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Nima Amjady, Member, IEEE, Davood Farrokhzad, and Mohammad Modarres ... D. Farrokhzad and M. Modarres are with the Department of Industrial En-.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2003

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Optimal Reliable Operation of Hydrothermal Power Systems With Random Unit Outages Nima Amjady, Member, IEEE, Davood Farrokhzad, and Mohammad Modarres

Abstract—A new model for long-term operation of hydrothermal power systems is introduced and a method for obtaining an optimal solution is also developed. We assume that reservoir inflows and energy demand are stochastic and all units are exposed to random outages. The objective is to minimize the total cost of the system as well as the expected interruption cost of energy (EIC) during a given planning horizon. This goal is reached through determination of hydroplant discharges, thermal units energy output, and the system reliability level simultaneously. In fact, we integrate long-term hydrothermal system operation planning and system reliability determination in a unified model. Since the resulting model is a large-scale stochastic nonlinear programming, an algorithm is especially developed to solve it. This algorithm that includes decomposition technique, Lagrangian relaxation, and nonlinear and dynamic programming finds an optimal solution within three stages. To test the method, it is implemented for the Khuzestan power system in Iran and the results are analyzed. Index Terms—Decomposition, hydrothermal, interruption cost, optimization, outage, reliability.

I. INTRODUCTION

O

PTIMIZATION of resources in hydrothermal power systems has been the subject of intensive research for more than four decades [1], [2]. Due to the complex nature of the hydrothermal coordination problem, which includes elements of stochastic, dynamic, nonlinear, and nonconvex programming, many simplifying assumptions and approximations are adopted to make the models more tractable but clearly less realistic. However, with the achievements in computational technology as well as the emergence of newly developed algorithms, the trend is toward improving the assumptions and considering further aspects of actual system operations. When long-term optimization of hydrothermal resources is considered, stochasticity of inflows, demand, and unit outages are so significant that deterministic treatment of these parameters is not realistic at all. Thus, one should be concerned about the high chance of demand satisfaction. Therefore, the reliability of a generating system is an indispensable component of the modeling process. Although in most long-term models, inflows are treated as random parameters but demand for energy is assumed to be deterministic [3]–[5]. In some studies, even Manuscript received March 19, 2002; revised July 1, 2002. This work was supported by the Tavanir Company. N. Amjady is with the Department of Engineering, Semnan University, Semnan 35195–363, Iran (e-mail: [email protected]). D. Farrokhzad and M. Modarres are with the Department of Industrial Engineering, Sharif University of Technology, Tehran 11365-9363, Iran (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TPWRS.2002.807070

though demand is considered as stochastic but in the process of calculating energy shortage, the supply is compared with the expected value of demand or its expected value as a “load duration curve” [6], [7]. In other words, they are practically dealing with deterministic demand. In [8], a long-term model for the operation of a hydrothermal power system with stochastic reservoir inflows and normally distributed energy demand with variable variance is investigated. In this paper, we extend the results of that model and consider the case in which both hydro and thermal units are exposed to random outages. The objective is to optimize long-term operation and determine the reliability level of this system simultaneously. In previous studies, the optimization of power system operations and determination of optimal level of system reliability are considered separately [9], while we incorporate both in a single model. Therefore, what makes this research distinguished from the previous ones is integrating the following assumptions in a unified model: a) inflow to reservoirs is assumed to be stochastic; b) demand for energy is assumed to be stochastic with variable variance; c) units are allowed to have random failures; d) reliability is incorporated in the model and its level is not given in advance as input data, but it is determined such that the total cost is minimized. Since our model turns out to be large nonlinear stochastic programming, the classical methods are not capable of solving it. Therefore, a new method is developed especially for this model in which the decomposition technique, Lagrangian relaxation, dynamic programming, and nonlinear algorithms are applied. To test the method, it is implemented for a Khuzestan power system, which has the largest hydrothermal system in Iran. The results are compared with the existing planning methods. There are a great number of published materials in the field of hydrothermal power systems as well as reliability evaluation. Due to space limitation, we only mention some of them that have similarity with ours although neither our model nor our method is the same as theirs. Reference [1] reported a comprehensive bibliography in the area of reservoir management and operations, which includes a section for models incorporating reliability. In [10], a method for reliability evaluation of limited energy hydrothermal power systems, based on the use of hydro units for peak shaving the load duration curve is presented, but no optimization for reliability level or optimal generation of hydro and thermal units is made. In [11] and [12], reliability indicators are calculated for a hydrothermal power system by obtaining an

