A Genetic Algorithm Solution Approach to the Hydrothermal ...

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A Genetic Algorithm Solution Approach to the Hydrothermal Coordination Problem Christoforos E. Zoumas, Student Member, IEEE, Anastasios G. Bakirtzis, Senior Member, IEEE, John B. Theocharis, Member, IEEE, and Vasilios Petridis, Member, IEEE

Number of hydroplants. Number of upstream hydroplants for the th reservoir. Number of available units of th hydroplant. Number of online units of th hydroplant. Power output of th hydroplant (in megawatts). Spinning reserve contribution of th hydroplant (in megawatts). Water discharge rate of th hydroplant in m /h (positive when generating/negative when pumping).1 Spillage rate over th reservoir (in m /h ). Mean net inflow rate in th reservoir in m /h . Water transport delay from th hydroplant (in hours). Volume of water stored in th reservoir at the end of time interval (in m ). Maximum water discharge of th hydroplant (in m /h ). Minimum water discharge of th hydroplant (in m /h ). Upper bound of volume variation of th reservoir (in m ). Lower bound of volume variation of th reservoir (in m ). Initial storage volume of th reservoir at the beginning of scheduling horizon (in m ). Target storage volume of th reservoir at the end of scheduling horizon (in m ).

Abstract—In this paper, a genetic algorithm solution to the hydrothermal coordination problem is presented. The generation scheduling of the hydro production system is formulated as a mixed-integer, nonlinear optimization problem and solved with an enhanced genetic algorithm featuring a set of problem-specific genetic operators. The thermal subproblem is solved by means of a priority list method, incorporating the majority of thermal unit constraints. The results of the application of the proposed solution approach to the operation scheduling of the Greek Power System, comprising 13 hydroplants and 28 thermal units, demonstrate the effectiveness of the proposed algorithm. Index Terms—Genetic algorithms, hydrothermal coordination, short-term generation scheduling.

NOMENCLATURE Time interval (hour) index. Total number of time intervals (scheduling horizon). Load demand forecast (in megawatts). Spinning reserve requirement (in megawatts). Total thermal production cost in . Number of available thermal units. Commitment state of th thermal; unit if if reserved. committed or Power output of th thermal unit (in megawatts). Fuel cost function of th thermal unit in /h; a func. tion of Spinning reserve contribution of th thermal unit (in megawatts). Maximum power output of th thermal unit (in megawatts). Minimum power output of th thermal unit (in megawatts). Maximum spinning reserve contribution of th thermal unit (in megawatts). Ramp rate of th thermal unit in (in megawatts per hour). Minimum-up time of the th thermal unit (in hours). Minimum downtime of the th thermal unit (in hours). Start-up cost of the th thermal unit in . Shut-down cost of the th thermal unit in . Manuscript received August 25, 2003. This work was supported in part by the General Secretariat of Research and Technology and in part by the Public Power Corporation (D.E.I.) under Grant 96SYN81. The authors are with the Department of Electrical and Computer Engineering, Aristotle University Thessaloniki, Thessaloniki 54006, Greece. Digital Object Identifier 10.1109/TPWRS.2004.825896

I. INTRODUCTION

T

HE objective of the hydrothermal coordination (HTC) is to determine the optimal operation schedule of thermal units and hydroplants that minimizes the total thermal production cost over a predefined short-term period, taking into account the system-wide (coupling) and unit-wise (local) operating constraints. The HTC problem is a mixed-integer, nonlinear optimization problem, requiring the determination of both integer and continuous decision variables representing unit on-off status and production levels of thermal and hydro units. The importance of efficient generation scheduling is well recognized. An efficient generation schedule not only reduces the production cost but also increases the system reliability, securing valuable reserves, regulating margins, and maximizing

1The standard measurement unit of water flow quantities is [m /s]. Here, with the exception of Section VI (Test Results), water flow quantities are expressed in [m /h] to avoid the use of conversion coefficients in equations.

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ZOUMAS et al.: A GENETIC ALGORITHM SOLUTION APPROACH TO THE HYDROTHERMAL COORDINATION PROBLEM

