Electric Power Systems Research 71 (2004) 91–98

An efficient stochastic self-scheduling technique for powerproducers in the deregulated power market

G.B. Shrestha∗, Song Kai, L. Goel

Nanyang Technological University, School of Electrical and Electronic Engineering, Nanyang Avenue, Singapore 639798, Singapore

Received 30 June 2003; received in revised form 2 December 2003; accepted 14 December 2003

Abstract

Generation scheduling and dispatch are determined by individual power producers’ bids in a deregulated power market. The benefitsobtained by a power producer will depend largely on how effectively it can incorporate the variation of the market price in its generationscheduling. A stochastic scheduling technique is presented in this paper for maximizing a producer’s benefit considering the stochastic natureof power price. Formulation of the problem has been presented in detail. Two approaches to the solution, namely two-stage and multi-stagestochastic methods, are presented which accommodate the features of the day-ahead and hour-ahead power markets, respectively.

Implementation of the technique is illustrated by applying the technique to scheduling by a power producer with 11 generators in twodifferent seasons. The price data from a real system has been used for the price forecasting and the formation of price scenario tree. It is shownthrough illustrative case studies that the proposed stochastic methods can enhance benefits if proper values are assigned for the probabilitiesof the price scenarios and if the selection of the probabilities of price scenarios are directly related to the variation of the actual power prices.The proposed technique is particularly effective when the uncertainty in the price is high and the price forecast is not very accurate.© 2004 Elsevier B.V. All rights reserved.

Keywords: Stochastic schedule and dispatch; Price uncertainty; Price scenario tree; Power market

1. Introduction

Decentralized generation schedule and dispatch areadopted in deregulated electricity markets such as in Nordpool, California Power Exchange, etc. The task of gener-ation scheduling is left to power producers themselves inorder for them to make dispatch decisions to maximize theirown benefits. Several modes of power trading are practicedin deregulated markets. Power producers can bid in thespot market and/or directly engage in bilateral contractswith the customers along with financial instruments to re-duce market risks. Thus, more uncertainties are inherent inderegulated power markets, which may affect the benefitsof the power producers. The risk-averse power producersmay prefer to engage in long-term bilateral contracts withcustomers, which would guarantee that they obtain certainbenefits by selling their power, while the risk-prone powerproducers may like to enhance their benefits by exploiting

∗ Corresponding author.E-mail address: egovinda@ntu.edu.sg (G.B. Shrestha).

the variation of price in the spot market[1]. In general,both methods may be adopted by power producers to max-imize their benefits using the available market information.Considering these new features of deregulated markets, itis essential for power producers to have new techniques forgeneration scheduling considering the uncertainties of themarket such as the electricity price and their own generationcapabilities to maximize their benefit in this deregulatedenvironment[2]. Strategic bidding studies using stochasticmethods have been reported in the literature. A novel for-mation for integrated bidding and hydrothermal schedulingis presented in[3]. The market-clearing price (MCP) ismodeled as a Markov chain and the risks are directly formu-lated in the objective function in terms of MCP variances.Bidding decision-making in a spot market was studied as aMarkov decision process from the viewpoint of a supplierin [4,5]. A discrete stochastic optimization method wasproposed where all other suppliers are modeled by theirbidding parameters with corresponding probabilities and asimplified market-clearing system was adopted.

The rest of this paper is organized as follows.Section2 explains the stochastic self-scheduling leading to the

0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.epsr.2003.12.019

92 G.B. Shrestha et al. / Electric Power Systems Research 71 (2004) 91–98

problem formulation including the formation of price sce-narios tree.Section 3presents the implementation issuesoutlining the method used to forecast the power price us-ing the ARIMA model, the formation of price scenarios ateach hour, and the solution techniques using two-stage andmulti-stage models for scheduling generation. It is followedby a numerical example to illustrate the effectiveness of theproposed techniques inSection 4. AndSection 5presents theconclusions.

