Optimal Bidding Strategy for Multi-unit Pumped Storage Plant in Pool-Based Electricity Market ...

8/14/2019 Optimal Bidding Strategy for Multi-unit Pumped Storage Plant in Pool-Based Electricity Market Using Evolutionary T

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ICSET 2008

Optimal Bidding Strategy for Multi-unit PumpedStorage Plant in Pool-Based Electricity Market

Using Evolutionary Tristate PSOP. Kanakasabapathy and K. Shanti Swarup

AbstractThis paper develops optimal bidding strategy foroperating multi-unit pumped storage power plant in day-aheadelectricity market. Based on forecasted hourly market clearingprice, a multistage looping algorithm to maximize the profitof multi-unit pumped storage plant is developed consideringboth spinning and non-spinning reserve bids and meeting the

technical operating constraints. The proposed model is adaptivefor the nonlinear three-dimensional relationship between thepower produced, the energy stored, and the head of the associatedreservoir. Evolutionary Tristate Particle Swarm Optimization(ETPSO) based approach is also proposed to solve the same

problem, combining basic Particle Swarm Optimization (PSO)with tri-state coding technique and mutation operation. Thediscrete characteristic of a pumped storage plant is modeled usingtri-state coding technique and genetics based mutation operationis used for faster convergence in getting global optimum. Theproposed approaches are applied with an actual utility consisting

of four units. Experimental results for different operating cyclesof the storage plant indicate the attractive properties of theETPSO approach in a practical application, namely, a highlyoptimal solution and robust convergence behaviour.

Index TermsEvolutionary Tristate Particle Swarm Optimiza-tion, ETPSO, Pumped Storage, Bidding Strategies, Optimal

Scheduling, Electricity Market.

I. INTRODUCTION

IN the new environment of competitive electricity mar-ket, power producers face challenging problems with theultimate goal of maximizing their profits. Pumped storage

hydro electric plant, the oldest kind of large-scale energy

storage technology since 1904, are in active operation and

new ones are still being built because of their operational

flexibility and ability to provide rapid response to changes

in system loading or spot price of electricity. In integrated

systems, the pumped-storage plants are used to serve the peak

load. In a competitive electricity market, pumped-storage unit

owner can buy and sell electricity through trading. The income

of a pumped-storage unit includes the revenue received by

selling energy when it is in the generating mode and by being

accepted in the non-synchronous reserve market when not in

the generating or pumping mode. The pumped-storage unitcan also be committed for synchronous reserve when it is in

the pumping mode because it can readily reduce its pumping

power and, consequently, reduce the system load.

Manuscript received on July 12, 2008.P. Kanakasabapathy and K. Shanti Swarup are with the Department of

Electrical Engineering, Indian Institute of Technology Madras, Chennai -600036, India. e-mail: [email protected], [email protected].

Thus, in deregulated market, there are strong incentives for

pumped-storage units to optimize their schedules. Initially, the

marginal cost method has been used for scheduling pumped

storage plant that is operated in combination with other plants

in the vertically integrated traditional systems [1]. The problem

was also addressed with different techniques like dynamic

programming [3] and genetic algorithm [4]. An evolutionary

computation technique, known particle swarm optimization

(PSO) [5], has recently become a candidate for many optimiza-

tion applications due to its high performance and flexibility.Hybrid and binary PSO [6] were successfully applied to solve

discrete problems. The proposed evolutionary tristate particle

swarm optimization (ETPSO) approach combines classical

PSO technique with tristate coding and a mutation operation.

Optimal bidding strategies for a pumped-storage plant in a

competitive electricity market, in which the Market Clearing

Price (MCP) is insensitive to the bid price of a single generator

has been developed in [7]. There are several characteristics

of the pumped storage plant like generating limits that are

strong function of head, discrete pumping loads and generating

schedules, which strongly affect the strategy that can be used

for bidding in the pool based day-ahead electricity market.