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effective load duration curve, where a given operation policy for hydro units is assumed. References [13] and [14] have given a general model for energy-limited units based on operational consideration but reliability levels are not considered as decision variables. In [9], a method for transformation of a hydrothermal power system initial load curve into a thermal one for reliability evaluation is introduced but the model does not allocate the total hydro energy to each time period. Although reliability is explicitly considered as a constraint in some models, the objective function does not include interruption cost explicitly [15]. In [16], a new concept of comprehensive satisfaction degree by using multiobjective fuzzy dynamic programming, with different weights of economy and reliability, is introduced. Reference [17] considers a multiarea power system for which reliability of each area measured by loss-of-load probability (LOLP) is included in the model as a constraint. In their model, the target value for loss-of-load probability (LOLP) [called ] is an input parameter for short-term scheduling. Due to the complexity of analytical methods, many analyze hydrothermal power systems by applying simulation techniques. Reference [18] applied stochastic sequential simulation to assess hydrothermal system reliability, but no optimization is followed in their proposed algorithm. Reference [19] considers the sum of operation and interruption costs and also a criterion for committed capacity reserve planning aimed at minimizing total cost in short-term planning. Amjady [22] applied a neural network technique for reliability analysis, without giving any consideration to the problem of hydrothermal coordination. The remainder of this paper is organized in the following way. We present the statement of the problem and its assumptions as well as the mathematical model in Section II. In Section III, our approach to the determination of reliability is discussed. The main theorem regarding integration of long-range planning and reliability is also presented in Section III. Then, in Section IV, the method as well as the details of the algorithm are discussed. Finally, we test the proposed method for a real case, a Khuzestan power system, and show the results in Section V. II. PROBLEM FORMULATION Consider a power system, which consists of hydroplants thermal units. Each unit has a given forced outage rate and (FOR). The inflow of water is a random variable and demand for energy is assumed to be stochastic. The objective is to determine discharge of the hydroplants as well as energy output of the thermal units in order to minimize the total expected cost of the system during a horizon of periods. Total cost of the system comprises of three parts as follows. a) Cost of energy generation by thermal units of the system. Power generation by hydroplants is almost expense free (except the fixed cost, which is independent of the output level), thus no cost associated with hydro power generation is considered in the objective function. However, it is also possible to consider price curves or bid history curves instead of cost curves, in which case the objective function will include them for thermal and hydroplants. b) Terminal cost of reservoirs associated with the level of water at the end of planning horizon.

c) The expected interruption cost of energy (EIC), which will be discussed in more detail later. A. Outage Probability Table The model describing the random outages of a group of units is usually expressed in terms of discrete probability functions and more specifically in the form of a table, called capacity outage probability table (COPT). It gives the various states of the system regarding possible combination of conditions of units. It is a common practice to model each unit by a two-state random variable. In fact, each row of COPT represents a subset of units that are down. The corresponding probability of each row is constructed by using a discrete convolution of probability functions of individual units [20]. The available capacity of each state of COPT is obtained by subtracting the capacity of down units of that state from the total generation capacity. B. Notation Consider this notation for the following equations: the planning horizon, number of periods; number of hours within period ; number of thermal units; number of hydroplants; output of thermal unit in period in megawatt hours; water released from hydroplant in period in million cubic meter (MCM); output of hydroplant in period in megawatt hours; water content of reservoir in the beginning of period in MCM; generating cost of thermal unit as a function of its output in U.S. dollars; terminal cost of reservoir as a function of its water content in U.S. dollars; inflow of water to reservoir in period in MCM; minimum and maximum allowable value for ; minimum and maximum allowable value for ; minimum and maximum allowable value for ; minimum and maximum allowable value for ; water spilled from reservoir in period in MCM; energy demand in period in megawatt hours; mean of energy demand in period ; standard deviation of energy demand in period ; distribution function of demand ; available capacity of the th row of capacity outage probability table (COPT) constructed using units operating in period in megawatts; Probability of the th row of COPT; energy shortage in period , for the th row of COPT, ; that is, interrupted energy assessment rate in period in U.S. dollars per megawatt hour; the level of reliability for period ; set of indices corresponding to the rows of COPT for period .

