Building Generation Supply Curves under Uncertainty

1

Building Generation Supply Curves under Uncertainty in Residual Demand Curves for the Day-Ahead Electricity Market David Berzal, José Ignacio de la Fuente and Tomás Gómez, Member, IEEE

Abstract--The restructuring process that took place in the Spanish electricity market in 1998 has caused the appearance of new procedures and rules, and the implementation of decentralized operation models. Under this new competitive framework, utilities pursue supply strategies to maximize their benefits. This paper proposes a stochastic building model of competitive supply curves under uncertainty, whose objective is to maximize the expected profit of a generation utility in the dayahead market of a deregulated electrical system. The model contemplates the global strategy of the utility, but also it specifies the bid for each generation unit. At the same time the model has a great flexibility characterizing different market conditions. Firstly, the formulation and the restrictions of the iterative optimization problem are described. Afterwards, a brief review of the model algorithm, based on successive optimizations for different scenarios, is showed. Next the building up procedure of the bidding supply curves is explained. Finally, a case study is presented that allows to check the validity of the proposed model. Index Terms--Day-Ahead Market, Supply Curves, Uncertainty in Residual Demand Curves, Stochastic and Iterative Optimization.

I. INTRODUCTION

D

URING the last decade electrical power systems all over the world have experienced a series of regulatory changes that, through the incorporation of various competitive mechanisms, have the objective to improve the economic efficiency of the electric business. In this new framework it is necessary for the power generation utilities to develop new bidding strategies which allow them to be competitive under the new environment. The new objective of the utilities in these competitive markets is maximizing their profits. The determination of the optimal supply strategy of a generation utility in a competitive market depends on several factors, like its size and type of technology and the one of its competitors, marginal price and technical restrictions of the generation units. Therefore it is necessary to develop tools which combine technical restrictions and economical conditions. This work was supported by Unión Fenosa Generación, S.A. D. Berzal, J. I. de la Fuente and T. Gómez are with the Instituto de Investigación Tecnológica, Universidad Pontificia Comillas (UPCo), Madrid, 28015 Spain (e-mail: [email protected]).

Three different scenarios taking care of energy market regulation and size and characteristics of the agents can be defined: monopoly, oligopoly and perfectly competitive market [1]. Nowadays, many electricity markets behave like oligopolies. Thus, the production supplied to the market by a determinate agent affects the clearing price. In order to model this effect, the supply curve of the considered agent facing the residual demand curve of the system is used. The residual demand function is obtained subtracting to the total demand curve the added supply curve of the rest of the competitors [2]. Nevertheless, the residual demand curves are not known, but there is uncertainty in the market conditions, the total demand of the system and the supplies of the competitors. Also, new structures of energy systems divide the power dispatch in several markets attending to the temporary horizon considered. This article is focused on the day-ahead electricity market in the Spanish system. In this wholesale market a set of generation and demand bids are matched. The bids for this market are represented by a pair quantity (MW)-price (c /kWh) and can be simple or complex (with technical or economic extra restrictions) bids [3]. The aim of this paper is to explain a novel approach for building up supply functions for the day-ahead energy market. The model proposed builds a robust supply curve against the uncertainty in the residual demand curves. The proposed method is based on the optimization of a range of scenarios using as decision variables the amount of power supplied by each generation unit of the utility facing the 24-hour residual demand curves of each scenario. The model builds up the supply strategy for a base scenario (the most likely) and for a set of upper and lower pricescenarios. As mentioned before, each scenario consists of a set of 24-hour residual demand curves. After the optimization of each scenario, the feasibility area of the supply curve for the optimization of the following scenario is defined. This is due to the fact that the supply curve should be an increasing curve. There are studies and models for profit optimization of an utility in a day-ahead market that define the complete supply strategy of the utility as a whole [4]. Beside this, the present model also defines the supply curve for individual generation units at each hour.

