Reliability-Constrained Unit Commitment using ...

Reliability-Constrained Unit Commitment using Stochastic Mixed-Integer Programming Masood Parvania, Mahmud Fotuhi-Firuzabad, Farrokh Aminifar, Amir Abiri-Jahromi Center of Excellence in Power System Management and Control Electrical Engineering Department, Sharif University of Technology Tehran, Iran Abstract— This paper proposes a stochastic mixed-integer programming (SMIP) model for the reliability-constrained unit commitment (RCUC) problem. The major objective of the paper is to examine both features of accuracy and efficiency of the proposed SMIP model of RCUC. The spinning reserve of generating units is considered as the only available reserve provision resource; however, the proposed formulation can be readily extended to comprise the other kind of reserve facilities. Expected load not served (ELNS) and loss of load probability (LOLP) are accommodated as the reliability constraints. Binding either or both reliability indices ensures the security of operation incorporating the stochastic nature of component outages. In this situation, the spinning reserve requirement is no longer considered explicitly. The Monte Carlo simulation method is used to generate scenarios for the proposed SMIP model. The scenario reduction method is also adopted to reduce computation burden of the proposed method. The IEEE reliability test system (RTS) is employed to numerically analyze the proposed model and the implementation issues are discussed. The simulations are conducted in the single- and multi-period bases and the performance of the model is investigated verses different reliability levels and various numbers of scenarios.

i t NG NT NS

NN i

Number of segments of piecewise linear cost function

SUCit

of generating unit i Startup cost of unit i at time t

MCi

Minimum production cost of unit i

I it

Commitment state of unit i at time t

Pit

Real power generation of unit i at time t

n

Pit

Real power generation of unit i in segment n at time t

Pi min

Lower limit of real generation of unit i

978-1-4244-5721-2/10/$26.00 ©2010 IEEE

Upper limit of real generation of unit i

Ki

Startup cost of unit i

SRit

Scheduled spinning reserve of unit i at time t

RU i

Ramp-up limit of unit i (MW/min)

RDi

Ramp-down limit of unit i (MW/min)

Ti

on

Minimum up time of unit i

Ti off

Minimum down time of unit i

X iton

On time of unit i at time t

X itoff

Off time of unit i at time t

Dt

Total system load at time t

μ

ps

Slope of the segment n of the piecewise linear cost function of unit i at time t Capacity cost of unit i for providing spinning reserve at time t Probability of scenario s

srit , s

Deployed spinning reserve of unit i at time t in

LCt , s

scenario s Load curtailment at time t in scenario s

n it

μitSR

Keywords- reliability-constrained unit commitment (RCUC); spinning reserve; stochastic mixed-iteger programming (SMIP); uncertainty management.

NOMENCLATURE Index of generating units Index of time period Number of generating units Number of scheduling hours Number of scenarios

Pi max

τ

T

λi μi

Spinning reserve market lead time (min) System lead time (h) Failure rate of generating unit i (failures/h) Repair rate of generating unit i (repair/h) I.

INTRODUCTION

Independent system operators (ISOs) of power systems procure the spinning reserve requirement (SRR) to cover unexpected unit outages and/or unforeseen load fluctuations. There are two different alternatives for ISOs to determine the SRR which are deterministic and probabilistic techniques. In the deterministic approach, a predefined amount of SRR is of interest to clear the market of this commodity. This value might be a certain percent of the system hourly load, capacity of the largest committed generating unit, or any sort of their combinations. These criteria, although are very comprehensive, lead to inconsistent operating risk levels. In contrast, the

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period problem are also conducted, and the accuracy and efficiency of the proposed SMIP model is analyzed clearly. The rest of the paper is organized as follows. Section II discusses about applicability of SP in the RCUC problem. Section III presents the problem formulation. In Section IV, the numerical analyses and implementation issues are discussed. Concluding remarks are drawn in Section V.

