 0.64% 0.92% 0.60% 0.94%

ARX GARCH Z 3.15% 3.04% 2.74% 2.83% 3.47%

ARMA GARCH 6.49% 6.16% 6.03% 6.54% 6.61%

ARMAX GARCH JNS 0.96% 0.64% 0.72% 1.04% 1.60%

ARMAX GARCH KB 3.30% 3.19% 3.29% 3.34% 4.41%

ARMAX GARCH Z 2.71% 2.74% 2.61% 2.71% 3.21%

ARIMA GARCH 5.73% 5.68% 6.04% 5.80% 6.15%

ARIMAX GARCH JNS 0.87% 0.55% 0.77% 0.85% 1.50%

ARIMAX GARCH KB 3.33% 3.17% 3.28% 3.36% 4.06%

ARIMAX GARCH Z 0.49% 0.65% 0.82% 0.77% 0.89%

62

Table 24: 2014 Fall Zonal MAPEs Part IModel Capital Central Dunwoodie Genesee Hudson Long Island

ARMA 10.69% 10.09% 9.54% 10.07% 9.33% 9.33%

ARMAX JNS 5.74% 2.79% 3.90% 3.06% 4.62% 4.62%

ARMAX KB 6.49% 6.36% 5.05% 5.68% 4.96% 4.96%

ARMAX Z 4.77% 2.86% 2.48% 2.98% 4.79% 4.79%

ARMA with Logs 10.17% 9.38% 9.13% 9.33% 8.72% 8.72%

ARMAX with Logs JNS 4.82% 2.65% 3.44% 2.79% 3.96% 3.96%

ARMAX with Logs KB 5.83% 4.56% 4.46% 4.62% 4.42% 4.42%

ARMAX with Logs Z 4.21% 2.51% 2.21% 2.64% 4.20% 4.20%

ARIMA 10.35% 9.56% 8.84% 9.84% 8.82% 8.82%

ARIMAX JNS 5.63% 2.89% 4.14% 3.28% 4.73% 4.73%

ARIMAX KB 6.41% 6.26% 4.99% 5.63% 4.91% 4.91%

ARIMAX Z 4.61% 2.91% 2.42% 3.05% 4.78% 4.78%

ARIMA with Logs 10.36% 9.18% 8.67% 9.76% 8.48% 8.48%

ARIMAX with Logs JNS 4.82% 2.63% 3.58% 2.79% 3.98% 3.98%

ARIMAX with Logs KB 5.76% 4.59% 4.49% 4.65% 4.45% 4.45%

ARIMAX with Logs Z 4.15% 2.51% 2.28% 2.72% 4.14% 4.14%

ARX GARCH Z 6.24% 4.56% 4.18% 4.50% 6.17% 6.17%

ARMA GARCH 10.21% 9.54% 9.22% 9.45% 8.49% 8.49%

ARMAX GARCH JNS 4.11% 2.60% 3.10% 2.74% 4.16% 4.16%

ARMAX GARCH KB 6.07% 5.03% 4.77% 5.04% 4.69% 4.69%

ARMAX GARCH Z 6.80% 4.67% 4.43% 4.54% 6.04% 6.04%

ARIMA GARCH 10.02% 9.14% 8.80% 9.11% 11.00% 11.00%

ARIMAX GARCH JNS 4.13% 2.59% 3.10% 2.77% 4.08% 4.08%

ARIMAX GARCH KB 6.08% 4.99% 4.79% 5.06% 4.84% 4.84%

ARIMAX GARCH Z 6.23% 2.37% 2.74% 2.67% 4.58% 4.58%

63

Table 25: 2014 Fall Zonal MAPEs Part IIModel Millwood Mohawk Val. North NYC West

ARMA 9.58% 10.25% 10.89% 9.49% 9.98%

ARMAX JNS 4.02% 2.77% 3.94% 4.11% 3.18%

ARMAX KB 5.27% 6.40% 7.38% 5.10% 5.81%

ARMAX Z 2.51% 2.86% 3.99% 2.69% 3.73%

ARMA with Logs 9.16% 9.51% 10.07% 9.09% 9.33%

ARMAX with Logs JNS 3.47% 2.65% 3.53% 3.46% 2.95%

ARMAX with Logs KB 4.44% 4.55% 5.63% 4.48% 4.88%

ARMAX with Logs Z 2.25% 2.50% 3.28% 2.25% 3.03%

ARIMA 8.88% 9.27% 10.00% 9.58% 9.45%

ARIMAX JNS 4.23% 3.01% 3.89% 4.25% 3.13%

ARIMAX KB 5.25% 6.15% 7.42% 4.95% 5.84%

ARIMAX Z 2.43% 2.94% 4.05% 2.66% 3.79%

ARIMA with Logs 8.71% 9.28% 10.26% 8.62% 9.12%

ARIMAX with Logs JNS 3.59% 2.61% 3.32% 3.59% 2.94%

ARIMAX with Logs KB 4.47% 4.53% 5.54% 4.51% 4.87%

ARIMAX with Logs Z 2.29% 2.54% 3.34% 2.30% 3.04%

ARX GARCH Z 4.11% 4.60% 4.25% 4.30% 4.91%

ARMA GARCH 9.26% 9.71% 10.16% 9.13% 9.28%

ARMAX GARCH JNS 3.14% 2.59% 3.31% 3.09% 2.92%

ARMAX GARCH KB 4.85% 5.04% 5.93% 4.81% 5.38%

ARMAX GARCH Z 4.42% 4.68% 4.34% 4.46% 4.99%