AMJADY et al.: OPERATION OF HYDROTHERMAL POWER SYSTEMS WITH RANDOM UNIT OUTAGES

C. Model The problem is represented by a mathematical model as follows:

(1)

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period. In other words, if the system is facing only alternative of the COPT, then with probability of we have . Therefore, using the concept of conditional probability and by considering all states of the COPT, the demand is satisfied if the following inequality, which is the same as constraint (7) holds

These stochastic constraints are equivalent to the following deterministic constraints: (8)

(2)

is the probability of the th row of COPT and where is the inverse distribution function of demand. For a given , the in period is calculated expected energy not supplied as follows:

(3) (4) (5) (6) (9)

(7) Note: 1) The first and second terms of objective function (in the first bracket) are the operating and terminal cost, respectively, while its last term indicates that the “EIC” will be discussed with more detail in the next section. 2) When price curves are used instead of generation cost term in the objective curves, the function should be replaced by

The cost of energy shortage is evaluated by introducing the concept of interrupted energy assessment rate (IEAR). As [21] shows, IEAR is an important factor in generating system reliability assessment and is quite stable for a given system, (i.e., it generally does not vary significantly when the system load level and other factors change). Consequently, for different load levels, EIC can be obtained by multiplying IEAR by EENS. Therefore, (10) which is, in fact, the second term of the objective function. E. Terminal Cost Curve

where PCT and PCH are price curves of thermal and hydroplants with respect to energy generation, respectively. D. EIC When the demand for energy is treated as deterministic quantity directly or indirectly, as is the case in some previously mentioned studies, then deterministic energy balance equation can relate demand and supply. However, due to the nature of energy demand as well as the generating unit’s failure, this assumption is not realistic. Therefore, we assume that demand for energy is stochastic and normally distributed. Furthermore, each unit is exposed to random outages. In our model, the demand is satisfied according to chance constraints (7). In other words, in period , the demand for energy is not always satisfied, but with probability of . Constraint (7) is derived as follows: In period , demand is satisfied if the total generation is not less than the demand in that

Our experience shows that the terminal cost function versus water content of reservoir is a convex function as shown in Fig. 1. The lower and upper limits and the desired amount of water content of the reservoir at the end of the planning horizon , and , are shown as respectively. F. Energy Generation and Water Inflow Model The generating cost of a thermal unit is approximated by a quadratic function with respect to its energy output , and is modeled as follows, within lower and upper bounds of :

where and are constant parameters. The energy output of a hydroplant is modeled as follows:

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where is the efficiency of the plant; and are average head in period . It is assumed that water inflow to a reservoir in period is independent of inflow to other ones but it depends on its inflow . More specifically, water inflow to a reservoir in period follows a Markov Chain pattern and is modeled as follows:

where and which represents actual inflow to reservoir is a lognormal random variable with and parameter . Furthermore, is a random mean ; is variable generated from normal distribution [3], [4]. the correlation coefficient between periods and

subject to constraints (2)–(6) and (8), where the expectation is taken with respect to random reservoir inflows and random unit failures. By taking derivatives from both terms of the objective function (1) with respect to and considering the fact that for optimal solution is neither zero nor one, it is implied that, for optimal solution

(14) using chain rule, we have

III. OPTIMALITY OF RELIABILITY LEVEL It is a common practice to select a reliability level for each period, as an input parameter of the system. However, in order to minimize the total cost of the system, we determine this important factor. The operation cost is clearly an increasing while EIC is a decreasing function of it. Thus, function of the total cost assumes its minimum value at a specific level, say . In fact, in this paper is considered a decision variable which, in effect, determines the reliability of the system. It is obvious that # 0 and # 1. Theorem 1: Let demand be normally distributed with mean and standard deviation of . If for a given feasible solution the following set of equations hold, then the solution is optimal

and applying the concept of shadow prices, we know that

also for normal distribution, observing the fact that

then

On the other hand, taking derivatives from (9) results in (11) where

By considering (12), which is the result of the integral inside the above bracket for normal distribution, this is equivalent to

and by substitution of the above terms in (14), the proof is complete. IV. ALGORITHM