Meeting these technical operating constraints, based on fore-

casted hourly MCP curves, this paper will focus on developingoptimal bidding strategies using multi-looping algorithm and

ETPSO approach to maximize the profit of multi-unit pumped-

storage operators to bid in the day-ahead market, considering

into account both spinning and non-spinning reserve bids.

II. OPTIMALO PERATION OFP UMPEDS TORAGE P LANT

Pumped storage stations usually have two reservoirs; the

upper reservoir having little inflow, the lower reservoir is used

to store the water after generation and will be pumped back

to upper reservoir whenever cheap and surplus power is avail-

able. Reversible turbine-pump system along with synchronous

machine is used for generating and pumping modes. In the

pump mode, because of inherent losses, the power requiredfor pumping water is more than the power that is generated

by the same volume of water. That is, for the plant cycle

efficiency of p (0 < p < 1), it is economical to bid forsellingpM W hof energy at a time period oftg with pumpedstorage generation, if there exists a time duration tp to bidfor buying 1 M W h, such that the ratio of the MCPs duringpumping and generating is less than p.

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Fig. 1. Market Clearing Price curve: (a) Daily MCP; (b) Composite MCP.

A. Optimal Operating Time

Considering a period ofThours operating cycle, the energystoredET inM W h over the period T is

ET =E0+ Ein+ Ep Eg (1)

Here E0 is the initial stored energy in the upper reservoirandEin is the inflow energy during the period T. If the unitgenerates Pg(i) MW for i = 1, 2, ... tg hours and pumpsat Pp(i) MW for i = 1, 2, ... tp hours, the pumped andgenerated energies are defined respectively by

Ep

= p

tp

i=1

Pp

(i) Eg

=

tg

i=1

Pg

(i) (2)

Considering pumping power remain same for the entire period

tp, the operating times tp andtg are related by

tgi=1

Pg(i) = Pptpp ET+ E0+ Ein (3)

The optimal operation during the operating cycle requireE0 =ET. If the unit generates at an average Pg MW for tg hours,the above equation (3) reduces to

tg =Pptpp+ Ein

Pg(4)

Assuming zero changeover time, using equation (4), maximum

pumping time within the operating cycle T =tp+tg can beestimated as

tpmax=T Ein

Pg

1 + pPpPg

= T

1 + pPpPg

(5)

when there is no inflow to the upper reservoir. Therefore tpmaxcan be used as the stopping criterion for optimization in order

to meet the energy balance requirement E0 = ET.

Fig. 2. Approximation of Equivalent Energy Curve; Piecewise linearapproximation of relationship between stored energy and height of water

B. Optimal Market Clearing Price

Consider a real time daily MCP curve for a typical restruc-

tured power system [11] shown in Fig. 1(a). Let Bg be theMCP in $/MWh above which the plant operates in generating

mode and sell energy to the market,Bpbe the MCP in $/MWh

below which the plant buys energy from market and operate inpumping mode. A composite MCP curve shown in Fig. 1(b)

can be obtained by sorting MCP in ascending order. When

the MCP is greater thanBg, the pumped-storage unit suppliespower to the grid. When the MCP is less than Bp, the unitpumps water for storage. For the plant efficiency ofp, to beeconomically profitable, the condition to be satisfied is

Bg Bp/p Bg 1.5 Bp (6)

Since typical value of plant cycle efficiency p is about 67%.

III. MODEL FORP UMPEDS TORAGE P OWERP LANT

A. Approximation of Equivalent Energy Curve

The relationship that exists at any time tbetween the energy

stored Es and the height of water in the reservoir h in apumped storage power plant is given by

Est= f1 (vt) = f2 (hrest ); t T (7)

Where Est = Energy stored in the reservoir at time t, vt =Volume of water contained in the reservoir andhrest =Waterhead at the instant of time. Equation (7) is a nonlinear relation.

Piecewise linear model provide appropriate approximation [2].A piecewise linear model shown in Fig. 2 can be implemented

through the following set of linear equations.