2. Stochastic self-scheduling

This paper presents a self-scheduling technique for thepurpose of a small power producer (IPP or a small utility)to optimize its benefit in the electricity market. The powerproducer is taken as a price-taker in the market, whichmeans that its generation bids are not expected to influencethe market price. The scheduling task for such a producerwould be straightforward if the future market prices wereknown or the scheduling is performed deterministically fora fixed estimated future price. The optimal generation ata particular moment is simply where the incremental gen-eration cost equals the market price. It is assumed in thisanalysis that such a price-taker is able to sell its genera-tion at whatever level it wants. One obvious way is to bidthe power it seeks to supply at a significantly low price toensure that it is scheduled in the market-clearing process.But its revenue is determined not by its bid price but by themarket-clearing price. However, the market price in the newemerging competitive environment shows wide variations,and there is no way for the supplier to know the precisemarket-clearing price in advance.

The estimation/determination of power price to be incor-porated in the scheduling problem can be done in two ways[6]. One approach is to regard it as an internal variable, i.e.,the forecaster simulates the market-clearing process to de-termine the power prices based on the known informationof the power system, the estimation of other participants’information and its own information. However, this methodrequires the maintenance of a much larger system, which be-comes quite difficult for a realistic system. The other methodis to regard the price as an external stochastic variable, and itis estimated using some forecasting technique(s) using rel-evant historical data. The latter approach is adopted in thisstudy. It should be noted that although the price forecast us-ing this approach should reflect the normal stochastic vari-ations in system operating conditions, these price forecastsmay not be accurate enough or valid when drastic systemicchanges occur. One simple solution for a price-taker will beto base the generation schedule/bid decisions on the meanforecasted prices, but the producer has to live with the pos-sible losses that arise due to the discrepancy between theforecasted price the actual price in the future. This could bequite substantial, which may not be acceptable. The resultswith the proposed stochastic self-scheduling are compared

to the results obtained with this simple approach in the casestudies discussed later inSection 4.

2.1. Formulation of the problem

A stochastic approaches to generation scheduling consid-ering price uncertainty is proposed and developed in thefollowing:

Generally, the spot market is a day-ahead market and thebids will be made 24 h in advance. In some markets, powerproducers can resubmit bids several hours or even just 1 hahead of market clearing. As the price estimation for thenext hour would be much more accurate that for the peri-ods further down the future, it would be desirable to preparethe bids and generation schedules only 1 h ahead. However,because of the minimum up and down time constraints ofgeneration units the generation schedule planning must bemade at least several hours in advance. This is a necessitydespite the increased uncertainty further down the future.These conflicting requirements are handled in the proposedapproach in the following manner. The scheduling generat-ing units (i.e., unit commitment) is optimized a day aheadfor 24 h on the basis of the forecasted prices for the 24 h forthe next day. Once, the feasible unit commitment is avail-able, the optimal power output will be determined only forthe next hour based on a more accurate estimate of the mar-ket price. Bid in this study means the generation level forthe next hour. The actual bid will be for the next hour only,while the unit commitment will be based on the forecast forthe next 24 h.

The formulation of optimal scheduling is based on theforecasted power prices along with the price forecast uncer-tainty. Price forecasting is used to estimate the prices (ex-pected values) for the entire scheduling period of 24 h alongwith their likely variation (the standard deviation). The fore-cast is used to identify a number of likely price scenarios ateach hour along with the likelihoods of their occurrences.Three scenarios (normal price, higher price, and low price)are used in this study. A price scenario tree is then formedfor the 24 h period as shown inFig. 1. The task of stochas-tic scheduling is to determine path in this tree from 1 to24 h in order to maximize the expected value of the benefitsover the entire 24 h period. The optimization problem willbe solved as a problem coupled for 24 h. This will yield theoptimal unit commitment for each hour on the basis of theprice forecast for 24 h. Then, the optimal output at each hour

1 2

...

...

3

1.0

0.3

0.3

0.4

123

45

6

7 8

9

4 5

...

High price

Normal Price

Low price

t

Fig. 1. Price scenario tree.

G.B. Shrestha et al. / Electric Power Systems Research 71 (2004) 91–98 93

is determined from the more precise estimate price for thenext hour.