probabilistic approach satisfies a desirable level a reliability index and the SRR is dealt with implicitly [1]. Several approaches have been proposed to accommodate reliability constraints in the generation scheduling problem; the resulting problem is often referred to as reliability-constrained unit commitment (RCUC). In the relevant literature of RCUC, comprehensive reviews around the proposed methods along with their pros and cons were provided. So, it is sufficient here to just briefly talk about the available techniques. References [1]-[2] are pioneers in instituting reliability criteria for the operating reserve evaluation. Reference [3] was the first to consider how the spinning reserve could be optimized within the UC problem. In [4], a continuous approximation method was proposed to estimate the capacity outage probability table (COPT) explicitly within the reserveconstrained UC as a function of the commitment variables. Reference [5] attacked the RCUC problem based on the priority list (PL) method. Reference [6] considered the uncertainty of the load forecast in addition to the unit unavailability in the RCUC model and simulated annealing (SA) algorithm was used for solving the problem. In [7], a technique has been suggested to balance the cost of providing spinning reserve against its benefits, which are measured in terms of expected energy not supplied (EENS) reduction. In [8]-[10], several models have been proposed in which both the reliability and performance records of the generators and interruptible loads were taken into consideration. Reference [11] proposed an innovative way to explicitly formulate LOLP and ELNS as linear functions of UC variables. Due to nonlinear and combinatorial nature of reliability indices, the third and higher order outages are neglected as well as the outage probabilities are replaced with their associated upper bounds. Accordingly, the reliability indices obtained by the applied formulation would be slightly different from exact values of reliability indices calculated by COPT. However, the numerical analysis illustrated that the induced error lies within an acceptable range. The other important point that should be emphasized here is owing to the computational efficiency of the proposed formulation. To convert the nonlinear expressions associated with products of binary variables or a continuous variable and a binary variable, some mathematical tricks were employed that led to significantly increase the number of decision variables and problem constraints. This characteristic restricts the applicability of the proposed formulation in multiperiod problems and real-scale systems. This approach was extended in [10] to model the interruptible load contribution incorporating the interruption notice time. Stochastic programming (SP) is a framework for the modeling of optimization problems that involve uncertainty [12]. Recently, the application of SP in power system problems is overly increasing [13]-[15]. This paper proposes a SP model for the RCUC problem. As the model contains both continues and binary variables, it is categorized as a stochastic mixedinteger programming (SMIP) problem. The Monte Carlo simulation method is employed to create the scenarios and scenario reduction method is used to reduce computation burden of the problem. The IEEE-RTS is employed for numerical analyses. The proposed model is first solved in a single-period basis. However, a set of simulations for the multi-

II.

APPLICATION OF SP IN THE RCUC PROBLEM

The SP is one of the most efficient mathematical tools to attack a particular class of problems, which are subjected to some kind of uncertainties in the problem data [18]. Uncertainty means that some of the problem data can be represented by a set of random variables. A probabilistic description of random variables is given by probability measures. Specific values that random variables will take are only known after executing random experiments. Thus, in SP, the decision on certain variables has to be made after the stochastic solution is disclosed, whereas others could be made before. In the UC problem, we have to decide about committing generating units here-and-now, with uncertainty about the real state of them after the decision, but knowing the probability distribution of their outages. So, the RCUC problem has been defined in which the random behavior of generating units and the likelihood of their multiple outage states are considered in the UC procedure and determination of the SRR. This aspect is realized with calculating the reliability index or indices. However, it usually results in very complex formulations, relatively huge computational burdens, and very long execution times. The SP can suggest an efficient way to solve this problem. Uncertainties we face in the RCUC problem are random outages of generating units in each hour of the scheduling time span. From the stochastic programming point of view, this problem can be presented as a SP problem with uncertainties only affects constraints [12]. The vector of all system random variables are designated by ξ ( s ) in which s denotes a possible realization of the random variables vector. Each s represents a possible state of generating units, which is usually called a scenario. Each scenario has a probability associated with it. As the first step to construct a SP model, the uncertainties of the problem should be model in an appropriate manner. The two-state continues-time Markov model, depicted in Fig. 1 is used in this paper for the modeling of generating unit outages [1].