Est=iS

i ht,i t T (8)

hrest =iS

ht,i t T (9)

Where S = no of steps used for piecewise linear approx-imation, i = slope of the i

th step of linearized function

and ht,i = variable defined for every step that expressesthe part of the step that is needed to obtain total height of

hrest . Equation (8) provides the piecewise linear approximationfor the energy stored and equation (9) the piecewise linear

approximation for height. Note that, for the model presented in

equation (8), only positive values of parameteriare realistic.

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Fig. 3. Estimation of generation bids. (a) Unit Limit Curves; (b) EquivalentEnergy Curve

B. Optimal Selection of Generation Bid

At the beginning of each hour, let EAM W hbe the energystored in upper reservoir and hA meters be the correspondinghead. From the equivalent energy curve and power limit curvesshown in Fig. 3,PgA M Wis the high power limit correspond-ing to the head hA. If the plant generates at PgA M W, at theend of the hour the energy stored is EB = EAPgA M W h.PgA can not be considered as power bid, because immediatelyafter starting generation the high limit constraint would be vi-

olated. From Fig. 3,PgB MW is the high limit correspondingto the headhB and stored energyEB. If the plant generates atPgB MW for an hour, at the end of this duration the energy

stored is EC = EA PgB MW h. PgC MW is the highlimit corresponding to the head hC and stored energy EC.In the equivalent energy curve EC lies below EA and aboveEB. The optimal power bid for generation lies in between theextreme values PgC andPgB , obtained using simple iterativealgorithm with stopping condition EC EB , where isthe minimum difference between blocks for power bids.

IV. BIDDINGS TRATEGY FORE NERGY ANDA NCILLARYSERVICES

A. Mathematical Model of Plant Operation

Revenue of pumped storage plant include incomes from

day ahead market by selling energy during generating mode,

synchronous reserve market by reducing power consumed for

pumping fromPp to(Pp Prs) at a price ofBrs $/MWh bysimply stopping the pumping for a period up to a maximum of

tp hours, where 0 Prs Pp and non-synchronous reservemarket at a price ofBrn $/MWh for the preiod of(Ttptg)when the MCP is between the two price thresholds, Bg and Bpand the unit is in off line. The expenditure include payments

for power needed to pump water into the upper reservoir,

operating and maintenance costs. The problem is

(10)

Max

tg

i=1

Pg(i)Bg(i) +

tpj=1

Prs(j)Brs(j)

+

(Ttptg)

k=1

Pg(k)Brn(k)

tpm=1

Pp(m)Bp(m)

(tg+tp)n=1

Co(n) Cm

To optimize the profit of the unit, the optimal period of

pumping tpopt and generating tgopt during a cycle can befound by increasing the pumping time of the pumped-storageunit from zero totpmaxand checking for the maximum profitsubject to the constraint of energy stored in the upper reservoir

E(t), has an upper and lower limits given by

Emin E(t) Emax t T (11)

and the generation for the plant must be in the intervals:

P LL(t) Pg(t) P HL(t) t T (12)

B. Optimal Operating Condition

Being a discrete function with a step of 1h, the condition

for maximum profit is obtained by making

(13)P

t =PgBgtg PgBrntg Cotg+ PrsBrstp

PgBrntp PpBptp Cotp = 0

From equation (4), tg =Pptpp

Pg(14)

IfPp is completely reduced to bid in syn-reserve market, i.e.ifPrs = Pp, solving equation (13) using the equation (14)

Bg = 1p

Bp+ Brn

p+ Pg

Pp

Brs+ Co

1Pp

+ pPg

(15)

From equation (15), it is evident that synchronous and non-

synchronous reserve market bids have significant impact on

deciding the margin between optimal bids for energy in gen-

erating and pumping mode. In order to solve the constrained

optimization problem given in equation (10), multistage loop-

ing algorithm is developed. For discrete price data points the

proposed sequential algorithm is efficient. Instead, line searchtechnique can also be applied to obtain faster solution.