The following notations are used in the following formu-lation:i index of the thermal units;t index of the time stages;s index of the price scenarios;N total number of thermal units;T number of stages (24);S number of price scenarios (3);q(i, s, t) output of uniti at scenarios and

staget (MW);db amount of bilateral contract (MW);pb price of the bilateral contract ($/MW);Sc(i, s) start-up cost for uniti from

full-cold state;u(i, s, t) decision variable indicating whether

unit i is up or down at staget (0,1);qmax (i) maximum output of uniti;qmin (i) minimum output of uniti;Ton/off (i) minimum up/down time of unitI;Xon/off (i, s, t) duration when uniti has been on/off

at staget;p(t, s) estimated price at scenarios at staget;w(s) probability of price scenarios;whl probability of high or low price scenario;a0(i), a1(i), quadratic cost coefficients for uniti;

a2(i)

The objective function to maximize is the expected valueof the power producer’s benefit.

maxS∑

s=1

[w(s)

T∑t=1

N∑i=1

(revenue(i, t, s) − cost(i, t, s))

]

=S∑

s=1

w(s)

T∑t=1

{p(t, s)∗

[N∑

i=1

q(i, t, s)∗u(i, t, s) − db

]

−N∑

i=1

[a0(i) + a1(i)∗q(i, t, s) + a2(i)

∗q2(i, t, s)]∗

×u(i, t, s) −N∑

i=1

sc(i, t, s)

}(1)

Both bilateral contracts as well as bidding in power mar-kets are considered in the development. If a producer hasa contract fordb MW at the fixed price ofpb $/MW for aperiod (mid-term or long-term), then the power producer’soutput must be no less thandb and it can submit bids to thepower only for the rest of its generation capacities.

The objective function (1) is to be maximized subject tothe following system constraints at every hour:

(i) Output limits for units

qmin(i)u(i, t, s) ≤ q(i, t, s) ≤ qsmax(i)u(i, t, s),

s = 1, . . . , S, t = 1, . . . , T (2)

(ii) Requirements of bilateral contract

N∑i=1

q(i, t, s)u(i, t, s) ≥ db,

s = 1, . . . S, t = 1, . . . , T (3)

(iii) Minimum cold time, up, and down time limits

(Xon(i, s, t − 1) − T on(i))(u(i, s, t − 1)

−u(i, s, t)) ≥ 0, (Xoff (i, s, t − 1) − T off (i))

(u(i, s, t) − u(i, s, t − 1)) ≥ 0,

s = 1, . . . , S, t = 1, . . . , T. (4)

(iv) Non-anticipativity constraints

u(i, t, s) = u(i, t, s′) = cs(i, t),

i = 1, . . . , N whens = s′ (5)

The progressive hedging policy is adopted as a decisionrule. Decision variables are introduced at each node of thescenario tree, i.e. if two different scenarioss ands′ are in-distinguishable at timet, the decision for scenarioss ands′at time t must be the same. These are the non-anticipativityconstraints, which are named scenario bundles.

2.2. Solution technique

The stochastic bidding model is a large-scale mixed-integer optimization problem. Lagrangian relaxation methodis applied to decompose the problem with regard to gener-ating units. Associating stochastic Lagrange multipliers (theLagrange multipliers are updated using proximal bundlemethod that is detailed in[7]) with the coupled constraint(3) leads to the following Lagrangian functionL:

L(λ) =S∑

s=1

w(s)

T∑t=1

{N∑

i=1

[revenue(i, t, s) − cost(i, t, s)]

+λ(t, s)∗[

N∑i=1

q(i, t, s)u(i, t, s) − db

]}

=S∑

s=1

w(s)

T∑t=1

{p(t, s)∗

[N∑

i=1

q(i, t, s)∗u(i, t, s) − db

]

−N∑

i=1

[a0(i) + a1(i)∗q(i, t, s) + a2(i)

∗q2(i, t, s)]∗

× u(i, t, s) −N∑

i=1

sc(i, t, s) + λ(t, s)∗

×[

N∑i=1

q(i, t, s)u(i, t, s) − db

]}

=N∑

i=1

Di(λ)−S∑

s=1

w(s)

T∑t=1

(p(t, s)∗db+λ(t, s)∗db) (6)

94 G.B. Shrestha et al. / Electric Power Systems Research 71 (2004) 91–98

where

Di(λ) =N∑

i=1

w(s)

T∑t=1

{p(t, s)∗q(i, t, s)∗u(i, t, s) − [a0(i)

+a1(i)∗q(i, t, s) + a2(i)

∗q2(i, t, s)]∗u(i, t, s)

−sc(i, t, s) + λ(t, s)∗q(i, t, s)u(i, t, s)}

(7)

Then the dual functionD is:

D(λ) = maxL(λ) (8)

Then the dual problem is

min{D(λ)} (9)

The minimization of (9) is subject to the single unit con-straints (2), (4), and (5).λ(t, s,) is the stochastic multipliercorresponding to the constraint (3) for scenarios of staget.