Figure 1. Two-state model of generating unit i

The model shown in Fig. 1 consists of two states for operating and failed conditions. The time-dependent probabilities of these two states are calculated as follows [2]: i Pfailed (T ) =

λi λi + μi

−

λi λi + μi

e

− ( λi + μi )T

i i Poperating (T ) = 1 − Pfailed (T )

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(1)

(2)

If the time considered in (1) is assumed to be very large, which is the case in planning and adequacy studies, the second term of (1) vanishes and the first term represents unavailability of generating unit i. This value is known as the unit forced outage rate (FOR) in power system applications. Some of the papers which study the short-term problems such as UC used FOR values for the unit outage probabilities. However, in the UC problem, the system lead time is relatively short such that the failed unit may not be repaired or replaced. Under this assumption, (1) and (2) can be approximated, respectively, by: i Pfailed (3) (T ) = 1 − e− λiT ≅ λiT = ORRi

• Generating units startup cost: SUCit ≥ K i ( I it − I i (t −1) )

∀i, ∀t

• Lead time limit on the spinning reserve: ∀i, ∀t 0 ≤ SRit ≤ RU i ×τ • Minimum up and down time constraints: ⎡⎣ X ion( t −1) − Ti on ⎤⎦ ( I i (t −1) − I it ) ≥ 0 ∀i, ∀t ⎡⎣ X − Ti ⎤⎦ ( I it − I i (t −1) ) ≥ 0 • Ramping up and down constraints: Pit − Pi ( t −1) ≤ ⎡⎣1 − I it (1 − I i ( t −1) ) ⎤⎦ RU i off i ( t −1)

i Poperating (4) (T ) = 1 − ORRi In the multi-period simulation conducted by this paper, we use the ORR values as the unit outage probabilities. However, for the single-period study, the FOR values are used to make the results comparable with those of [11].

III. FORMULATION OF THE PROPOSED SP MODEL The formulation of the proposed SMIP model for the RCUC problem is presented in this section. The formulation of the problem is divided to the objective function, base-case constraints, scenario-based constraints, and reliability constraints, which are presented below, respectively.

off

∀i, ∀t

+ I it (1 − I i (t −1) ) Pi min

∀i, ∀t

Pi ( t −1) − Pit ≤ ⎡⎣1 − I i ( t −1) (1 − I it ) ⎤⎦ RDi + I i (t −1) (1 − I it ) Pi min

∀i, ∀t

(11) (12) (13) (14)

(15)

(16)

Constraint (9) expresses that the spinning reserve provided by each unit is restricted by the unit available capacity. The other constraint on spinning reserve is the unit ramping capability within the spinning reserve market lead time, which is stated in (12). It is essential to emphasize that expressions (6)-(12) are linear. In contrast, expressions (13)-(16) are not in the linear format as they include product of decision variables. This issue does not make any concern since [16] has presented linear representations of these constraints. So, the model is still MIP based.

A. The Objective Function In the objective function of the proposed model, ISO minimizes the summation of the energy and spinning reserve procurement costs, over the scheduling horizon. This statement is mathematically presented as follows: NT NN ⎧ NG ⎛ ⎞ min ∑ ⎨∑ ⎜ SUCit + MCi I it + ∑ μitn Pitn ⎟ t =1 ⎩ i =1 ⎝ n =1 ⎠ (5) NG ⎫ SR + ∑ ( μit SRit ) ⎬ i =1 ⎭ In (5), the first line represents generating units’ startup cost and energy production cost function. Note that the formulation is in the MIP format [10]. The second line denotes the capacity cost of the spinning reserve provided by generating units. The objective function (5) is subjected to a set of constraints to model the system condition in both normal and scenario situations as well as to exhibit physical restrictions of units.

C. Scenario-based Constraints There are other constraints in the proposed model which are subjected to data uncertainty. These constraints have to be satisfied during realization of each scenario. These constraints are, therefore, defined over all system scenarios. They are listed in the following. • Demand-supply power balance in each scenario: NG

NG

∑ ξ ( s ) P + ∑ sr i

i =1

B. Base-Case Constraints Some constraints of the UC problem are related to basecase condition of the system. The most important characteristic about these constraints is that they do not contain uncertain data. These constraints are as follows: • Demand-supply power balance in the base case:

it

it , s

+ LCt , s = Dt

∀t , ∀s

(17)

i =1

• Deployed spinning reserve limit in each scenario: 0 ≤ srit , s ≤ ξiG ( s ) SRit ∀i, ∀t , ∀s

(18)

• Load curtailment limit: 0 ≤ LCt , s ≤ Dt

(19)

∀t , ∀s

∀i, ∀t

(7)

0 ≤ Pitn ≤ Pi n ,max

∀n, ∀i, ∀t

(8)

max

I it − SRit

∀i, ∀t

(9)