2 The paper is organized as follows: in section II the formulation and constraints of the optimization problem are explained. Afterwards, in section III, the methodology of the proposed model is showed. In section IV the building up supply function is showed. Finally, a case study with the determination of the optimal supply curve for a fictitious generation utility is analyzed. The model has been successfully implemented in a Pentium III PC, 550 MHz, and has been programmed in GAMS language, using CPLEX 7.0 program as optimizer. II. MODELING OF THE OPTIMIZATION PROBLEM The main objective of the optimization of the supply strategy of a generation utility is the maximization of its profit in the day-ahead energy market. In order to achieve this goal, the strategic decision variables which are handled are the hourly production (MW) of each generation unit and the hourly prices (c /kWh). The real objective function is a quadratic function, because when an oligopolist market is considered, the price in every period of the energy market depends on the total production of the utility. The effect that the supply of a generation utility exerts on the electricity market price is modeled by the residual demand curves. These curves also represent the market conditions and behavior. Therefore, they relate the amount of power supplied by the generating utility to the market price. García et al. explain the calculation and mathematical meaning of these curves [2]. The residual demand curves are different for each utility. García et al. [4] have proposed an interesting linear approximation of the problem through an approach of the income curve. The model proposed in this paper uses an alternative approach. An iterative process with a linear approximation of the objective function around the residual demand curve working point is used, obtaining therefore a resoluble mixed-integer linear optimization problem. After each iteration, it is necessary to calculate the prices and slopes of the hourly residual demand curves around the utility production point fixed by the optimization [1], [5]. The problem solved at each iteration is the maximization of the utility benefit considering simultaneously the 24-hour residual demand curves and the technical constraints of the generation units and the strategic restrictions of the utility. The model contemplates the possibility of optimizing the schedule of the generating units with thermal and hydraulic groups. The special treatment of the hydraulic units is explained in detail in [5] and [6]. The optimization also considers the possibility of modeling in a linear way economic signals like subventions to special generation (coal, solar, wind, etc.), stranded costs (transition to a competitive market costs, etc.), or percentage of recovery of start-up and shutdown costs, and the profits of generation units whose property is shared by several companies [5]. Candiles et al. [1] and Berzal et al. [5] explain in detail the convergence procedure of the iterative optimization process.

III. MODELING OF THE UNCERTAINTY There is a great uncertainty about the behavior and conditions in the electricity markets. Therefore it also exits about residual demand curves. Generation utilities do not know them with accuracy. In order to face this uncertainty, the approach explained in this paper purposes building up the supply curves in a stochastic way through the optimization of a set of scenarios of residual demand curves. The proposed model is based on the optimization of a first scenario, called base scenario, and the later optimization of several upper and lower residual demand curve-scenarios. For the generation of scenarios of residual demand curves used by the model, and the determination of the base scenario, statistical methods like univariate regression, clustering and time series have been used. A great database has been used in the research. This database has data about residual demand curves and explaining variables as demand, market price, runoff-the-river hydropower and nuclear power. Fixing the base scenario is also an important item because his optimization is the only one unconstrained as is explained in section IV. A procedure to determine the base scenario based on explaining variables has been used. IV. PROPOSED METHODOLOGY TO BUILD UP THE SUPPLY CURVE The supply curve of a generating utility and the curves of each unit should be increasing curves. For this reason the points obtained when optimizing each one of the scenarios must fulfill certain restrictions. The optimization of the base scenario is unconstrained, that is, it does not have restrictions for the optimal values of power and price, except of the own technical and strategic restrictions of the optimization problem. The optimal point calculated in the optimization of the base scenario defines feasible regions for the later optimizations. Feasible regions are necessary in order to get an increasing curve. The feasible region in the optimization of each scenario determines the minimum and maximum levels of power variables for the optimization of the next scenario. A. Upper Scenarios In the case of the first upper scenario, the feasible region is defined by the optimal point of the base scenario optimization, as shown in Fig. 1: price ( Pta / kWh )

Feasible Region

pbase

RDCu 1 RDCbase

qbase

qumax 1

Residual Demand ( MW )

Figure 1. Feasible Region of the First Upper Scenario.