From (6), it is seen that the dual function can be expressedas the sum of single unit sub-functions. Therefore, the dualproblem is decomposed into some single unit sub-problems.The sub-problem for uniti is.

maxDi(λ) = maxN∑

i=1

w(s)

T∑t=1

{p(t, s)∗q(i, t, s)∗u(i, t, s)

−[a0(i) + a1(i)∗q(i, t, s) + a2(i)

∗q2(i, t, s)]∗

×u(i, t, s) − sc(i, t, s) + λ(t, s)∗

×q(i, t, s)u(i, t, s)} (10)

It is subjected to the single unit constraints (2); (4); and(5). The sub-problem is mixed-integer stochastic program,which is solved using stochastic dynamic programming(SDP) method. The approach used is very similar to thatdetailed in[3]. Detailed description of the SDP method ofsolution applied to this problem has been reported in[8].

3. Implementation

3.1. Price forecast and formation of price scenarios

Autoregressive integrated moving average (ARIMA)model is utilized to forecast the price based on the recentactual historical data. Only the historical price data are re-quired for this model. The variances can also be obtainedin addition to the mean values of the price for every fore-cast interval. The price forecasting using ARIMA model isreported in the literature, for example[9–11]. This methodis very suitable to form different price scenarios based onthe forecasted expected prices and the standard deviationsobtained from the corresponding variances. Of course, thereare other approaches to forecast prices and to generate pricescenarios, which may also be investigated in the future.

In the ARIMA model, the seasonal part of 1 week (168 h)is first considered as the electricity prices are inherently sea-soned by weeks. In addition, the time difference is consid-ered based on the feature of the historical data.

Table 1Discrete price values with probabilities at an houri

Node no.

1 (high price) 2 (normal price) 3 (low price)

Discrete price yi + 1.1616σi yi yi − 1.1616σi

Probability 0.3 0.4 0.3Price range >yi + 0.52σi yi ± 0.52σi <yi − 0.52σi

Using the ARIMA model, the variancesσ2i along with

expected valuesyi of the forecast prices are obtained for ev-ery forecast intervali. This continuous distribution can beconveniently used to identify a suitable number of discreteprice scenarios. Generally, a larger number of discrete pricescenarios would yield more accurate results, but with addi-tional complexity of the problem and heavier computationaltasks. In this study, for each hour three discrete price stateshigh price, normal price, and low price, with probabilities ofwhl; (1–2 whl); andwhl, respectively are used. The volatil-ity of the market price can be represented by an appropriatechoice of the value ofwhl. The values of discrete prices fora givenwhl can be easily computed andTable 1shows thesevalues forwhl = 0.3.

It is noted that since only three discrete prices are obtainedfor each hour, the infeasible task of exponential computationof 24th power is avoided. Other ways for discretization andscenario generation are possible.

Detailed stochastic scheduling studies are presented inthe next section for a period of 30 days in the summer andthe winter. Price forecast is therefore performed for thesetwo periods. The data used for forecasting are the historicalhourly prices. Historical data of the previous 5 weeks areutilized to forecast the price of a particular day.Fig. 2showsthe forecasted prices for a typical day in winter (5 February2000) and a typical day in summer (30 August 1999). Theactual historical prices and the high and low prices com-puted asyi + 1.1616σi and yi − 1.1616σi are also shownin the figure. When the pricesyi ± 1.1616σi do not exceedthe price limitsymax, i andymin, i (maximum and minimumprice values observed in historical data) the three values aredistinct. When the discretized low or high prices exceed themaximum or minimum observed value, the price is cappedat this value.

It may be noted fromFig. 2 that the forecasted expectedprices are quite close to the real prices in the winter. Butthe forecasted expected prices at 13–17 h in the summerare significantly lower than the real prices. The implicationof these forecast accuracies will be discussed in the nextsection. The price scenario tree shown inFig. 1 is formedwith these three price scenarios at each stage except at 1 hwhere the price is considered known.