ARIMA GARCH 8.83% 9.25% 10.16% 8.66% 9.06%

ARIMAX GARCH JNS 3.12% 2.55% 3.30% 3.08% 2.84%

ARIMAX GARCH KB 4.88% 5.10% 6.06% 4.84% 5.39%

ARIMAX GARCH Z 2.82% 2.20% 2.06% 3.35% 3.24%

64

C.3 Exogenous Variables

Figure 23: Fall 2014 Target Models Results North Zone

65

Figure 24: Fall 2014 Target Models Results NYC Zone

Figure 25: Fall 2014 Target Models Results Capital Zone

66

Figure 26: Spring 2014 Target Model Results North Zone

Figure 27: Spring 2014 Target Models Results NYC Zone

67

Figure 28: Spring 2014 Target Models Results Capital Zone

68

C.4 Peak Analysis

Table 26: Peak Period Analysis all TestsWinter Fall Winter Fall

Model MAPE MAE MAPE MAE Model MAPE MAE MAPE MAE

Markov Chain KB 13.50% 21.66 13.50% 21.66 ARIMA 13.93% 21.86 13.93% 21.86

Markov Chain JNS 13.13% 21.39 13.13% 21.39 ARIMAX JNS 6.04% 9.31 6.04% 9.31

Markov Chain Switching KB 14.21% 22.91 14.21% 22.91 ARIMAX KB 10.67% 15.84 10.67% 15.84

Markov Chain Switching JNS 13.95% 22.83 13.95% 22.83 ARIMAX Z 3.40% 5.16 3.40% 5.16

Linear Regression JNS 12.86% 17.80 12.86% 17.80 ARIMA with Logs 14.96% 0.15 15.00% 0.15

Linear Regression KB 12.86% 19.72 12.86% 19.72 ARIMAX with Logs JNS 6.83% 0.07 6.80% 0.07

Linear Regression Z 12.86% 14.60 12.86% 14.60 ARIMAX with Logs KB 10.65% 0.11 10.60% 0.11

Linear Regression with Logs JNS 12.27% 0.12 12.30% 0.12 ARIMAX with Logs Z 3.90% 0.04 3.90% 0.04

Linear Regression with Logs KB 15.00% 0.15 15.00% 0.15 ARFIMA 13.03% 20.72 13.03% 20.72

Linear Regression with Logs Z 17.41% 0.17 17.40% 0.17 ARFIMAX JNS 6.03% 9.32 6.03% 9.32

AR 10.34% 15.39 10.34% 15.39 AR GARCH 15.26% 23.89 15.26% 23.89

ARX JNS 8.74% 12.75 8.74% 12.75 ARX GARCH JNS 6.82% 10.93 6.82% 10.93

ARX KB 9.66% 14.31 9.66% 14.31 ARX GARCH KB 11.17% 16.85 11.17% 16.85

ARX Z 9.66% 14.31 9.66% 14.31 ARX GARCH Z 11.61% 18.78 11.61% 18.78

AR with Logs 10.24% 0.10 10.20% 0.10 ARMA GARCH 14.21% 22.72 14.21% 22.72

ARX with Logs JNS 8.75% 0.09 8.80% 0.09 ARMAX GARCH JNS 6.86% 10.96 6.86% 10.96

ARX with Logs KB 9.76% 0.10 9.80% 0.10 ARMAX GARCH KB 10.77% 16.43 10.77% 16.43

ARX with Logs Z 4.43% 0.04 4.40% 0.04 ARMAX GARCH Z 10.73% 17.51 10.73% 17.51

ARMA 13.71% 21.67 13.71% 21.68 ARIMA GARCH 14.18% 22.55 14.18% 22.55

ARMAX JNS 6.16% 9.51 6.16% 9.51 ARIMAX GARCH JNS 6.32% 10.09 6.32% 10.09

ARMAX KB 10.55% 15.91 10.55% 15.91 ARIMAX GARCH KB 11.21% 16.52 11.21% 16.52

ARMAX Z 4.03% 6.00 4.03% 6.00 ARIMAX GARCH Z 4.10% 5.93 4.10% 5.93

ARMA with Logs 15.04% 0.15 15.00% 0.15 VARX JNS 11.28% 16.50 7.68% 7.36

ARMAX with Logs JNS 7.21% 0.07 7.20% 0.07 VARX KB 11.11% 16.35 10.67% 10.00

ARMAX with Logs KB 10.72% 0.11 10.70% 0.11 VARX Z 7.67% 11.65 6.24% 6.07

ARMAX with Logs Z 4.22% 0.04 4.20% 0.04

69

Table 27: 2014 Winter Peak Hour Zonal MAPEs IModel Capital Central Dunwoodie Genesee Hudson Long Island