The value of the power of the whole utility in the optimization of the first upper scenario should be in the

3 feasible region. This value is limited between qbase and the value of power in the first upper scenario with a price pbase. Feasible region is updated in the successive upper scenarios optimization according to the optimal point of the previous optimization. The new feasible region is represented in Fig. 2: price ( Pta / kWh )

price ( Pta / kWh )

Infeasibility pbase

RDCu 2 RDCu 1 RDCbase

Infeasibility

Feasible Region

RDCl 1 RDCl 2

pu 1

qbase


Figure 5. Scenarios Infeasibilities.

pbase


qbase

qu1

qumax 2


Figure 2. Feasible Region of the Successive Upper Scenarios.

B. Lower Scenarios As previously described, the resulting optimal point of the residual demand curves of the base scenario defines a feasible zone for the optimization of the first lower scenario. Maximum and minimum power that can be supplied by the utility, each hour, are fixed by the optimal point (qbase and pbase) of the base scenario (see Fig. 3). price ( Pta / kWh )

pbase

Feasible Region

RDCbase RDCl 1

qlmin 1

qbase


Figure 3. Feasible Region of the First Lower Scenario.

In order to obtain an increasing supply function, it is necessary to update the feasible region before the optimization of each lower scenario, as shown in Fig. 4. price ( Pta / kWh )

pbase pl 1

RDCbase

Feasible Region

qlmin 2

RDCl 1 RDCl 2 ql 1

qbase


Figure 4. Feasible Region of the Successive Lower Scenarios.

C. Scenarios Infeasibilities The use of feasible regions forces the residual demand curves of the different scenarios not to intersect. The model optimizes a base scenario and some upper and lower scenarios and they can not intersect. This constraint is necessary to avoid infeasibilities for optimizations of the upper and lower scenarios, in order to know which should be the next scenario to optimize. These infeasibilities are represented in Fig. 5. Using this restriction is not a problem because the great uncertainty that exits about the residual demand curves.

D. Building Up the Supply Curve Pairs quantity (MW)-price (c /kWh) obtained with the optimization of all the scenarios allows the construction of a increasing supply curve for each hour for the utility and each individual unit. Nevertheless any supply function which crosses these points maximizes the utility profit for the considered scenarios [2]. The model presented in this paper proposes the construction of the final curve to be submitted to the market, joining the optimal points in a increasing way by a integer number N of equal segments. These segments are placed in order to minimize the average quadratic error between the built curve and the piece-wise linear curve which links the optimal points (see Fig. 6). The model also elaborates the 24 individual supply curves of each generation unit in the same way that the whole utility curve. This bid is defined by the value of power variables of thermal and hydraulic units in the optimal points. The construction of the segments between optimal points of the supply curves of the units is made assigning the corresponding power in an efficient way. In case a group starts up between two consecutive points, the minimum power of this group is bid in one indivisible segment and the rest in N-1 segments (see Fig. 7). The power dispatched of any group between the origin and the optimal point for the lowest scenario should be bid in an efficient way. The proposed model bids this amount at the variable cost of the group, if it is lower than the price of the optimal point of the lowest scenario. If not, the bidding price should be this lowest-scenario price (see Fig. 7). The model complies with the condition that the supply function of the utility is the sum of the individual unit supply curves. Fig. 7 illustrates the procedure that follows the model to build up the supply curve of the generation utility for a certain period. price ( Pta / kWh )

N Segments

pu 1

Pice-wise linear curve

pbase

RDCu

1

RDCbase qbase

qu1


Figure 6. Building Up the N Segments between Optimal Points.

4 price ( Pta / kWh )

Minimum Power

N segments

Base Scenario

Optimal Points

Base Scenario

Optimal Points


pl 2

Price (c /kWh)

pbase pl 1

Lowest-Scenario Price

Price (c /kWh)

pu 2 pu 1

RDCl 1 RDCl 2

Variable Cost ql 1 = ql 2 qbase

Origin

qu 1

qu 2

Residual Demand (MW)

(b) Base Scenario

V. APPLICATION CASE Next a study case in which the production of a fictitious company has been optimized is presented. The generating system is made up of 15 thermal units (three nuclear units, four coal units and three fuel units) and three hydraulic units. Generators input data are shown in the Appendix (Tables II and III). 17 scenarios have been considered: one base scenario, ten upper scenarios and six lower scenarios. Each scenario is made up of 24 residual demand curves, one for every hourly period. Fig. 8 shows the residual demand curves of all the scenarios for off-peak, plateau and peak hours.