Here, ξi ( s ) is a binary number to represent the state of generating unit i in scenario s. If the unit is healthy in scenario s, ξi ( s ) is one, and otherwise it is zero. The active power generation of units is determined in base case such that the demand-supply power balance equation (6) is satisfied. As any scenario s is realized, active power generation variables Pit do not change. Instead, the spinning reserve facilities should appropriately be deployed to compensate the generation shortage. However, in the case of outage of large producers or having inadequate spinning reserve capacity, it is possible to have to interrupt a portion of load. This means that it is impossible to have one hundred percent reliability in the power system. The term LCt , s models such a situation. In each

min

I it

(10)

scenario s, spinning reserve variables srit , s and load curtailment

NG

∑P

it

= Dt

∀t

(6)

i =1

• Generating units power relations and constraints: NNi

Pit = Pi min I it + ∑ Pitn n =1

Pit ≤ Pi

Pit ≥ Pi

∀i, ∀t

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would be a number a bit greater than 0, while the right-hand side would be a number a bit greater than 1. As the LOLI is a binary variable, it would be set to 1 in this case. Inspecting (25) when the load curtailment is 0 in scenario s, we find that the LOLI would be 0 in this case. Thus, the inequality (25) leads to a linear equivalent representation of (23).

variables LCt , s , are utilized such that the scenario demandsupply balance equation (17) is satisfied. The relationship between the scheduled and the deployed reserve variables is expressed in (18). The deployed spinning reserve variables in scenarios are bounded by scheduled ones. Equation (18) also ensures that only generating units which are healthy in scenario s can provide spinning reserve service. Another scenario-based constraint is that the load curtailment in each scenario is limited to the maximum curtailable load in each scheduling hour which is considered here to be the system hourly load.

IV. STUDY RESULTS The effectiveness of the proposed SMIP model is demonstrated here on the IEEE-RTS [17]. The data of the IEEE-RTS are directly extracted from [17]. The capacity cost of spinning reserve provided by generating units is assumed to be 10% of their maximum incremental cost of producing energy. For the sake of comparison of the results with those of [11], fuel costs are extracted from [11]. The random variables vector contains 32 random variables, associated with availability status of 32 generating units. The model was solved using the mixed-integer programming solver CPLEX 11.2.0 under GAMS [18] environment, on a DELL vostro 1500 computer with a 2.2 GHz dual-core processor and 2 GB of RAM.

D. Reliability Constraints

As the load curtailment variables LCt , s does not have any associated costs in the objective function, the optimization process will compensate units’ outages only by utilizing load curtailment. However, in the RCUC problem, the load curtailment in system is limited by bounding the reliability index or indices. The expected load not served (ELNS) and loss of load probability (LOLP), are used in this paper as measures of the system reliability in the UC problem. The bounds on these two indices are expressed in (20) and (21), respectively. By adding (20) and (21) to the problem, the load curtailment variables LCt , s are determined such that these two constraints are satisfied. ELNSt ≤ ELNStmax ∀t (20)

A. Single-Period Problem In the first case study, a single-period scheduling problem with the system total load of 2000 MW is considered. As it was used in [11], instead of ORR, in this study we consider the FOR of units as their failure probability. In this case, the Monte Carlo simulation method generates 27191 unit outage scenarios after one million iterations. The backward reduction method [19] is then used to reduce the number of the generated scenarios. In this study, only ELNS is bounded in the model. Table I shows the results of the proposed model for two different values of reduced scenarios. In each case, the model is run considering 3 different values of ELNSmax. Note that in this table, the ELNS index is not given because it is, as expected, equal to the associated ELNSmax.

LOLPt ≤ LOLPt max ∀t (21) The ELNS represents expected amount of load curtailment in each scheduling hour which can be calculated by: NS

ELNSt = ∑ ( ps × LCt , s ).

(22)

s =1

The LOLP is the probability that the available generation including spinning reserve cannot meet the system load. A set of binary indicator variables, called loss of load indicators (LOLI), are defined over the system scenarios in each scheduling hour as follows: if LCt , s ≠ 0 ⎪⎫ ⎪⎧1, LOLI t , s = ⎨ (23) ⎬ ∀t , ∀s . if LCt , s = 0 ⎪⎭ ⎪⎩0, Using these binary variables, the LOLP can be calculated by:

TABLE I. Number of ELNSmax Scenarios (MW) 80 50 50 5 80 100 50 5

NS

LOLPt = ∑ ( ps × LOLI t , s ).