ARMA 8.56% 18.76% 8.54% 20.64% 8.62% 8.75%

ARMAX JNS 6.47% 6.38% 4.46% 8.61% 4.27% 5.26%

ARMAX KB 7.68% 14.79% 6.03% 16.77% 5.76% 5.84%

ARMAX Z 3.18% 4.44% 2.08% 7.67% 2.02% 4.99%

ARMA with Logs 10.18% 4.00% 9.80% 21.43% 9.79% 10.12%

ARMAX with Logs JNS 7.55% 4.00% 5.08% 9.56% 4.80% 6.56%

ARMAX with Logs KB 7.57% 4.00% 6.36% 16.59% 6.03% 6.21%

ARMAX with Logs Z 2.68% 4.00% 1.95% 6.25% 1.85% 6.19%

ARIMA 7.98% 18.22% 8.55% 21.39% 8.67% 8.32%

ARIMAX JNS 6.06% 6.55% 4.20% 8.43% 4.06% 5.42%

ARIMAX KB 6.29% 14.75% 5.87% 17.53% 5.59% 6.11%

ARIMAX Z 2.73% 4.10% 1.86% 4.96% 1.87% 4.55%

ARIMA with Logs 8.51% 19.17% 9.06% 22.95% 9.16% 8.81%

ARIMAX with Logs JNS 6.69% 7.83% 4.42% 9.14% 4.21% 6.09%

ARIMAX with Logs KB 7.29% 14.41% 6.26% 15.98% 5.95% 6.31%

ARIMAX with Logs Z 2.11% 5.64% 1.64% 5.35% 1.67% 4.93%

ARX GARCH Z 7.15% 15.95% 6.59% 17.13% 6.66% 7.90%

ARMA GARCH 9.71% 21.04% 8.72% 18.41% 8.75% 10.77%

ARMAX GARCH JNS 7.21% 7.29% 5.18% 8.00% 5.03% 5.60%

ARMAX GARCH KB 6.96% 15.33% 6.24% 16.63% 6.10% 6.24%

ARMAX GARCH Z 7.22% 13.80% 6.65% 15.01% 6.68% 7.89%

ARIMA GARCH 8.74% 21.37% 8.78% 20.49% 8.88% 8.95%

ARIMAX GARCH JNS 6.51% 7.07% 4.31% 7.92% 4.20% 5.42%

ARIMAX GARCH KB 6.47% 16.25% 6.00% 17.78% 5.75% 6.48%

ARIMAX GARCH Z 1.05% 5.77% 1.17% 6.55% 1.15% 4.96%

70

Table 28: 2014 Winter Peak Hour Zonal MAPEs IIModel Millwood Mohawk Val. North NYC West

ARMA 8.53% 19.00% 22.94% 8.38% 18.09%

ARMAX JNS 4.39% 6.51% 8.97% 4.38% 8.04%

ARMAX KB 5.89% 15.36% 18.32% 5.82% 13.80%

ARMAX Z 2.13% 4.25% 3.68% 2.06% 7.84%

ARMA with Logs 9.77% 19.79% 25.43% 9.63% 19.85%

ARMAX with Logs JNS 4.80% 8.04% 12.44% 4.88% 7.60%

ARMAX with Logs KB 6.23% 14.82% 19.64% 6.32% 13.65%

ARMAX with Logs Z 1.94% 5.42% 4.60% 1.87% 8.02%

ARIMA 8.54% 21.17% 24.68% 8.42% 17.27%

ARIMAX JNS 4.20% 6.69% 9.12% 4.12% 7.64%

ARIMAX KB 5.88% 16.13% 18.69% 5.90% 14.65%

ARIMAX Z 1.93% 3.74% 3.32% 1.87% 6.42%

ARIMA with Logs 9.14% 22.22% 28.08% 8.82% 18.68%

ARIMAX with Logs JNS 4.48% 8.01% 12.70% 4.30% 7.25%

ARIMAX with Logs KB 6.00% 15.05% 20.36% 6.22% 13.31%

ARIMAX with Logs Z 1.73% 5.51% 5.59% 1.53% 7.14%

ARX GARCH Z 6.58% 16.55% 18.87% 6.45% 17.88%

ARMA GARCH 8.68% 21.13% 21.92% 8.58% 18.55%

ARMAX GARCH JNS 4.98% 7.82% 11.30% 5.18% 7.86%

ARMAX GARCH KB 6.14% 15.47% 17.52% 6.12% 15.66%

ARMAX GARCH Z 6.66% 14.12% 16.91% 6.51% 16.56%

ARIMA GARCH 8.78% 19.76% 21.38% 8.66% 20.21%

ARIMAX GARCH JNS 4.33% 7.65% 10.96% 4.26% 6.86%

ARIMAX GARCH KB 6.19% 15.75% 19.07% 6.08% 17.54%

ARIMAX GARCH Z 1.18% 4.64% 2.87% 1.10% 14.65%

71

Table 29: 2014 Fall Peak Hour Zonal MAPEs IModel Capital Central Dunwoodie Genesee Hudson Long Island