(12) is the optimal Lagrange multiplier for period and where is standard normal CDF. Summary of Proof: By our notation, the total cost of the , where is objective function (1) is defined by (10) and

(13)

Obtaining an optimal solution is quite complicated because the model is a large nonlinear stochastic programming problem. Therefore, we propose an algorithm on the basis of Lagrangian relaxation and decomposition technique to solve the problem. A. Framework The problem is solved hierarchically in three stages, moving forward from first to third stage and then backward to the first stage. Stage 1: For a given level of , we have a nonlinear programming problem with the objective function (13) subject to constraints (2)–(6) and (8).

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Stage 2: By applying Lagrangian relaxation (LR) for constraint (8), the corresponding dual problem is

(15) subject to constraints (2)–(6). is a linear combination of Stage 3: Note that and for all hydro and thermal units, such that the coefficient of each generation level for hydro and thermal units is either one or zero, depending on whether the unit is up or down in the th state of COPT. Therefore, the problem of stage two is separable in terms of hydro and thermal unit generations. The is thermal subproblem for

Minimize (16) ; subject to where is a subset of , consisting of the rows of COPT is present. But note that is just for which unit (1-FOR ), where FOR is the FOR of the thermal unit and

(17) The hydro subproblem for

is

(18) is a subset of subject to constraints (2) and (4)–(6), where ; consisting of the rows of COPT for which plant is up; also is just (1-FOR ), where FOR is the FOR note that of the th hydroplant. Assuming that plant failure and reservoir inflows are independent, it follows that

plant up plant down

(19)

where the expectation is taken with respect to reservoir inflows. It is also assumed that when hydroplants are in down state, bypass gates are used to discharge the required water release. When price curves are used, the hydro subproblem will have in its objective additional terms with respect function. The expected value of to plant failure is calculated similarly to the calculation of with respect to plant failure. expected value of , The solution of thermal subproblem for period , say is found by equating the derivative of to zero and comparing the result with the lower and , upper bounds. Then the optimal solution is

Fig. 1.

Terminal cost function of reservoir.

, . Note that when price curves are used instead is used instead of GC of generation cost curves, PCT in the calculations from before. Dynamic programming is applied to solve the hydro subproblem, starting from the terminal cost function as illustrated . in Fig. 1 and moving backward to period After obtaining hydro and thermal subproblem optimal solutions, they are coordinated by updating Lagrange multipliers. This can be done by applying any of existing methods, such as a subgradient or variable metric method. As a matter of fact, we go back to stage two and solve the updated problem. As long as the convergence is not reached, moving between the second and third stages continues. When convergence is reached, we should follow a feasibility phase since solutions of LR technique are generally infeasible in the primal problem. In order to make solutions feasible, we should manipulate them to satisfy constraints (8) since other constraints are already satisfied in subproblems. To do this, we should note that only thermal unit generation levels could be changed after convergence of LR technique since hydroplant generations in different intervals are not independent of each other. Among thermal units that have not reached their upper bounds of generation, we increase the generation of the thermal unit, which has the lowest difference with its upper bound. If by increasing the generation of this unit up to its upper bound and demand is not satisfied, we proceed to the next unit with the lowest difference with its upper bound and so on. If the tolerance used for convergence of LR technique were chosen sufficiently small, increasing generation of the first thermal unit will usually make the solution feasible. Generation levels obtained after feasibility phase are used to construct COPT and then optimality is tested according to Theorem 1. If optimality conditions are not satisfied, moving between stages one and two continues. Thus, coordination between the original problem (1) and the main problem (13) is carried out through an optimality equation (11). For a large-scale system, the COPT tends to be very large, and hence, a dimensionality problem may occur. In order to address this dimensionality problem, the following procedures may be followed [23]: 1) truncating COPT for capacity values which have a probability of occurrence less than some specified value, for instance, values having probabilities less than 10 may be truncated; 2) aggregating several rows of COPT into a single row (clustering of COPT); and 3) using an approximate COPT in which only single outages are incorporated. If the last

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Fig. 3. Convergence behavior of the proposed method. TABLE I DATA FROM KHUZESTAN POWER SYSTEM