Optimal Points

Price (c /kWh)

Figure 7. Building Up Methodology.


(c) Figure 9. Optimal Points and Residual Demand Curves at hour 3 (a), 15 (b) and 21 (c).

The resultant supply function has been built up in this study case with five segments between every two consecutive optimal points. Fig. 10 represents the final supply curves of the utility matched with the residual demand curves of the base scenario at hours 3, 15 and 21.


Price (c /kWh)

Price (c /kWh)

Price (c /kWh)

Base Scenario

Price (c /kWh)

Base Scenario


(a)


Residual Demand (MW) Residual Demand (MW)

(a)

(b)

(a)

(b)

Price (c /kWh)

Price (c /kWh)

Base Scenario



(c) Figure 8. Residual Demand Curves at hours 3 (a), 15 (b) and 21 (c).

The optimization process of all the scenarios is shown in Fig. 9 where are represented the calculated optimal points for the studied hours.


(c) Figure 10. Supply Curve and Residual Demand Curve of Base Scenario at hour 3 (a), 15 (b) and 21 (c).

The model also builds up the supply function for each unit. VI. CONCLUSIONS The paper has presented a methodology of building supply competitive curves for the day-ahead market of energy. The proposed model allows the profit maximization of a generation utility from a set of 24-hour residual demand scenarios that represent the conditions of the market. We assume that the utility is able to generate this set. This paper has shown a stochastic building model of competitive supply curves that are a robust solution to face the uncertainty in the market behavior. For each scenario, profit is maximized deterministically. The final result is an optimal, global and competitive function for the generation utility,

5 specifying the bid for each unit. A study case which has been presented has allowed to check the validity of the proposed model.

power of thermal and hydraulic groups in the new optimization are upper limits:

ql 1jt ≤ qbase jt

VII. APPENDIX A. Mathematical Formulation 1) Upper Scenarios In the case of the first upper scenario the feasible region is defined by (1) and (2) for the whole power supplied by the company at each period: 1

qu t ≥ qbaset

∀t

(1)

≥ pbaset ∀t (2) Maximum power of the utility qumax is defined by the power value for the first upper scenario which corresponds to pbase (see Fig. 1): qu t1 ≤ qumax 1 = RDCu1 ( pbaset ) ∀t (3) pu t1

As the supply curve of the whole utility, the functions of each one of the thermal and hydraulic units of the generating system also should be increasing. Equations (4) and (5) define the inferior limits for the values of the variables of amount of each one of the thermal and hydraulic groups of the generating system: 1

qu jt ≥ qbase jt

∀j ∀t

(4)

1 qu ht ≥ qbaseht ∀h ∀t (5) Minimum levels of price and power are updated in the successive upper scenarios optimization according to the feasible region that forces the supply curve to be increasing:

quti ≥ qu ti −1 ≥

i −1 pu t

∀t

(6)

∀t

(7)

Optimal price in the previous scenario determines the maximum power value of the utility, as shown in (8) (see Fig. 2):

(

)

quti ≤ qumax i = RDCui pu i −1 ∀t t

t

(8)

New minimum power constraints of thermal and hydraulic groups also update when optimizing each one of the upper price-scenarios:

qu ijt ≥ qu ijt−1

∀j ∀t

(9)

i i −1 qu ht ≥ qu ht

∀h ∀t

(10)

2) Lower Scenarios Maximum power that can be supplied by the utility and maximum price to which it can be supplied for the first lower scenario are defined in (11) and (12):

qlt1 plt1

≤ qbaset

∀t

(11)

≤ pbaset ∀t (12) Minimum power of the utility is qlmin defined by the power value for the first lower scenario which corresponds to pbase (see Fig. 3):

(

)

qlmin1 = RDCl1 pbaset ∀t (13) t In the case of lower scenarios, the limits for quantity of qlt1

≥

(14)

≤ qbase ht ∀h ∀t (15) Maximum levels of price and power in the successive upper scenarios optimization are:

qlti ≤ qlti −1

∀t

(16)

plti −1

≤ ∀t (17) Optimal price in the previous scenario determines the minimum power value of the utility, as shown in (18) (see Fig. 4): qlti ≥ qlmin i = RDCli pl i −1 ∀t (18) plti