ARMA 27.36% 26.24% 14.19% 27.97% 6.80% 6.80%

ARMAX JNS 10.87% 5.56% 6.31% 5.02% 2.91% 2.91%

ARMAX KB 12.63% 10.07% 6.48% 10.98% 5.58% 5.58%

ARMAX Z 9.99% 5.77% 3.52% 5.48% 1.89% 1.89%

ARMA with Logs 29.13% 31.06% 15.47% 33.25% 6.52% 6.52%

ARMAX with Logs JNS 10.66% 5.32% 5.40% 4.88% 4.87% 4.87%

ARMAX with Logs KB 16.58% 12.36% 7.82% 12.28% 5.29% 5.29%

ARMAX with Logs Z 13.05% 4.85% 4.17% 4.53% 3.60% 3.60%

ARIMA 32.02% 27.37% 14.22% 33.30% 7.23% 7.23%

ARIMAX JNS 11.59% 5.82% 6.52% 5.25% 3.05% 3.05%

ARIMAX KB 12.66% 10.07% 6.42% 11.19% 5.76% 5.76%

ARIMAX Z 10.45% 5.84% 3.59% 5.56% 2.35% 2.35%

ARIMA with Logs 39.39% 32.45% 15.00% 41.79% 41.79% 7.25%

ARIMAX with Logs JNS 11.88% 5.54% 5.86% 4.92% 4.92% 3.99%

ARIMAX with Logs KB 17.02% 12.14% 7.35% 12.35% 12.35% 5.06%

ARIMAX with Logs Z 13.18% 5.03% 4.65% 4.76% 4.76% 4.10%

ARX GARCH Z 23.17% 18.39% 11.06% 18.86% 7.20% 7.20%

ARMA GARCH 25.66% 26.43% 14.69% 27.39% 7.32% 7.32%

ARMAX GARCH JNS 5.55% 6.01% 4.58% 5.58% 3.23% 3.23%

ARMAX GARCH KB 16.72% 12.26% 7.66% 11.76% 4.82% 4.82%

ARMAX GARCH Z 21.29% 16.42% 10.31% 17.22% 6.51% 6.51%

ARIMA GARCH 31.87% 24.35% 13.92% 25.38% 9.03% 9.03%

ARIMAX GARCH JNS 6.17% 6.08% 5.02% 5.68% 3.56% 3.56%

ARIMAX GARCH KB 16.96% 12.13% 7.92% 11.80% 4.31% 4.31%

ARIMAX GARCH Z 10.59% 5.28% 3.77% 5.15% 1.70% 1.70%

72

Table 30: 2014 Fall Peak Hour Zonal MAPEs IIModel Millwood Mohawk Val. North NYC West

ARMA 14.19% 26.02% 33.05% 20.02% 25.45%

ARMAX JNS 6.38% 4.93% 2.38% 6.66% 6.45%

ARMAX KB 6.50% 12.62% 9.75% 7.25% 11.15%

ARMAX Z 3.45% 5.59% 2.78% 4.34% 6.43%

ARMA with Logs 15.46% 30.70% 42.90% 20.78% 29.93%

ARMAX with Logs JNS 5.41% 5.47% 2.66% 5.96% 5.35%

ARMAX with Logs KB 7.65% 13.04% 14.08% 10.50% 9.96%

ARMAX with Logs Z 4.10% 5.00% 1.93% 6.07% 4.84%

ARIMA 14.22% 25.01% 32.18% 21.13% 26.06%

ARIMAX JNS 6.63% 5.04% 2.35% 6.79% 6.54%

ARIMAX KB 6.26% 12.48% 9.64% 6.86% 11.45%

ARIMAX Z 3.54% 5.81% 2.99% 4.52% 6.55%

ARIMA with Logs 15.00% 28.25% 11.16% 4.50% 7.07%

ARIMAX with Logs JNS 5.99% 5.82% 0.68% 1.27% 1.18%

ARIMAX with Logs KB 7.53% 12.75% 2.73% 2.09% 2.10%

ARIMAX with Logs Z 4.57% 5.37% 0.59% 1.30% 1.14%

ARX GARCH Z 11.08% 19.28% 25.52% 13.63% 16.92%

ARMA GARCH 14.68% 26.15% 33.59% 19.26% 25.94%

ARMAX GARCH JNS 4.32% 6.21% 6.36% 3.64% 5.36%

ARMAX GARCH KB 7.52% 11.97% 16.87% 7.55% 13.36%

ARMAX GARCH Z 10.36% 17.43% 24.48% 13.17% 15.56%

ARIMA GARCH 13.92% 23.94% 33.53% 19.77% 25.17%

ARIMAX GARCH JNS 4.74% 6.35% 6.55% 3.79% 5.52%

ARIMAX GARCH KB 7.82% 12.18% 16.74% 7.67% 13.01%

ARIMAX GARCH Z 3.80% 5.15% 3.81% 4.32% 7.08%

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C.5 Hourly Analysis

Table 31: 2014 NYISO Hourly MAPEs Targeted ModelsHour ARIMAX JNS ARIMAX Z ARMAX JNS ARMAX Z