B. Model Implementation Fig. 2. Flow diagram of the proposed method.

procedure is used then for a system with N units, COPT will rows. We have found the last procedure to be sufhave N ficiently accurate; however, since the real system used in our study was relatively small, we have used exact COPT without any approximations. The method is summarized as follows and is shown in the flow diagram of Fig. 2. 1) Select a given level of reliability of supply and set up as in the model of (13). 2) Select initial values for Lagrange multipliers, which are close to the thermal system marginal cost. 3) Solve hydro and thermal subproblems with a given reliability level and the corresponding Lagrange multipliers. 4) If the convergence is not reached, modify Lagrange multipliers by applying a subgradient method and go to step 3. 5) With optimal values obtained after the feasibility phase followed by step 3, construct COPT and check for the optimality condition of Theorem 1 [(11)]. This is done . by finding the roots of (11) which are , , If in iteration and for stop. Otherwise, go back to step 3 with the new reliability , . levels which are The convergence behavior of the proposed method is shown schematically in Fig. 3.

In order to implement the model, some preprocessing is necessary to transform the raw data obtained from power companies into the form required by the model. Most of the transformations involve statistical inferences from historical data associated with various components of the system. In this section, we summarize the main preprocessing required on input data to be used by the model. 1) By applying the maximum likelihood estimation (MLE) method, mean and standard deviation of energy demand are estimated for from historical data. 2) Estimate distribution of reservoir inflows for each period and correlation coefficients for successive periods from historical data. 3) Estimate the FOR of hydro and thermal units from historical operation data using the MLE method. If necessary, include unit derated states in calculation of FOR. 4) Obtain cost curves for thermal energy generation of each unit from operation data and by applying the MLE method. 5) Given seasonal limitations, fuel constraints, hydroplant characteristics, availability, and maintenance schedule for generating units, determine the corresponding lower and upper bound constraints for each period. 6) Determine the modified parameters for demand probability distribution by considering the information

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TABLE II GENERATING UNIT DATA

regarding energy interchange contracts with neighboring regions. by considering the customer damage 7) Calculate function obtained from regional surveys. This can be done, for example, by the Monte Carlo simulation does not vary technique. As mentioned before, significantly when the system load level and other factors change.

TABLE III OPTIMAL SOLUTION FOR KHUZESTAN POWER CO.

V. CASE STUDY Several examples are tested to examine the robustness of the proposed algorithm. Among them is an interconnected power system located in Khuzestan, which is described in detail in [5]. There exist 12 thermal units and two hydroplants (including 12 hydro units) in this power system with a total capacity of 3778 MW of which 1520 MW is produced by hydroplants. This province has a maximum load of 2887 MW with a load factor of 81.8%, at the peak period. The energy demanded at peak period has a mean value of 1636 GWh. It is assumed that the standard deviation of demand distribution is linearly increasing with respect to time, and in the first month, it has a value of 2% of demand mean, which increases to 5% at the end of the planning horizon. Table I gives the energy demanded and the corresponding for various months. In the fifth column, the minus sign indicates the import of energy. Table II provides hydro and thermal unit characteristics. A. Results In order to implement the model, software is developed within the MATLAB environment and applied to the Khuzestan power are system. The results are shown in Table III, where and the optimal reliability level and optimal Lagrange multiplier in period , respectively. Total cost for this case is equal to U.S.$ 200 932, with a CPU time of 4 h, 34 min, and 16 s, using a Pentium pro 200, 33-MHz personal computer (PC). This CPU time is the result of 54 iterations of LR technique with six overall iterations for updating reliability levels. Hence, on the average, there have been nine iterations of an LR technique for each iteration of updating reliability level. To see the difference between stochastic and deterministic treatment of demand, we also implemented the proposed method to the above system for the following cases:

TABLE IV OPTIMAL SOLUTION FOR CASE 1), DETERMINISTIC DEMAND

a) considering the energy demand as a deterministic quantity, which is equal to its expected value; b) considering an extra supply of 10% of average demand as reserve for each period; c) considering an extra supply having a magnitude equal to one standard deviation (one sigma) for each period. The results for the above cases are summarized in Tables IV–VI, respectively. The CPU times and the number of iterations are given in Table VII for each case. Note that since in these cases the value of demand to be satisfied is specified in advance, no optimization for reliability is carried out, and hence, the CPU times of Table VII are associated with the LR part of the algorithm. Total costs for cases 1)–3) are equal to U.S.$525 679, U.S.$207 032, and U.S.$247 587, respectively. It is seen that there is an increase of 162% in total cost if