V. EVOLUTIONARY T RISTATEPARTICLE S WARM

OPTIMIZATION M ETHODOLOGY

Classical particle swarm optimization concept [8] consistsof, at each time step, changing the velocity vk of each particle

ptowards its potential solutionpbestand global optimumgbestusing specific update rules. In this approach, classical updation

rule is modified and specific algorithm given in section V-B

is designed for formation of particle substrings in the initial

phase and updation of particle substrings during the iterations

of ETPSO methodology.

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Fig. 4. Tri-state coded particle string for a pumped storage power plant.(a) Particle string for T hours operation (b) Details of a substring (N units)

A. Tri-state Coding of Particle String

Control variables that optimize the objective function are

plants operating mode and the power output/input. Sincepumping loads and generating schedules are discrete and theyare handled on hourly basics, instead of plants water discharge

rate, the plants generation output/power input is used asanother control variable. Water dynamics and storage reservoir

constraints are handled in terms of energy in MWh. Fig. 4(a)

presents a particle string for T hour operation of a plant [9].Fig. 4(b) shows control variable details of a substring for a

plant with N units.

B. Algorithm for Formation of Particle Strings

1) For each strings S = 1 to P op, Initialize the tri-statebit of all T substrings in the particle string by randomlyselecting1,1 or 0 with appropriate probability.

2) Read the Initial stored energyEo.3) For substringt = 1to T, for each block in the substringu= 1 to N, select the control variables.

4) If the tri-state bit is 1, store the possible Pg corre-sponding to Es. Adjust the Es accordingly.

5) If the tri-state bit is -1, then store Pp if the reservoirlimits are not violated. Increase Es accordingly.

6) If the tri-state bit is 0, store the optimalPg values. Donot adjust Es since this block is used only for biddingin non-synchronous reserve market.

C. Mutation Operation

Conventional PSO approach converges rapidly during the

initial search period, and then slows down during the later

stage. Mutation operation is capable of overcoming this short-coming [10]. Mutation is an occasional operation with a small

probability, make random alternation of tri-state bit of the

pbeststring, as shown in Fig. 5.

D. Scoring Function

The scoring function adopted is based on the corresponding

plant profit which is normalized into a range of0 1.

Fig. 5. Mutation operation. (a) Substring of pbeststring, (b) Substring ofnew particle

SCORE(p) = 1 + kp

profit(p)

prof it(gbest) 1

(16)

where SCORE(p) is the score (fitness value) of the pth

particle string, profit(p) is the corresponding profit of thepth particle,profit(gbest)is the profit of the highest rankingparticle string, i.e.gbest, and kp is a scaling constant.

E. ETPSO Solution Methodology

The following algorithm is used for evolving the optimal

bidding strategy and corresponding scheduling for multi-unit

pumped storage power plant using ETPSO approach.

1) Read the data and plant constraints.2) Initialize swarm using algorithm given in section V-B.

3) Evaluate the particles.

(a) Estimate the profitof each particle.

(b) Scoreeach particle using equation (16).(c) Initialize pbest with current position of particles.(d) Initialize thegbestas the best among all thepbest.

4) Update the position of particle string.

(a) Perform mutation for one bit in the particle string.

(b) Using steps 2-6 of algorithm given section V-B,

update the position of particle string.5) Estimate profitand score the new particle position.

6) Updatepbest if the new position is better than pbest.

7) Updategbest if the new position is better than gbest.

8) Perform steps 4-7 for all the particles in the swarm.

9) Every ten iterations perform advanced mutation opera-

tion for three tri-state bits in the gbeststring as follows.

(a) Randomly select three tri-state bits stored with

1, -1, and 0.

(b) Mute the bits respectively with -1, 0, and 1

or 0, 1, and -1. Go to step-3.

10) Perform steps 3-9 for required number of iterations. Print

gbestas the optimal solution.

V I. CAS ES TUDYWe consider Blenheim-Gilboa Pumped Storage Plant of

New York Power Authority connected with New Yorks elec-tricity transmission grid. The New York Independent System

Operator (NYISO) [11] manages the transmission network and

electricity market. The plant details are Capacity=4260MW,Active head=300 to 330m, Emin = 1000M W h , E max =8000MWh, E0 = 1000MW h and p= 0.6667 [12].