0 9.95% 6.03% 9.98% 6.47%

1 4.62% 3.26% 4.44% 2.92%

2 4.12% 2.57% 3.76% 2.22%

3 3.46% 2.50% 3.15% 1.97%

4 2.60% 2.19% 2.92% 2.05%

5 3.88% 3.34% 3.86% 3.36%

6 5.43% 4.28% 5.41% 4.49%

7 6.12% 4.18% 6.20% 4.28%

8 3.97% 2.85% 4.11% 2.98%

9 3.06% 2.26% 3.11% 2.39%

10 2.84% 2.27% 2.88% 2.25%

11 2.44% 1.80% 2.59% 1.98%

12 2.36% 1.75% 2.56% 1.85%

13 1.84% 1.46% 2.16% 1.53%

14 2.04% 1.80% 2.26% 1.77%

15 1.77% 1.39% 2.12% 1.62%

16 3.81% 3.04% 3.85% 3.40%

17 4.33% 3.26% 4.23% 3.28%

18 4.08% 2.65% 4.19% 2.74%

19 2.70% 2.06% 2.87% 2.20%

20 3.23% 2.69% 3.39% 2.50%

21 3.97% 2.72% 3.99% 2.82%

22 3.98% 3.53% 4.01% 3.55%

23 5.22% 4.11% 5.08% 4.08%

Average 3.83% 2.83% 3.88% 2.86%

Hour 16-20 3.63% 2.74% 3.71% 2.82%

Hour 22-3 5.47% 3.69% 5.28% 3.53%

C.6 ARIMA and ARMA with Log Specifications

Table 32: NYISO Average Target Exogenous Model MAPE and MAE with Log SpecificationsModel MAPE MAE

ARMAX with Logs 3.13% 0.03

ARIMAX with Logs 3.09% 0.03

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Table 33: NYISO Exogenous Model MAPEs and MAEs Seasonally with Log SpecificationsSeason Winter Spring Summer Fall

Model MAPE MAE MAPE MAE MAPE MAE MAPE MAE

ARMAX with Logs 5.17% 0.05 2.61% 0.03 1.36% 0.01 3.37% 0.03

ARIMAX with Logs 5.02% 0.05 2.51% 0.03 1.37% 0.01 3.45% 0.03

Table 34: Selected Tests Exogenous Variable MAPEs with Log SpecificationsWinter Spring Summer Fall

Model MAPE MAE MAPE MAE MAPE MAE MAPE MAE

ARMAX JNS 6.21% 4.437 4.62% 1.75 0.59% 0.158 3.90% 1.923

ARMAX KB 11.58% 7.446 8.45% 2.98 3.21% 0.826 5.05% 2.470

ARMAX Z 3.91% 2.800 2.19% 0.85 0.64% 0.167 2.48% 1.191

ARIMAX JNS 5.87% 4.267 4.82% 1.81 0.60% 0.159 4.14% 2.014

ARIMAX KB 9.53% 6.617 7.93% 2.81 3.20% 0.824 4.99% 2.431

ARIMAX Z 4.00% 2.700 1.56% 0.65 0.62% 0.163 2.42% 1.183

Table 35: Select Model Peak Period Accuracy with Log SpecificationsWinter Fall Winter Fall

Model MAPE MAE MAPE MAE Model MAPE MAE MAPE MAE

ARMA with Logs 15.04% 0.15 15.00% 0.238 ARIMA with Logs 14.96% 0.15 15.00% 0.293

ARMAX with Logs JNS 7.21% 0.07 7.20% 0.055 ARIMAX with Logs JNS 6.83% 0.07 6.80% 0.057

ARMAX with Logs KB 10.72% 0.11 10.70% 0.104 ARIMAX with Logs KB 10.65% 0.11 10.60% 0.107

ARMAX with Logs Z 4.22% 0.04 4.20% 0.051 ARIMAX with Logs Z 3.90% 0.04 3.90% 0.055

D. Literature AppendixThis appendix provides additional and supporting detail on the existing literature avail-

able for electricity price forecasting. Discussed are models not incorporated in the paper,but the ideas put forward in the models may be significant for the study of electricity priceforecasting.

D.1 Jump Diffusion Models(Bunn and Karakatsani 2003) examine a variety of statistical and financial forecasting

techniques in their overview of electricity price forecasting. First they examines jump diffu-sion models. These models have been borrowed from financial modeling to predict electricityprices. The specifications are provided below. Bunn looks at Random-walks, mean rever-sion models, Mean-reversion with jumps, with both constant parameters and time-varyingparameters. (Bunn and Karakatsani 2003) provide the following equations:

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Random Walk:dPt = µPtdt + �dWt (19)

Mean-Reversion

dPt = a(µ� Pt)dt + �dWt (20)

Mean-reversion with JumpsConstant Jump Parameters:

dPt = a(µ� Pt)dt + �dWt + kdqt(�) (21)

Time Varying Jump Parameters:

Pt = ft +Xt (22)

dvt = kv(✓v � vt)dt + v

1/2�dZt, �t = v

1/2t ) (23)

�t = �1Winter + �2Spring + �3Summer + �4Fall (24)

In Equations 19 through 24, Pt denotes the spot price, Wt a Weiner process, ft a deter-ministic component such as fuel prices, load and weather, qt is a Poisson process, � describesthe seasonal jump occurrences, and k = k(µj�j) is a random variable that describes thejump magnitude. This allows the model to adjust for seasonality. (Bunn and Karakatsani2003) highlight that these models do a poor job disentangling mean reversion from reversalof spikes to normal levels, which restricts actual short-term predictions. He argues that theassumed Poisson distribution of jumps in prior literature is not the issue with these models.The problem is that the models do not pick up the tight demand-supply conditions andspikes that occur out of season. He puts forward regime-switching as an alternative to jumpdiffusion.

(Bunn and Karakatsani 2003) show that these models perform better but caution againstthe use of financial techniques in electricity price forecasting because of the above-mentioneddifferences. (Bunn and Karakatsani 2003) advocate for the structural modeling discussedbelow.