( )

t

t

Equations (19) and (20) show the variation between lower consecutive scenarios of the maximum levels for the variables of thermal and hydraulic groups power.

ql ijt ≤ ql ijt−1

t

i pu t

∀j ∀t

1 ql ht

∀j ∀t

(19)

ql hti ≤ ql hti −1 ∀h ∀t (20) 3) Building Up the Supply Curve Equations (21) and (22) define the power of each one of the segments and its supply price in the case of optimal points of two upper scenarios: qui −qui −1 qtn = t N t ∀n ∀t (21)

ptn =

puti − puti −1 N

∀n ∀t

(22)

For two consecutive lower scenarios, the power of each one of the segments and its supply price are: ql i −1 −ql i n qt = t N t ∀n ∀t (23) n

pt =

plti −1 − plti N

∀n ∀t

(24)

In order to define the segments between the adjacent points (upper and lower) to the optimal point of the base scenario, it is considered qbase and pbase as qu, pu, ql and pl when i=0.

RDCbase RDCui

TABLE I INDEXES, PARAMETERS AND VARIABLES Definition Index of the scenarios Index of the period (24 hours) Index of thermal groups Index of hydraulic groups Index of the number for segments between optimal points Residual demand curve of the base scenario Residual demand curve of the upper scenario i

RDCli

Residual demand curve of the lower scenario i

qbaset

Power generated by the utility at the period t in the base scenario Bid price of the utility at the period t in the base scenario

Notation

I T J H N

pbaset i

qu t

i

pu t

qumaxti

Power generated by the utility at the period t in the upper scenario i Bid price of the utility at the period t in the upper scenario i Maximum power generated by the utility at the period t in the upper scenario i

6 Power generated by the utility at the period t in the lower scenario i Bid price of the utility at the period t in the lower scenario i

qlti i

pl t

Minimum power generated by the utility at the period t in the lower scenario i Power generated by the thermal unit j at the period t in the base scenario Power generated by the hydraulic unit h at the period t in the base scenario Power generated by the thermal unit j at the period t in the upper scenario i Power generated by the hydraulic unit j at the period t in the upper scenario i Power generated by the thermal unit j at the period t in the lower scenario i Power generated by the hydraulic unit j at the period t in the lower scenario i Power generated by the utility at the period t in the segment n Bid price of the utility at the period t in the segment n

qlminti qbase jt qbaseht i

qu jt i

qu ht i ql jt i

ql ht n

qt

n

pt

P P E E

Maximum rated capacities of units

u l a b c v

Upper ramp rate limits of thermal units Lower ramp rate limits of thermal units

[2]

[3]

[4]

[5]

[6]

[7]

Minimum rated capacities of units Maximum energy limits of hydraulic units

O. Candiles, J.I. de la Fuente, T. Gómez, J. Contreras, “Bidding Strategies in Day-Ahead Electricity Markets”, in Proc. Distributech DA/DSM Europe & Power Delivery Europe Congress, Madrid, Spain, 1999. J. García, J. Barquín, J. Román, “Building Supply Functions under Uncertainty for a Day-Ahead Electricity Market”, in Proc. 6th Probabilistic Methods Applied to Power Systems Conference, Madeira, Portugal, Septiembre 2000. Compañía Operadora del Mercado Español de Electricidad, “Reglas de Funcionamiento del Mercado de Producción de Energía Eléctrica”, Marzo 1998. Available: http://www.omel.es. J. García, J. Román, J. Barquín, A. Gonzalez, “Strategic Bidding in Deregulated Power Systems”, in Proc. 13th Power Systems Computation Conference, Trondheim, Noruega, June-July 1999. D. Berzal, J. I. de la Fuente, T. Gómez, “Elaboración de Estrategias Competitivas de Oferta para el Mercado Diario de Energía”, presented at the VII Jornadas Hispano Lusas de Ingeniería Eléctrica, Madrid, Spain, 2001. J. Bushnell, “Water and Power: Hydroelectric Resources in the Era of Competition in the Western US”, in Proc. Power Conference on Electricity Restructuring, University of California, Energy Institute, 1998. M. Ilic-Spong, J. Christensen, K.L. Eichorn, “Secondary Voltage Control Using Pilot Point Information”, in Proc. 1987 Power Industry Computer Applications Conference, pp 128-135.