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Fig. 6. Energy storage with respect to time for daily and weekly operatingstrategies (22-28 June 2008): Results from Sequential Optimization Approach

TABLE IOPTIMAL BIDDING STRATEGY AND CORRESPONDING SCHEDULE USING

SEQUENTIAL M ETHODOLOGY FROM22-28 JUNE 2008

Let the daily operating cycle starts from 0:00 AM and ends

at 0:00 AM of the following day and the weekly operatingcycle start on Sunday with E0 = 1000 MW h and endson the following Sunday with ET = 1000 M W h. Priceforecasts are obtained from the New York Independent System

Operator (NYISO) website [11]. MCP of NYC region for a

week starting from 22 to 28 June 2008 are considered. It is

assumed that Brs = 6 $/MWh and Brn = 0.5 $/MWh asconstants. The optimal bidding strategy developed in sectionIV is implemented and both the algorithms are simulated in

MatLab 7.6 R (R2008a) and applied for daily and weeklyoperating modes. Both spinning and non-spinning reserve

biddings are also considered for optimizing the generating and

pumping power bids.

A. Sequential Methodology

The energy storage in the upper reservoir with respect to

time is shown in Fig. 6. In daily operating mode energy

balance is satisfied at the end of each day, whereas in theweekly operating mode it is satisfied only at the end of

the week. This allows weekly operating mode to operate at

higher level of head and hence unit can be bid for higher

generating power than the daily operating mode. Optimal

bidding strategy and corresponding schedule obtained usingsequential methodology for the period from 22-28 June 2008

is shown for daily operating mode in Table I(a) and for weeklyoperating mode in Table I(b). Optimal power bids for the same

period with respect to daily schedule are given in Fig. 7(a)

and weekly schedule are given in Fig. 7(b). It is seen that

the pumping mode fall on valley MCP period and generation

mode fall on peek MCP period.

Fig. 7. Optimal power bids for the period from 22-28 June 2008 (a) Dailyschedule (b) Weekly Schedule: Sequential Optimization Approach

Fig. 8. Energy storage with respect to time for daily and weekly operatingstrategies (22-28 June 2008): Results from ETPSO Approach

B. ETPSO Methodology

The energy storage in the upper reservoir with respect to

time for the same period obtained form ETPSO approach is

shown in Fig. 8. Optimal bidding strategy and correspondingschedule obtained are shown for daily and weekly operating

modes in Table II (a) and (b). Corresponding optimal power

bids are given in Fig. 9(a) and (b). Fig. 10 illustrates the

convergence characteristics of the proposed ETPSO for seven

different cases of daily operating mode with 8 particles and

shows the impact of particle initialization and provide anindication for robustness of the ETPSO. The ETPSO with

particle population of 50 is applied for solving the case of

weekly operating mode. Fig. 11 illustrates the convergence

characteristic of the proposed method for this case.

C. Comparison and Observations

Form the Figures 7 and 9, it is seen that both the algorithms

are adaptive for the nonlinear three-dimensional relationshipbetween the power produced, the energy stored, and the head

of the associated reservoir. During the generating period, asthe head decreased the generating power bids are accordingly

reduced. Performance of sequential and ETPSO methodologies

in view of profit, plant operation parameters and average CPU

execution time are given in Table III. For weekly scheduling

both the methodologies yield almost same profit, but ETPSO

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TABLE IIOPTIMAL BIDDING STRATEGY AND CORRESPONDING SCHEDULE USING

ETPSO METHODOLOGY FROM22-28 JUNE 2008

Fig. 9. Optimal power bids for the period from 22-28 June 2008 for (a)Daily schedule (b) Weekly Schedule: Results from ETPSO Approach

Fig. 10. Convergence characteristics of ETPSO for daily operating modewith 8 particles