D.2 Similar Day and Exponential SmoothingSimilar day forecasting is a very popular benchmark model in electricity price forecasting

(Weron 2014). Similar characteristics include day of the week, day of the year, holidays,weather and consumption figures (Shahidehpour, Yamin, and Li 2002). The most commonsimilar-day approach is the naïve approach used by (Nogales et al. 2002) and (Conejo et al.2005). The naïve approach specifies a rule that a Monday is similar to the Monday of theprevious week, the same for Saturdays and Sundays. Tuesdays are similar to the Mondayof the same week; Wednesday is similar to Tuesday and so on through Friday. The naïveapproach is highly accurate in the absence of price volatility, hence its use as a benchmarkmethod (Conejo et al. 2005).

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Another simplistic benchmark model is exponential smoothing. Exponential smoothinghas routinely been applied to electricity load forecasting and has only recently been appliedto electricity price forecasting (Weron 2014). In exponential smoothing the prediction isconstructed from exponentially weighted averages of past observations, each smoothed valueis the weighted average of the previous observations and the weights decrease exponentiallydepending on the value of the parameter (Weron 2014). There are more complex modelsthat deal with seasonal and other trend components (Gardner 2006). The leading edge ofexponential smoothing is the THETA model established and tested by (Assimakopoulos andNikolopoulos 2000) and (Hyndman and Billah 2003). The THETA model attempts to adda drift method to traditional exponential smoothing to help the basic model account forextreme volatility and price spikes. The model has only been tested in limited circumstancesand it is unclear how it would perform for daily or hourly electricity prices. Overall, ex-ponential smoothing models perform slightly better than ARIMA models absent large pricespikes but worse than dynamic regressions and transfer functions (Weron 2014). Nonethe-less, exponential smoothing along with similar day forecasting offer important benchmarksfor all electricity price forecasts.

Exponential Smoothing:x̂t = st = axt + (1� a)t�1 (25)

There are several variations of equation 25. However, as put forward in (Gardner 2006)and (Weron 2014) st is the weighted average of previous observations. The weights decreaseexponentially according to the parameter a 2 (0,1). With this model, forecasts are not com-puted from consecutive previous observations alone. They show an independent smoothedtrend that allows the addition of seasonal components. The model can be applied to bothload and price. This paper applies the model to energy prices. Therefore, the energy x̂ isthe weighted average of past hourly day-ahead electricity spot prices.

D.2.1 Smoothing Results

Exponential smoothing tests were administered on the NYISO data. However, theirresults were far weaker than all of the tests run. They may provide an interesting benchmark MAPE, however the Linear Regression tests all do this with greater accuracy. Belowis the summary of the Exponential Smoothing MAPE. The forecasts were computed usingthree types of smoothed data. One forecast forecasted non-adjusted data, one where thesample period had gone through a moving-average adjustment and one with an exponentially-smoothed adjustment.

Table 36: 2014 NYISO Sample Expoentiopnal Smoothing ResultsWinter Spring Summer Fall

Smoothing MAE MAPE MAE MAPE MAE MAPE MAE MAPE

Non-Adjusted 59.75 115.13% 19.39 38.36% 6.57 25.70% 19.39 38.36%

MA-Adjustment 59.05 113.65% 17.26 33.73% 6.50 26.55% 17.26 33.73%

Smoothed-Adjustment 66.35 137.96% 13.56 27.25% 7.22 31.62% 13.56 27.25%

Smoothed data takes away from peak periods which is why the absolute error is so high

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in periods of high volatility with price greatly deviating from the mean such as the winterperiod. During the most stable period, the Summer, the baseline forecast still misses a lot ofintraday trends and movements leading to larger MAPEs than other baseline forecasts suchas the Linear regression based forecast.

D.3 Dynamic Regressions and Transfer FunctionsDynamic regressions are commonly used subset of ARIMA in electricity price forecast-

ing. (Nogales et al. 2002) applied dynamic regressions and transfer functions to the wholesaleelectricity markets of Spain and California. The dynamic regression model overcomes serialcorrelation in the data. The model used by (Nogales et al. 2002) relates the price at aspecific hour t to the past price hours t � 1, t � 2, which leads to uncorrelated the errors.(Nogales et al. 2002) found that overall dynamic regressions carried lower errors than thetransfer function models.

The transfer function model is another subset of ARIMA models that assumes both priceand demand series are stationary. This model includes serially correlated errors. In this way,the model relates actual prices.

In both models (Nogales et al. 2002) apply logarithmic transformations to price anddemand data to obtain homogeneous variance. The study adjusted for outliers. Finally toadd robustness to the forecasts, (Nogales et al. 2002) add half of the variance to the forecastbefore taking its exponential to eliminate the logarithmic transformation that occurs whenthe equations are log transformed.

(Nogales et al. 2002) find that Spain had higher mean standard errors than Californiain both studies because there was less natural competition. Spanish markets had higherdispersion, which lead to greater uncertainty, adding difficulty to the forecast. Anotherimportant note is that California relies more heavily on long-term contracts, so their marketis allowed greater fluctuations by regulators because it is far less of a share of total electricitythan the Spanish Market.

(Conejo et al. 2005) also compared dynamic regressions and transfer functions. Theestablishment of the models follow’s the same four steps as the ARIMA. (Conejo et al.2005) established the dynamic regression and the transfer function similarly to (Nogales etal. 2002). However, he found that the transfer function outperformed both the ARIMA andDynamic regression.