(24)

s =1

NG

NG

i =1

i =1

Dt NG

≤

≤ LOLI t , s NG

Dt − ∑ ξi ( s ) Pit − ∑ ξ i ( s ) SRit

The Proposed Model COPT Results LOLP Reserve Cost Solution ELNS LOLP (%) (MW) ($/h) time (s) (MW) (%) 41.23 66.23 20220.07 0.130 90.03 44.27 32.83 195.63 20409.16 0.189 57.28 30.74 3.79 644.92 22887.65 0.248 5.39 3.59 41.51 87.9 20251.77 0.408 83.54 39.98 33.35 213.46 20435.22 0.435 53.98 30.20 3.87 655.17 22902.65 0.518 5.27 3.45

It can be seen in Table I that, as the bound on ELNS is reduced, the total cost of the system and the amount of scheduled spinning reserve increase. It means that a more reliable system necessitates more spinning reserve to be procured and the operating cost consequently increases. As indicated in the Abstract of the paper, one of the goals of this paper is to investigate the accuracy of SMIP-based RCUC model. Accordingly, we need to calculate the exact values of reliability indices and compare them with those obtained by reduced scenarios. The last two columns of Table I show ELNS and LOLP calculated by the COPT. As expected, the indices calculated in the proposed stochastic model have some errors because, the scenario reduction feature induces some approximation. The error level varies with ELNSmax. For

The equation (23) is nonlinear and should be represented in a linear fashion to preserve the format of the model as MIP. Equation (23) can be replaced by the following linear inequalities: Dt − ∑ ξ i ( s ) Pit − ∑ ξi ( s ) SRit

RESULTS FOR THE SINGLE-PERIOD SCHEDULING

(25)

i =1 +1 Dt Consider a situation when load curtailment occurred in scenario s. In this case, the left-hand side of the inequality (25) i =1

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the MIP format to possess the maximum efficiency. The number of scenarios was reduced by a reduction technique and the test results on the IEEE RTS with 32 generating showed the efficiency and accuracy of the proposed formulation. In all simulated case studies, the error level induced because of scenario reduction was less than 10% which is quite acceptable in the reliability studies. The execution time was less than 1 s for the one-hour cases and less than 20 s for 24-hour simulations. Numerical studies illustrated that increasing the number of scenarios, although raises the execution time to some extent, leads to a better accuracy. These two conflicting aspects of the model should be appropriately compromised for the specific problem in question.

example, in the cases with 50 scenarios, the errors of ELNS are 12.54% and 7.8%, for ELNSmax=80 MW and ELNSmax=5 MW, respectively. The maximum error level is for the case with ELNSmax=80 MW. It is obvious that with increasing the number of scenarios, the accuracy of the model improves. For the cases with 100 scenarios, the maximum error level is associated with the case with ELNSmax=50 MW which is equal to 7.96%. The solution time of the proposed model is discussed here. The solution time in all simulated cases is less than one second which manifests the efficiency of the proposed model. Comparing the solution times in cases with 50 scenarios with those with 100 scenarios reveals that with increasing the number of scenarios, the solution time goes up. In both cases with 50 and 100 scenarios, the solution time changes inversely with ELNSmax. In other words, the tighter reliability index is considered, the more solution time is expected. The solution time and the accuracy of the results are also compared with those associated with the model proposed in [11]. In order to do this, the model of [11] is implemented in GAMS and run using the same computer used to produce results of Table I. Referring to Table I in [11], the errors of cases with ELNSmax=100 and ELNSmax=5 MW are 30.3 and 32.8%, respectively, which are slightly higher than those of the proposed model. However, reliability data are subject to significant errors as well as they have instinct uncertainty because of stochastic nature of failures. Accordingly, the error levels of even around 30% are acceptable. The solution times associated with the model of [11], but executed on our computers, are 52.98 and 34.21 s, respectively, for the cases with ELNSmax=50 and 5 MW. Comparing these amounts with those reported in Table I exhibits a significant difference in the efficiency of these two models.