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TABLE V OPTIMAL SOLUTION FOR CASE 2), CONSIDERING 10% EXTRA SUPPLY

nonlinear and dynamic programming. The model is solved within three stages. To see the efficiency of the proposed method, it is tested for a real case and the results obtained indicate the efficiency of the proposed method. An extension of this research is to consider the case that includes cascaded reservoirs in multiarea power systems. Other system features, such as fuel and transmission limitations as well as emission constraints, can also be included. ACKNOWLEDGMENT The authors would like to thank two anonymous referees for their constructive suggestions that greatly improved the paper. REFERENCES

TABLE VI OPTIMAL SOLUTION FOR CASE 3), CONSIDERING ONE  EXTRA SUPPLY

TABLE VII NUMBER OF ITERATIONS AND CPU TIMES FOR VARIOUS CASES

the demand is dealt as a deterministic quantity, case 1). If case 2) is adopted, then the cost is 3% higher than that of the proposed method, while the cost of case 3) is 23% more than the proposed method. Case 2) seems to have near optimal cost, but the situation varies if we change the forecast uncertainty. In fact, if we linearly increase demand standard deviation from 7 to 10%, then case 3) becomes near optimal. It is claimed that the optimal solution of the proposed method has the least cost among other operating policies, given the assumptions made by the method. VI. CONCLUSION In this research, we developed a new model for long-term planning of a power system, which consists of hydro and thermal generation plants. In the proposed model, not only have we obtained an optimal amount of discharge of hydro units and energy output of thermal units but we have also determined the system reliability level, simultaneously, in order to minimize the total cost. To solve this model, a method is also developed, applying decomposition technique, Lagrangian relaxation, and

[1] W. Yeh and W.-G. Chen, “Reservoir management and operation models: A state of the art review,” Water Resources Res., vol. 21, no. 12, pp. 1797–1818, 1985. [2] M. E. El-Hawary, “An overview of scheduling functions in hydro-thermal electric power systems,” in Proc. 1991 IASTED Int. Conf., Mar. 1991, pp. 1–37. [3] C.-a. Li, R. Yan, and J.-y. Zhou, “Stochastic optimization of interconnected multi-reservoir power systems,” IEEE Trans. Power Syst., vol. 5, pp. 1487–1496, Nov. 1990. [4] V. R. Sherkat, R. Campo, K. Moslehi, and E. O. Lo, “Stochastic long-term hydro-thermal optimization for a multi-reservoir system,” IEEE Trans. Power Syst., vol. PAS-104, pp. 2040–2050, May 1985. [5] R. Ardakaniaan, “Long-term optimal operation of hydro thermal power systems,” Ph.D. dissertation, McMaster University, Hamilton, ON, Canada, 1997. [6] R. W. Ferrero, J. F. Rivera, and S. M. Shahidehpour, “A dynamic programming two-stage algorithm for long-term hydro thermal scheduling of multi-reservoir systems,” IEEE Trans. Power Syst., vol. 13, pp. 1534–1540, Nov. 1998. [7] A. Conejo and J. Medina, “Long-term hydro-thermal coordination via hydro and thermal subsystem decomposition,” in Proc. 7th Mediterranean Electrotechnical Conf., vol. 3, 1994, pp. 921–924. [8] M. Modarres and D. Farrokhzad, “Optimization of reliable hydro-thermal power systems,” Sharif University of Technology, Scientia Iranica, Working Paper, Mar. 2002. [9] J. Juan and I. Ortega, “Reliability analysis for hydro-thermal generating systems including the effect of maintenance scheduling,” IEEE Trans. Power Syst., vol. 12, pp. 1561–1568, Nov. 1997. [10] R. Billinton and P. G. Harrington, “Reliability evaluation in energy limited generating capacity studies,” IEEE Trans. Power App. Syst., vol. PAS-6, pp. 2076–2085, Dec. 1978. [11] M. Caramanis, J. V. Stremmel, W. Fleck, and S. Daniel, “Probabilistic production costing,” Int. J. Elect. Power Energy Syst., vol. 5, pp. 75–86, 1983. [12] C. Dechamps, Z. A. Yamayee, and M. Chan, “Analytical treatment of hydro units energy in stochastic production simulation,” IEEE Trans. Power App. Syst., vol. PAS-101, pp. 113–119, Aug. 1982. [13] C. Singh and Q. Chen, “An analytical technique for the reliability modeling of generating systems including energy limited units,” IEEE Trans. Power Syst., vol. PWRS-2, pp. 123–128, Feb. 1987. [14] , “Modeling energy limited units in the reliability evaluation of multi-area electrical power systems,” IEEE Trans. Power Syst., vol. 5, pp. 1364–1373, Nov. 1990. [15] N. de Franco, F. A. Ventura, J. Kelman, P. A. S. Pegado, and A. Vian, “Impact of construction delays on the reliability of a hydro system. A Brazilian planning experience,” in CIGRE Conf., Paris, France, 1982, Paper 31-14. [16] X. Ying, C. Bin, Y. Yonghua, and G. Jianbo, “Short-term scheduling of pumped-storage plant using multi-objective fuzzy dynamic programming,” in Proc. Int. Conf. Elect. Eng., vol. 2, 1996, pp. 1108–1112. [17] C. Wang and S. M. Shahidehpour, “Power generation scheduling for multi-area hydro-thermal systems with tie line constraints, cascaded reservoirs and uncertain data,” IEEE Trans. Power Syst., vol. 8, pp. 1333–1340, Aug. 1993. [18] J. R. Ubeda and R. N. Allan, “Reliability assessment of hydro thermal composite systems by means of stochastic simulation techniques,” Rel. Eng. Syst. Safety, vol. 46, pp. 33–47, 1994.