The probabilities of the three price scenarios shown inFig. 1 are forwhl = 0.3 for the first period. However, theprobabilities of reaching a scenario at the successive hour(i+1) from a scenario at the previous hour (i) for anywhl canbe easily calculated and are shown inTable 2for whl = 0.3.

G.B. Shrestha et al. / Electric Power Systems Research 71 (2004) 91–98 95

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4

1 0

1 5

2 0

2 5

3 0

3 5

4 0

4 5

5 0

H ig h P r ic e

E x p e c te d P r ic e

L o w P r ic e

R e a l P r ic e

Price

($

/MW

)

H o u r N o(a)

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4

0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

1 0 0

1 1 0

H ig h P r ic e

E x p e c t e d P r ic e

L o w P r ic e

R e a l P r ic e

Price

($

/MW

)

H o u r N o(b)

Fig. 2. Forecasted price for forming price scenarios (a) forecasted price for a winter day: 5 February 2000; and (b) forecasted price for a summer day:30 August 1999.

3.2. Solution procedure

Two approaches to the solution of the problem have beenattempted to represent different levels of flexibility in thescheduling.

3.2.1. Two-stage stochastic procedureThe spot market is usually a day-ahead market. The sup-

plier submits bids 24 h ahead, and then they have limitedadjustments in their operation in different price scenarios inthe same time period, allowing only the outputs of the unitsto change. A two-stage stochastic procedure is appropriateto represent this feature of bidding by power producers. Inthe first stage, only the unit commitment schedule is deter-mined, which is kept unchanged for all the scenarios at thatperiod. In other words, the variableu(i, t, s) determined in

Table 2Arc probabilities

Arc no. Probability

1 0.092 0.123 0.094 0.125 0.166 0.127 0.098 0.129 0.09

the first stage, will have the same value for all the scenariosat any one time intervalt. The unit outputs are specified only1 h ahead based on a more accurate estimate of the price forthe next hour.

3.2.2. Multi-stage stochastic procedureSince the bids are committed only 1 h ahead it may be

desirable to adjust generation significantly, beyond the ca-pacity of a fixed unit commitment for all scenarios adoptedin the above approach if and when there is sudden orfast change in the price. Multi-stage stochastic procedureaccommodates this feature as both unit commitment andunit outputs can be adjusted in this procedure for differ-ent scenarios at the same time interval. As different unitcommitment decisions can be made for different scenariosfor any one time-interval, this approach can respond to thechange in power price more effectively than the two-stageapproach. In the two-stage procedure, only the output levelsof the units can be adjusted, while in the multi-stage proce-dure, the outputs as well as the unit commitment decisionsof all the units can be adjusted while subject to the on/offtime delay constraints. Therefore, the multi-stage procedureis expected to produce more expected benefits.

4. Case studies and findings

The proposed stochastic self-scheduling technique is ap-plied to the operation of a power producer with 11 generating

96 G.B. Shrestha et al. / Electric Power Systems Research 71 (2004) 91–98

Table 3Benefits obtained using different optimization procedures forwhl = 0.3

Time period Total benefit ($) Benefit difference ($)

R T M D T−D M−D M−T

Summer 28,156,205.33 27,942,355.85 27,994,068.2 27,913,752.53 28,603.31 80,315.65 51,712.34Winter 24,112,467.90 23,964,416.56 23,966,349.08 23,973,666.66 −9,250.09 −7,317.58 1,932.51

units to optimize the generation schedule (bids) in aday-ahead and an hour-ahead power markets. The generatordata used in these studies are obtained from[12] with nec-essary modifications to suit the problem and are listed inTable A.1in theAppendix A. Historical power prices wereobtained from the website of a real power market for theperiods 1 July–30 September 1999 and 1 January–31 March2000 to represent two different seasons, summer and win-ter, respectively. A bilateral contact ofdb = 1750 MW at afixed pricepb $/MW has been incorporated in the studies.The issues of how to determinepb anddb are not exploredin this study. The average price of the known historical datais taken as the contract pricepb.

The performance of the proposed techniques were evalu-ated by comparing the benefits obtained using the proposedtechniques with those obtained using other standard andideal conditions. As the proposed techniques are stochasticin nature, it would not be meaningful to compare the ben-efits over a short period of time. Therefore the benefits arecalculated for a relatively long time period of 30 days forcomparison. The benefits derived by the power producer us-ing the following four methods were computed.