[1] [2]

[3]

[4]

[5]

[6]

[7]

B. Multi-Period Scheduling The proposed model is also tested on a 24-hour scheduling case on the IEEE-RTS. The hourly demand data corresponds to a weekday in the summer while the peak load of the day is 2000 MW. In this study, the model presented in Section II is considered as of the outage probability of generating units. The Monte Carlo simulation method generates 811 unit outage scenarios after one million iterations. The backward reduction method [19] is then used to reduce the number of the generated scenarios. The ELNSmax is considered to be 0.3 MW in each scheduling hour. Two different cases are simulated on the model, in which 50 and 100 reduced scenarios are considered, respectively. Table II shows the detailed scheduling results for the case with 50 scenarios. It has to be noted that units which are committed in none of the scheduling hours, are omitted in this Table. The solution times and accuracy of results of the model are so noteworthy. The solution times are 11.57 s and 18.79 s, respectively for the cases with 50 and 100 reduced scenarios. The amounts of scheduled spinning reserve for the system are compared in Table III for two cases.

[8]

[9]

[10]

[11]

[12] [13]

[14]

[15]

V. CONCLUSION This paper considered the reliability constraints along with the UC problem by applying the SP. The probabilistic reliability criteria, i.e., ELNS and LOLP, were bound with predetermined values and other prevailing UC constraints were included as well. The proposed model was developed based on

[16]

[17]

REFERENCES R. Billinton and R. N. Allan, Reliability Evaluation of Power Systems, 2nd ed. New York: Plenum, 1996. L. T. Anstine, et. al, ‘‘Application of probability methods to the determination of spinning reserve requirement for the Pennsylvania-New Jersey-Maryland interconnection,’’ IEEE Trans. Power App. Syst., vol. PAS-82, no. 68, pp. 720---735, Oct. 1963. H. B. Gooi, D. P. Mendes, K. R. W. Bell, and D. S. Kirschen, ‘‘Optimal scheduling of spinning reserve,’’ IEEE Trans. Power Syst., vol. 14, no. 4, pp. 1485---1492, Nov. 1999. D. Chattopadhyay and R. Baldick, ‘‘Unit commitment with probabilistic reserve,’’ in Proc. IEEE Power Eng. Soc. Winter Meeting, New York, Jan. 2002, vol. 1, pp. 280---285. R. Billinton and M. Fotuhi-Firuzabad, ‘‘A basic framework for generating system operating health analysis,’’ IEEE Trans. Power Syst., vol. 9, no. 3, pp. 1610---1617, Aug. 1994. D. N. Simopoulos, S. D. Kavatza, and C. D. Vournas, ‘‘Reliability constrained unit commitment using simulated annealing,’’ IEEE Trans. Power Syst., vol. 21, no. 4, pp. 1699---1706, Nov. 2006. M. A. Ortega-Vazquez and D. S. Kirschen, ‘‘Optimizing the spinning reserve requirements using a cost/benefit analysis,’’ IEEE Trans. Power Syst., vol. 22, no. 1, pp. 24---33, Feb. 2007. J. Bai, H. B. Gooi, L. M. Xia, G. Strbac, and B. Venkatesh, ‘‘A probabilistic reserve market incorporating interruptible load,’’ IEEE Trans. Power Syst., vol. 21, no. 3, pp. 1079---1087, Aug. 2006. J. Bai, H. B. Gooi, and L. M. Xia, ‘‘Probabilistic reserve schedule with demand-side participation,’’ Elect. Power Comput. Syst., vol. 36, no. 2, pp. 138---151, Feb. 2008. F. Aminifar, M. Fotuhi-Firuzabad, and M. Shahidehpour, ‘‘Unit commitment with probabilistic spinning reserve and interruptible load considerations,’’ IEEE Trans. Power Syst., vol. 24, no. 1, pp. 388---397, Feb. 2009. F. Bouffard and F. D. Galiana, ‘‘An electricity market with a probabilistic spinning reserve criterion,’’ IEEE Trans. Power Syst., vol. 19, no. 1, pp. 300---307, Feb. 2004. P. Kall and S. W. Wallace, Stochastic Programming. New York: Wiley, 1994. S. Takriti, J. R. Birge, and E. Long, ‘‘A stochastic model for the unit commitment problem,’’ IEEE Trans. Power Syst., vol. 11, pp. 1497--1508, Aug. 1996. F. Bouffard, F. D. Galiana, and A. J. Conejo, ‘‘Market-clearing with stochastic security-part I: Formulation,’’ IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1818---1826, Nov. 2005. L. Wu, M. Shahidehpour, and T. Li, ‘‘Stochastic security-constrained unit commitment,’’ IEEE Trans. Power Syst., vol. 22, no. 2, pp. 800-811, May 2007. M. Carrion and J. M. Arroyo, “A computationally efficient mixedinteger linear formulation for the thermal unit commitment problem,” IEEE Trans. Power Syst., vol. 21, no. 3, pp. 1371–1378, Aug. 2006. Reliability Test System Task Force, “The IEEE reliability test system1996,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 1010–1020, Aug. 1999.