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[19] J. Nahman and S. Bulatovic, “Power system operating cost and committed generation capacity planning,” IEEE Trans. Power Syst., vol. 11, pp. 1724–1729, Nov. 1996. [20] A. M. Leite da Silva, F. A. F. Pazo Blanco, and J. Coelho, “Discrete convolution in generating capacity reliability evaluation—LOLE calculations and uncertainty aspects,” IEEE Trans. Power Syst., vol. 3, pp. 1616–1624, Nov. 1988. [21] R. Billinton, J. Oteng_Adjei, and R. Ghajar, “Comparison of two alternate methods to establish an interrupted energy assessment rate,” IEEE Trans. Power Syst., vol. PWRS–2, pp. 751–757, Aug. 1987. [22] N. Amjady and M. Ehsan, “Evaluation of power systems reliability by an artificial neural network,” IEEE Trans. Power Syst., vol. 14, pp. 287–292, Feb. 1999. [23] R. Billinton and R. N. Allan, Reliability Evaluation of Power Systems. New York: Plenum, 1996.

Nima Amjady (M’97) was born in Tehran, Iran, on February 24, 1971. He received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Sharif University of Technology, Tehran, in 1992, 1994, and 1997, respectively. Currently, he is an Assistant Professor in the Electrical Engineering Department at Semnan University, Semnan, Iran. He cooperates with the National Dispatching Department of Iran and Regional Electric Company of Semnan. His research interests include security assessment of power systems, reliability evaluation of power networks, load forecasting of power systems, artificial intelligence, and its application in the problems of power systems.

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Davood Farrokhzad received the B.S., M.S., and Ph.D. degrees in electrical engineering from Sharif University of Technology, Tehran, Iran. Currently, he is with Iran Power Generation and Transmission Company (TAVANIR), where he has been responsible for power system planning and reliability evaluations. He has also been with the Iranian Ministry of Energy, where he was involved in hydrothermal system studies. His research interests include the application of optimization methods to power system planning and operation problems as well as in simulation techniques for stochastic modeling of power systems.

Mohammad Moddares received the M.S. degree in electrical engineering from the University of Tehran, Tehran, Iran, and the M.S. and Ph.D. degrees in operations research from the University of California, Los Angeles. Currently, he is Professor of Industrial Engineering Department at Sharif University of Technology, Tehran. His research interests are in applied operations research, dynamic programming, and supply chain management. He is the author of a book in queuing theory and has had several articles published.