(Conejo et al. 2005) summarizes the differences between the basic ARIMA, dynamicregressions and transfer functions well. The basic ARIMA relates current prices to past pricesand current errors to previous errors. Dynamic regressions relate current and past pricesand current and past demands. Transfer functions relate price to past prices, demand anderrors. The validity and robustness of the three forecasting models depends on many factorsincluding the time period, volatility and regulatory structure of the market studied (Weron2014). These models were excluded because from final analysis due to the market structureof the NYISO. The NYISO functions similarly to the Ontario Power Market discussed in(Zareipour 2012) where it was found that the basic ARIMA model generally outperformedthe other two specifications. From a comparative standpoint it was more methodologicalto compare strictly AR models: The AR, ARMA, and ARIMA, therefore the DynamicRegression and Transfer Function have been excluded from the analysis. Tests were down

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incorporating these models and the accuracy was substantially less than the ARIMA models.Dynamic Regression:

ph = c+ u(B)ph + v(B)dh + "h (26)

u(B) =nX

k=1

ukBk (27)

v(B) =nX

k=1

vkBk (28)

Transfer Function:

ph = c+ w(B)dh +Nh (29)

w(B) =nX

k=1

wkBk (30)

Nh =✓(B)

F(B)"h (31)

Log Dynamic Regression:

log(ph) = c+ u(B)log(ph) + v(B)log(dh) + "h (32)

Log Transfer Function:

log(ph) = c+ w(B)log(dh) +Nh (33)

In equation 26 through 28, the Dynamic Regression: ph is hourly price dh is hour demandor load, c is a constant and "h is is the error term. u(B) and v(B) are polynomial back-shiftoperators.uk and vk are polynomial coefficients that are selected and estimated to achievewhite noise errors. For the Transfer Function, equations 29 through 31: cis a constant, phremains the price, w(B) is the back-shift operator, same as in the Dynamic Regression modeland Nhis a disturbance term following the ARMA model form. However, price and demandare both assumed to be stationary (Nogales et al. 2002). Dynamic Regressions and TransferFunctions relate actual prices to past demands through w(B) and past errors through ✓(B)and actual prices to past prices with F(B), they both also lead to uncorrelated errors. Asmentioned above (Nogales et al. 2002) used a logarithmic transformation to achieve a morehomogenous variance (see equations 32 and 33). Equations 26-33 are tested and developedin (Nogales et al. 2002) and (Conejo et al. 2005).

D.4 AR GARCH Type ModelsA common method to deal with the volatility of electricity prices is through a Gener-

alized Auto Regressive Conditional Heteroskedasticity (GARCH) Process combined with anARIMA model (Weron 2014). GARCH specifications are more suitable to capture dynamicsof a time-series’ conditional variance (Tan et al. 2010). As applied by (Tan et al. 2010), the

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model includes four phases: data preparation, model identification, parameter estimationand diagnostic checking. The same process (Conejo et al. 2005) (Tan et al. 2010) and(Nogales et al. 2002) use for all other non-linear models.

ARMA:Xt = F1Xt�1 � ..+ Ft�p + ✓1"t�1 + ..+ ✓q"t�q (34)

GARCH:Xt = "t�t (35)

Any GARCH process, even applied to an ARIMA model is an adaption of an ARMAmodel which is a non-linear expansion of an AR model. Xt is an ARMA process or price andload as specified in 34. The GARCH specifciation of X can then be applied to the ARIMAand AR as in (Koopman, Ooms, and Carnero 2007).

(Koopman, Ooms, and Carnero 2007) compares GARCH specifications to basic ARIMAbased regressions over the EEX power market of Germany, APX market of the Netherlandsand PowerNext in France. (Koopman, Ooms, and Carnero 2007) include GARCH models be-cause they take volatility clustering and extreme observations into account. GARCH modelsdo not stand well alone for short term electricity price forecasting, they must be applied withARIMA type models (Koopman, Ooms, and Carnero 2007)(Weron 2014). ARIMA-GARCHmodels outperform typical ARIMA models when high volatility and price spikes are present(Weron 2014). If volatility has been accounted for through alternative methods, GARCHspecifications do not add to the robustness of forecast (Weron 2014).

D.4.1 GARCH Results

GARCH specifications were tested for robustness. GARCH specifications show that theAR, ARMA, and ARIMA results were robust and consistent.

Table 37: Baseline GARCH ResultsModel MAPE MAE Model MAPE MAE

AR GARCH 8.69% 4.55 ARMAX GARCH KB 5.42% 2.89

ARX GARCH JNS 3.14% 1.85 ARMAX GARCH Z 4.06% 2.45

ARX GARCH KB 5.27% 2.85 ARIMA GARCH 7.93% 4.31

ARX GARCH Z 4.14% 2.49 ARIMAX GARCH JNS 2.95% 1.75

ARMA GARCH 7.94% 4.33 ARIMAX GARCH KB 5.10% 2.78

ARMAX GARCH JNS 3.06% 1.82 ARIMAX GARCH Z 2.42% 1.32

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Table 38: GARCH Specifications with Exogenous Variables ResultsWinter Spring Summer Fall