the energy capability of the reservoirs. During the last two decades, the HTC problem solution attracted the attention of the research community [1]–[27]. Until now, the majority of the proposed solutions use decomposition schemes based on two specific characteristics of the problem [1]. • Only unit commitment state variables are restricted to be integers while the remaining problem is considered as a continuous process. • Power balance and security requirements are the only “coupling” constraints linking the operation of different generating units. The original large-scale problem is usually decomposed into a set of subproblems of reduced dimension and complexity that are easier to solve. A coordination scheme, combining the outcomes of the subproblems, is introduced to deal with the coupling constraints and manage the exchange of information between the subproblem solutions. The basic HTC methods can be classified into four categories. A. Heuristic Decomposition Methods Heuristic decomposition methods [2], [3] decompose the HTC problem into hydro and thermal subproblems. The hydro-optimization subproblems use either the thermal cost functions or the thermal system marginal cost to efficiently allocate the water resources within the scheduling horizon. Having the hydro generation and reserve contributions subtracted from the load and reserve requirements, the thermal subproblem solves a standard unit commitment (UC) problem. The hydro part of the optimization process is usually solved with fast linear network flow (LNF) methods. Priority listing or even more simplified aggregate representation methods have been employed to reduce the computational requirements of the thermal subproblem. Heuristic methods are fast and easily implemented but, in general, give production schedules with relatively high costs because of the numerous modeling simplifications they impose. B. Bender’s Decomposition Methods Bender’s decomposition methods [4], [5] decompose the main HTC problem into one master problem dealing with integer variables (unit commitment states) and a subproblem dealing with the optimization of continuous variables (unit outputs). A set of dual values is returned to the master problem after the solution of the subproblem. An iterative solution of the master problem and the subproblem is required for the convergence to the optimal solution. The optimization of the hydro system is incorporated into the subproblem that is further decomposed into a thermal and a hydro part. The main disadvantage of the Bender’s decomposition method comes from the nature of the master problem, which remains a large integer-programming problem. Moreover, handling of the fuel-constrained thermal units increases the number of dual variables and the complexity of the optimization task. C. Dynamic Programming Methods Several dynamic programming (DP) [6]–[8] methods have been used for solving the thermal and hydro subproblems in

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numerous decomposition schemes. For the hydroscheduling problem, DP provides accurate modeling of the majority of hydroelectric plant characteristics [6] (variable head, discrete operating zones, constant load operation for pumping and various hydraulic constraints). The “curse of dimensionality” still remains the main drawback of using DP for a realistic system with multiple river basins and cascaded hydro plants. The dual dynamic programming (DDP) [8] overcomes the “curse of dimensionality” of traditional DP by approximating the future cost function with an iteratively created piecewise linear function, thus avoiding volume state discretization. D. Lagrangian Relaxation Methods During the last two decades, Lagrangian Relaxation (LR) gained the research interest because of its flexibility in dealing with different types of constraints and became the popular solution approach for the HTC problem [9]–[17]. LR uses Lagrange multipliers for the system-wide constraints of load and reserve requirement and adds the associated terms in the objective function, thus forming the Lagrangian function. For fixed values of the multipliers, which are interpreted as energy and reserve prices, the initial large-scale problem is decomposed in unit-wise thermal and river-wise hydro subproblems that can be easily solved with conventional optimization techniques such as DP and LNF. The solutions of the subproblems are coordinated through a price selection mechanism in order to locate a feasible solution near the dual maximum. A subgradient algorithm is usually employed to update the coordinating prices. There are two major disadvantages of the LR methods: 1) convergence of the commonly employed subgradient algorithms to the dual maximum is very slow and the solution of the subproblems may be very sensitive to variations of the prices, 2) due to the nonconvexity of the problem search space, the values of the multipliers that maximize the dual function do not guarantee feasibility of the primal problem. Usually, application of heuristic procedures is required to obtain a feasible solution and the duality gap is used as a measure of the quality of the solution obtained. Although extensively investigated, the HTC problem still attracts the attention of researchers because of the strong need for lower cost operating schedules. During the last decade, genetic algorithms (GAs) have been successfully applied to various generation scheduling problems such as economic dispatch (ED) [18], [19], UC [19]–[22], and hydrothermal scheduling (HTS) [23]–[27]. In [23], the generation scheduling of hydraulically coupled plants is formed as a thermal production cost minimization problem where the decision variables represent water discharge rates and the thermal part is solved independently. In [24], the authors introduce a more detailed problem formulation taking into account both water discharges and thermal UC states. In [25], a multistage GA with varying precision and time steps is presented for a small-sized hydro system and an equivalent unit model is selected for the thermal system. An advanced encoding scheme resembling the “diploid” chromosome structure of living beings is presented in [26]. In [27], the GA was also used successfully for the midterm scheduling of large hydro-dominated power systems.