TABLE IIIPERFORMANCE OFS EQUENTIAL AND ETPSO METHODOLOGIES

Mode tp tg P

maxg P

avg Profit E

maxs CPU

(hrs) (hrs) (MW) (MW) ($) (MWh) time

Sequential Methodology

Daily 42 35 270 203.93 575746 5760 1.08Weekly 45 30 290 239.67 577276 8000 533.94

ETPSO Methodology

Daily 41 35 270 199.14 565241 5760 4.38Weekly 42 28 290 235.71 574473 8000 350.74

takes about 60% of the run time taken by the sequential

approach. Whereas for daily scheduling sequential approach

gives faster and better solution.

Fig. 11. Convergence characteristics of ETPSO for weekly operating modewith 50 particles

VII. CONCLUSION

This framework provides a tool that allows a multi-unit

pumped storage hydro generating company to optimally deter-

mine the short-term self-scheduling of its plant. Optimal bid-

ding strategies for pumped-storage power plant in a pool based

competitive electricity market, in which the market clearing

price is insensitive to the bid price is investigated. A model

to account for the nonlinear three-dimensional relationshipbetween the reservoir head, the power output, and the water

discharged is proposed. A multistage-looping optimization and

ETPSO had been carried out to meet the constraints within

each time segment. The methodologies has been tested on a

typical pumped-storage power plant and had proved effective

in finding optimal daily and weekly operation schedules. The

results obtained are reported and comparative study has been

carried out.

REFERENCES

[1] C. D. Galloway and R. J. Ringlee, An Investigation of Pumped StorageScheduling,IEEE Transactions on Power Apparatus and Systems , Vol.PAS-85, No. 5, pp. 459-465, May 1966.

[2] Sebastian de la Torre and Antonio J. Conejo, Optimal Self-Schedulingof a Tidal Power Plant, Journal of Energy Engineering, Vol. 131, No.1, pp. 26-51, April 1, 2005.

[3] L.H. Jeng, Y. Y. Hsu, B. S. Chang, and K. K. Chen, A linearprogramming method for the scheduling of pumped-storage units withoscillatory stability constraints, IEEE Transactions on Power Systems,vol. 11, no. 4, pp. 1705-1710, Nov. 1996.

[4] P. H. Chen and H. C. Chang, Pumped-storage scheduling using agenetic algorithm, in Proc. 3rd IASTED Int. Conf. Power EnergySystems, Nov. 1999, pp. 492-497.

[5] J. Kennedy and R. Eberhart, Particle swarm optimization, in Proc.IEEE Int. Conf. Neural Networks, 1995, vol. 4, pp. 1942-1948.

[6] J. Kennedy and R. Eberhart, A discrete binary version of the particleswarm algorithm, in Proc. Int. Conf. Systems, Man, and Cybernetics ,Piscataway, NJ, 1997, pp. 4104-4109.

[7] Ning Lu, Joe H. Chow, and Alan A. Desrochers, Pumped-StorageHydro-Turbine Bidding Strategies in a Competitive Electricity Market,IEEE Trans on Power Systems, Vol. 19, No. 2, pp. 834-841, May 2004.

[8] R. C. Eberhart and Y. Shi, Particle swarm optimization: Developments,applications and resources, in Proc. Congress on Evolutionary Com-puting, 2001, vol. 1, pp. 81-86.

[9] P. H. Chen, Pumped-storage scheduling using evolutionary particleswarm optimization, IEEE Transactions on Energy Conversion, vol.23, no. 1, pp. 294-301, March. 2008.

[10] David E. Goldberg Genetic Algorithms in Search, Optimization, andMachine LearningAddison-Wesley, 1989.

[11] New York Independent System Operator Website. Available online:http://www.nyiso.com/public/market data/pricing data.jsp

[12] Dept of the Army, US Army Corps of Engineers, Washington, Eval-uating Pumped-Storage Hydropower, Engineers Manual: Engineeringand Design - Hydropower, Chapter.7, No.1110-2-1701, 31 Dec 1985.

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