(a) First, the benefits were calculated by applying deter-ministic optimal generation scheduling methods usingthe actual market prices and are denotedR. Obviously,the unit schedule and the power dispatch so determinedyield the maximum possible benefit. It should be noted,however, that the real prices can only be known after theactual event and cannot be used in practice. The benefitscomputed here are purely for comparison purposes.

(b) A convenient way of generation scheduling is to use theexpected values of the market prices. It will not be com-putationally demanding as it requires only the expectedvalues of prices for the next 24 h at the beginning ofthe day and then schedules its generating units deter-ministically based on the forecasted prices to maximizeits benefits. The unit outputs are adjusted in response tothe real market prices hour by hour as the time goes onwhile the UC is unchanged. The benefits obtained bythis method are denoted asD.

(c) The benefits enjoyed by the power producer using thetwo-stage stochastic scheduling described earlier are de-noted asT.

(d) The benefits obtained by applying the multi-stagestochastic scheduling described earlier are denoted asM.

The benefits calculated using the four methods for a pe-riod of 30-days in summer and winter seasons are listed inTable 3. The high/low price probabilitywhl is taken as 0.3,which means that the probability of high price, normal price,and low price scenarios are 0.3, 0.4, and 0.3, respectively.

As expected, the benefits using the actual price,R, arelargest in all cases. This is because the valuesR are themaximum possible benefits as described above. It shouldalso be noted that the benefits obtained using the multi-stagestochastic scheduling (M) are always larger than the benefitsobtained using the two-stage stochastic scheduling (T). Thebenefits derived by the power producer in an hour-aheadmarket are $51,712 and $1932 more in the summer and thewinter, respectively compared to the benefits derived in aday-ahead market.

The benefits achieved using the proposed stochastic meth-ods M andT exceed the benefit achieved applying the de-terministic method using the expected values of pricesD inthe summer by $80,350 and $28,603, respectively. However,despite the additional complexities involved in the stochas-tic methods, the benefits obtained using these methods arefound to be less than that obtained by the deterministicmethod in winter by $7317 and $9250. Thus, it appears thatspecific consideration of the variability of the power pricesin the optimization process may not necessarily enhance thetotal expected benefits.

A careful look at the forecasted price inFig. 2arevealsthat the expected value of the forecast values were veryclose to the actual prices in the summer and the fluctuationin price is very little. Therefore, ad-hoc assignment of highprobabilities to the high or low prices (whl) may not bequite appropriate. The forecasted price in summer shownin Fig. 2b exhibits a much larger variation from the actualprices. The proposed stochastic techniques seem to enhancethe benefits under these conditions. Thus, it appears that itis important to assign proper values towhl, which reflectsthe variability of the market prices.

This hypothesis was tested by computing the benefits us-ing the stochastic methods by varying the high/low priceprobabilitywhl from 0.02 to 0.46 in steps of 0.04. The vari-ation of the difference between the benefits obtained usingthe two-stage stochastic method (T) and the deterministicmethod (D) with different values ofwhl is shown inFig. 3.

It is seen that the benefit difference declines with highervalues ofwhl in the winter when the variation of the fore-cast price was very small. The benefit difference becomesnegative for(whl) ≥ 0.14. On the other hand, the benefit

G.B. Shrestha et al. / Electric Power Systems Research 71 (2004) 91–98 97

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44

-10000

-8000

-6000

-4000

-2000

0

2000

4000

Benefit

diffe

rence (

$)

Probability of low/high price

(a) winter

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

Be

ne

fit

Diffe

ren

ce

($

)

Probability of low/high price

(b) summer

Fig. 3. Variation of benefit difference (T−D) with whl.

difference increases withwhl in the summer when the pricevariation was high.

The variation of the difference between the benefits ob-tained using the multi-stage stochastic method (M) and thedeterministic method (D) with different values ofwhl isshown inFig. 4. These curves also exhibit the same patternas those ofFig. 3. The benefit difference decreases with anincrease inwhl in winter when the variation in actual pricewas low, and the benefit difference increases with an in-crease inwhl in summer when the variation of the actualprice was high.

Thus, it is evident that the proposed stochastic methodscan enhance the benefits of a power producer if the high/lowprice probabilitywhl is properly assigned. For the examplesystem, the value ofwhl should be kept≤0.10 in winterand ≥0.10 in the summer. It is further observed that thisprobability is directly related to the variability of the actualmarket prices. However, further studies would be requiredto investigate these relationships in more detail.