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TABLE II.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

G4

G7

G8

G20

G21

G22

G23

G24

G25

G26

G27

G28

G29

G30

G31

G32

Type

U76

U76

U76

U76

U155

U155

U400

U400

U50

U50

U50

U50

U50

U50

U155

U155

U350

Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit Pit SRit

34.4 0 15.2 0 15.2 0 15.2 0 15.2 0 15.2 0 15.2 0 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 38 38

15.2 0 15.2 0 15.2 0 15.2 0 15.2 0 15.2 0 15.2 0 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 38 38

15.2 57.9 15.2 0 15.2 0 15.2 0 15.2 0 15.2 0 34.4 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 56 13.9

15.2 0 15.2 0 15.2 0 15.2 0 15.2 0 15.2 0 15.2 42.8 47.6 28.4 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 60.8 15.2 38 38

155 0 155 0 124 31 124 0 124 0 147.2 7.8 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0

155 0 140.4 14.6 147.2 7.8 124 19.5 124 19.5 124 31 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 100 33.8 186.8 0 200 12.5 200 42.9 200 12.5 212.5 30.4 200 42.9 200 0 200 0 200 0 146.8 5.2 126.8 22.2 126.8 22.2 138.5 4.4 126.8 6.9 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 100 58.6 126.8 39.2 200 104.9 266.8 133.2 286.8 113.2 266.8 133.2 274.3 125.7 286.8 113.2 226.8 125.3 206.8 115.1 206.8 115.1 200 86 200 66 200 66 148.3 77.7 100 66 0 0

50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0

50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0

50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0

50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0

50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0

50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0

155 0 124 27.4 124 0 124 0 124 0 124 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0

155 0 139.8 15.2 124 7.8 124 0 124 0 124 7.8 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0 155 0

280 70 280 70 280 70 263.2 86.8 263.2 86.8 280 70 280 70 350 0 350 0 350 0 350 0 350 0 350 0 350 0 350 0 350 0 350 0 350 0 350 0 350 0 350 0 350 0 350 0 350 0

1280 127.9 1200 127.2 1160 116.6 1120 106.3 1120 106.3 1160 116.6 1280 128 1600 132.6 1740 133.8 1900 165.7 1980 206.5 2000 216.9 1980 206.5 2000 216.9 2000 216.9 1940 186.1 1920 175.9 1920 175.9 1860 152 1840 149 1840 149 1800 142.9 1740 133.7 1440 127.9

TABLE III. Hour 50 Scen. 100 Scen. Hour 50 Scen. 100 Scen. Hour 50 Scen. 100 Scen.

G3

Total

Hour 1

Unit

RESULTS FOR THE 24-HOUR UC PROBLEM

1 127.98 130.49 9 133.77 135.83 17 175.94 181.36

SCHEDULED SPINNING RESRVE FOR THE SYSETM 2 127.23 130.42 10 165.7 171.35 18 175.94 181.36

3 116.62 121.80 11 206.48 211.21 19 152.05 153.86

4 106.26 112.75 12 216.86 221.39 20 149 150.86

5 106.26 112.75 13 206.48 211.21 21 149 150.86

6 116.62 121.80 14 216.86 221.39 22 142.91 144.85

7 127.98 130.49 15 216.86 221.39 23 133.77 135.83

[18] R. E. Rosenthal, GAMS: A User’s Guide. Washington, DC: GAMS Development, 2006. [19] J. Dupačová, N. Gröwe-Kuska, and W. Römisch, “Scenario reduction in stochastic programming: An approach using probability metrics,” Math. Programm., vol. A 95, pp. 493–511, 2003.

8 132.65 135.09 16 186.09 191.26 24 127.98 130.49

205

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