Model MAPE MAE MAPE MAE MAPE MAE MAPE MAE

ARMAX GARCH JNS 6.06% 4.620 2.60% 1.04 0.50% 0.133 3.10% 1.496

ARMAX GARCH KB 9.90% 6.837 3.89% 1.49 3.13% 0.813 4.77% 2.410

ARMAX GARCH Z 6.38% 5.806 2.80% 1.02 2.63% 0.537 4.43% 2.451

ARIMAX GARCH JNS 5.48% 4.254 2.66% 1.06 0.55% 0.142 3.10% 1.524

ARIMAX GARCH KB 8.68% 6.404 3.85% 1.47 3.10% 0.809 4.79% 2.431

ARIMAX GARCH Z 5.16% 3.264 1.20% 0.51 0.58% 0.158 2.74% 1.336

Table 39: GARCH Peak Period PerformanceWinter Fall

Model MAPE MAE MAPE MAE

ARMA GARCH 14.21% 22.72 20.77% 19.302

ARMAX GARCH JNS 6.86% 10.96 4.92% 4.604

ARMAX GARCH KB 10.77% 16.43 10.48% 9.833

ARMAX GARCH Z 10.73% 17.51 14.48% 13.673

ARIMA GARCH 14.18% 22.55 20.90% 19.589

ARIMAX GARCH JNS 6.32% 10.09 5.18% 4.877

ARIMAX GARCH KB 11.21% 16.52 10.44% 9.782

ARIMAX GARCH Z 4.10% 5.93 4.76% 4.389

Table 40: Daily GARCH MAPEsDay ARIMAX GARCH JNS ARIMAX GARCH Z ARMA GARCH JNS ARMA GARCH Z All Test Average

1 2.68% 2.63% 2.80% 5.29% 3.58%

2 2.58% 2.18% 2.39% 5.12% 3.35%

3 2.45% 2.48% 2.22% 5.45% 3.04%

4 2.03% 2.18% 2.42% 4.47% 2.78%

4 3.40% 3.48% 3.45% 6.49% 3.26%

6 4.76% 4.14% 4.72% 5.42% 4.37%

7 5.23% 3.54% 5.24% 5.23% 4.92%

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D.5 Literature Accuracy Comparison

Table 41: Literature MAPEsModel Paper Winter Spring Summer Fall

Baseline (Bordignon et al. 2013) 7.61% 4.94% 4.15%Lin Reg (Karakatsani and Bunn 2008) 11.80%

(Karakatsani and Bunn 2008) 8.20%(Zareipour 2012) 48.75% 35.00% 34.25% 41.50%

Markov Chain (Karakatsani and Bunn 2008) 9.40%(Karakatsani and Bunn 2008) 5.69%

AR (Karakatsani and Bunn 2008) 10.17%(Karakatsani and Bunn 2008) 13.14%(Crespo Cuaresma et al. 2004) 32.94%

AR with Logs (Crespo Cuaresma et al. 2004) 35.08%ARMA (Crespo Cuaresma et al. 2004) 25.40%

ARMA with Logs (Crespo Cuaresma et al. 2004) 25.40%ARIMA (Conejo et al. 2005) 6.60% 15.36% 16.14% 11.19%

(Tan et al. 2010) 11.15% 14.26% 10.37% 13.93%(Zareipour 2012) 18.10% 17.20% 17.55% 17.60%

(Contreras et al. 2003) 12.06% 19.37% 8.17% 7.32%ARIMAX (Contreras et al. 2003) 9.97% 14.68% 9.39% 9.88%

ARIMA-GARCH (Tan et al. 2010) 10.00% 12.30% 10.37% 9.98%

GlossaryA. New York Electricity Sector Terms

1. FERC - Federal Energy Regulatory Commission2. LSE - Load Serving Entity3. IESO - Independent Electricity System Operator of Ontario4. ISO - Independent System Operator5. ISONE - Independent System Operator of New England6. NYISO - New York Independent System Operator7. NYPP - New York Power Pool8. NYPSC - New York Public Service Commission9. NYSRC - New York State Reliability Council

10. PJM - Pennsylvania-New Jersey-Maryland Interconnection

B. NYISO Terms

82

1. da-lbmp - Day Ahead Locational Based Marginal Price2. da-load - Day Ahead Forecasted Load3. dd (hdd/cdd) - Degree Day, the deviation in temperature from 65ºF4. energy-bid-load - Day Ahead Committed Load5. ICAP - Installed Capacity Market6. integrated-load - Real Time Committed Load7. LBMP - Locational Based Marginal Pricing8. rt-lbmp - Real Time Locational Based Marginal Price

C. Modeling Terms

1. AR - Autoregressive Model2. ARX - Autoregressive Model with exogenous variables3. ARIMA - Autoregressive Integrated Moving Average Model4. ARIMAX - Autoregressive Integrated Moving Average Model with exogenous vari-

ables5. ARMA - Autoregressive Moving Average Model6. ARMAX - Autoregressive Moving Average Model with exogenous variables7. VAR - Vector Autoregressive Model

D. Analysis Terms

1. MAPE - Mean Absolute Percentage Error2. MAE - Mean Absolute Error3. JNS Exogenous Variables - Jacob Silver selected variables including non-power

sector variables such as weather, fuel price, and neighbor price with past priceand load

4. KB Exogenous Variables - Karakatsani and Bunn selected variables includingprice and load volatility and fluctuation with past price and load

5. Z Exogenous Variables - Zareipour selected variables including future values ofprice and load instead of strong reliance on past price and load

83

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