The benefit of using the multi-stage over the two-stagestochastic procedure is highlighted inFig. 5, which showsthe variation of the benefit difference (M−T) for differentvalues ofwhl. It is clearly seen that the multistage methodconsistently gives more benefit in summer when the pricevariation was high. But, there is hardly any difference be-tween the benefits produced by the two stochastic methodsin winter when the price variation was very small. This isexplained by the fact that the ability to adjust the UC andunit outputs in a spot market will not produce additionalbenefits when the market prices do not vary significantly.

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

Ben

efit

Diff

eren

ce (

$)

Probability of high/low price

0.40 0.44

(a) winter

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44

-20000

0

20000

40000

60000

80000

100000

Ben

fit D

iffer

ence

($/

MW

)

Probability of high/low power price

(b) summer

Fig. 4. Variation of benefit difference (M−D) with whl.

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44-10000

0

10000

20000

30000

40000

50000

60000

Benefit Different in Winter Benefit Different in Summer

Ben

efit

Diff

eren

ce (

M -

T)

($)

Probability of high/low price

Fig. 5. Variation of benefit difference (M−T) with whl.

5. Conclusions

A stochastic scheduling technique has been presented inthis paper for maximizing a producer’s benefit consider-ing the stochastic nature of power price. Formulation of theproblem has been presented in detail. Two approaches to thesolution, namely two-stage and multi-stage stochastic meth-ods are described which accommodate the features of theday-ahead and hour-ahead power markets, respectively.

Implementation of the technique has been demonstratedby applying the technique to scheduling by a power pro-ducer with 11 generators. The price data from a real systemhave been used for the price forecasting which are used inthe formation of the price scenario tree in the implemen-tation of the proposed technique. The ability of the tech-nique to enhance the benefits are illustrated by comparing the

98 G.B. Shrestha et al. / Electric Power Systems Research 71 (2004) 91–98

Table A.1Generator unit data

Unit no. Output (MW) Cost function coefficients Start upcost ($)

Minimumup time (h)

Minimumdown time (h)

Minimumcold time (h)

Initialstate (h)

Minimum Maximum a b c

1 150 660 192.00 7.80 0.0026 440 9 9 9 92 160 450 264.00 17.40 0.0029 1500 6 5 5 53 100 400 253.09 9.00 0.0023 1000 8 5 5 84 140 350 212.47 13.03 0.0018 500 8 5 6 85 75 250 168.00 16.32 0.0014 70 4 4 5 −16 20 200 206.40 32.40 0.0312 250 5 5 6 −37 45 120 263.73 23.04 0.0084 160 4 3 3 −38 68.9 197 310.96 27.60 0.0031 400 5 4 4 −49 10 80 98.40 19.20 0.0276 120 3 1 2 −1

10 12 60 153.33 46.26 0.0384 60 1 1 1 −111 12 60 153.73 44.00 0.0391 60 1 1 1 −1

Note: for the initial state,−1 means that the unit has been ‘off’ for 1 h and 8 means that the unit has been ‘on’ for 8 h.

benefits achieved by the technique to: (i) the maximum pos-sible benefit from schedules obtained on the basis of theactual price data, and (ii) the benefits from schedules basedon the easily computed mean values of the forecast prices.It was found from the initial studies that the technique en-hanced the benefits in summer but not in winter. It is shownthrough further studies that the proposed technique will en-hance the benefits in both seasons if proper values are as-signed to the probabilities of the price scenarios used. Theprobabilities of the price scenarios should be selected on thebasis of the variation of the power price in the market.

It was also demonstrated that the multi-stage stochas-tic method, which takes into account the advantages of anhour-ahead market, may not produce additional benefits un-less the market price exhibits some variability.

It is noted that there are different methods to forecastprices and to generate price scenarios. Except the ARIMAmodel for price forecasting and the discretizing process forprice scenario generation used in this study (through whichthe effectiveness of the benefit enhancement has alreadybeen shown in the paper), it is possible to investigate the ef-fects of other price forecast and scenario generation methodsby incorporating these methods into the developed stochas-tic self-scheduling process.

Appendix A

(SeeTable A.1)

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