Minimum energy limits of hydraulic units

X. BIOGRAPHIES David Berzal was born in Madrid, Spain, in 1975. He received the degree of Electrical Engineer in 1999, from Universidad Pontificia Comillas (UPCo), Madrid, Spain. At present he is a research fellow at the Instituto de Investigación Tecnológica (IIT-UPCo) and a Ph.D. student. His areas of interest include operations, planning and economy of electric energy systems and application of operations research in electric energy markets.

Start-up costs of thermal units Fixed costs of thermal units Variable costs of thermal units Water value of hydraulic units TABLE II THERMAL UNITS DATA

UNIT 1 UNIT 2 UNIT 3 UNIT 4 UNIT 5 UNIT 6 UNIT 7 UNIT 8 UNIT 9 UNIT 10 UNIT 11 UNIT 12 UNIT 13 UNIT 14 UNIT 15

IX. REFERENCES [1]

u

P

P

MW

MW

1000.0 1000.0 600.0 500.0 500.0 400.0 350.0 350.0 250.0 125.0 50.0 350.0 300.0 300.0 100.0

l

a

b

MW/h MW/h

300.0 300.0 100.0 100.0 250.0 200.0 175.0 150.0 130.0 75.0 30.0 100.0 100.0 60.0 40.0

200.0 200.0 150.0 150.0 130.0 100.0 100.0 90.0 85.0 50.0 15.0 130.0 120.0 120.0 35.0

200.0 200.0 150.0 150.0 130.0 100.0 100.0 90.0 85.0 50.0 15.0 130.0 120.0 120.0 35.0

c /h

0.000 0.000 0.000 0.000 11000.000 12250.000 9000.000 15500.000 4500.000 17000.000 7500.000 2200.000 5000.000 5500.000 5500.000

0.000 0.000 0.000 0.000 1400.000 700.000 2250.000 440.000 500.000 150.000 660.000 1000.000 710.000 450.000 1250.000

c /kW h 0.400 0.350 0.400 0.750 1.500 2.250 1.600 1.400 1.800 3.000 3.100 2.500 5.800 4.200 3.300

TABLE III HYDRAULIC UNITS DATA

UNIT 1 UNIT 2 UNIT 3

v

P

P

E

E

MW

MW

MWh

MWh

300.0 150.0 135.0

100.0 25.0 15.0

6000.0 2200.0 1500.0

0.0 0.0 0.0

c /kW h 1.350 3.880 0.700

VIII. ACKNOWLEDGMENT The authors gratefully acknowledge the contributions of the staff at Centro de Gestión de la Energía of Unión Fenosa Generación, S.A. for their work and collaboration in this research.

José Ignacio de la Fuente was born in Soria, Spain, in 1967. He obtained both his B.S. and Ph.D. degrees in Electrical Engineering from Universidad Pontificia Comillas (UPCo), Madrid, in 1990 and 1997 respectively. Currently, he is a researcher at the Instituto de Investigación Tecnológica (IIT-UPCo). He also teaches Power Systems at the Engineering School ICAI, in the same university. His interest areas include power systems control and optimization, ancillary services procurement and remuneration, energy bidding strategies and power quality issues. Tomás Gómez San Román obtained the Degree of Doctor Ingeniero Industrial from the Universidad Politécnica, Madrid, in 1989, and the Degree of Ingeniero Industrial in Electrical Engineering from the Universidad Pontificia Comillas (UPCo), Madrid, in 1982. He joined Instituto de Investigación Tecnológica (IIT-UPCo) in 1984 where he has been Director from 1994 to 2000. Currently he is the Vice Chancellor of Research, Development and Innovation at UPCo. Dr. Gómez has a large experience in industry joint research projects in the field of Electric Energy Systems in collaboration with Spanish, Latinoamerican and European utilities. His areas of interest are operation and planning of transmission and distribution electrical systems, power quality assessment and regulation, and economic and regulatory issues in the electrical power sector.