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In this paper, the hydro part of the HTC problem is formulated as a nonlinear, mixed-integer optimization problem and solved using an enhanced GA (EGA). The modeling of the hydro system includes operating limits, head dependent efficiencies, and discrete pumping operation. The thermal subproblem is solved by means of a priority list method incorporating minimum up and downtimes, operating limits, and ramp-rate constraints. In contrast to other approaches, an efficient modeling of the hydro system at the unit level is introduced, allowing more accurate operation modeling and reducing the number of constraints to be handled by the GA. A set of problem-specific genetic operators is introduced in order to enhance the GA performance. The results of the proposed method on the operation scheduling of the Greek power system demonstrate the effectiveness of the proposed approach. II. MATHEMATICAL FORMULATION OF HTC The HTC problem can be formulated as a mathematical optimization problem as follows: min

(16) where power output of the th unit; water discharge rate through the th unit; volume of stored water at the th reservoir. Available hydroplant data are usually in the form of tables giving the unit efficiency for different, discrete, values of the reservoir stored water volume. The hydro unit I/O characteristic used in our model is, thus, of the form

(1)

(17)

(2)

function (representing the head-deThe values of the pendent efficiency of the hydro turbine) are obtained from the available tables using linear interpolation. In our model, we have assumed that all units in a hydroplant are identical; thus, the hydroplant discharge and power output are given as

subject to • system-wide (coupling) constraints Power balance

output. The start-up cost is expressed as an exponential function of the reservation time while the shut-down cost is considered constant. Thermal unit minimum up, minimum downtimes, ramp-rate limitations, and spinning reserve constraints are also modeled through (5)–(8). 2) Hydro System Modeling: The power output of a hydro unit is, in general, a nonlinear function of the turbine discharge rate and the net head or, equivalently, the volume of the stored water in the reservoir. This function is the input/output (I/O) characteristic of the th hydro unit in the th hydroplant

(18) Reserve Requirement

(3)

• Thermal unit constraints Operating limits Spinning reserve if then for Minimum uptime if then for Minimum downtime Ramp rates

(4) (5) (6) (7) (8)

• Hydro system constraints Hydroplant discharge limits

(9)

Hydroplant power output (10) Hydroplant spinning reserve (11) Reservoir operating limits (12)

Water balance (13) Reservoir initial volume Reservoir target volume

(14) (15)

1) Thermal System Modeling: The operating cost of a thermal unit is expressed as a quadratic function of the unit

When generating, hydro units can adjust their discharge rate within operating limits (9). When pumping, the flow of water through the turbine is constant due to the constant rotation frequency (r/min) of the pumped-hydro turbine. III. GENETIC ALGORITHMS Genetic algorithms (GAs) [28] are general-purpose stochastic search techniques based on natural genetic and evolution mechanisms. They combine the survival of the fittest law with a structured, yet randomized information exchange among a population of artificial creatures, resembling samples of the search space of the problem in hand. During the last two decades, GAs have been successfully applied to several complex optimization problems in business, science, and engineering. One of the most interesting aspects of GAs is that they do not require any prior knowledge, space limitations, or special properties of the function to be optimized such as smoothness, convexity, unimodality, or existence of derivatives. They only require the evaluation of the so called “fitness function” to assign a quality value to every solution produced. Another interesting feature of GAs is that they are inherently parallel. Instead of using a single point and local gradient information, they evolve a population of candidate solutions where each individual represents a specific solution not related to other solutions. Therefore, their application to large-scale optimization problems can be easily implemented on parallel machines resulting in a significant reduction of the required computation time. GAs are generally considered as offline optimization algorithms due to

ZOUMAS et al.: A GENETIC ALGORITHM SOLUTION APPROACH TO THE HYDROTHERMAL COORDINATION PROBLEM

the large amount of the CPU time required to converge to an optimal solution. However, their performance can be significantly improved when a suitable combination of the basic genetic with other problem-specific operators is employed. Assuming a randomly generated initial population, genetic evolution proceeds by means of three basic genetic operators: Parent selection, crossover, and mutation.

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Fig. 1. Hydroplant encoding scheme.

IV. GA SOLUTION TO HTC A. Basic Genetic Evolution Operators 1) Parent Selection: Parent selection is a simple procedure where two individuals are selected from the parent population based on their fitness value. Solutions with high fitness values have a higher probability of contributing new offsprings to the next generation. In our approach, a simple roulette-wheel selection rule is employed. 2) Crossover: Crossover is an extremely important operator for the GA. It is responsible for the structure recombination and the convergence speed of the GA; it is usually applied with high probability. The genetic material of the two parents selected is combined to form new individuals that inherit segments of information stored in parent chromosomes. Several crossover schemes have been suggested in the literature, such as single point, multipoint, or uniform crossover. In our approach, a uniform crossover operator with a probability of 0.9 was adopted. 3) Mutation: While crossover is the main genetic operator exploiting the information included in the current generation, it does not produce new information. Mutation is the operator responsible for the injection of new information. With a small probability, random bits of the offspring genotypes flip from 0 to 1 and vice versa, thus giving new characteristics not existing in the parent population. The mutation probability was chosen equal to 0.0001.

B. Additional Operators Apart from the basic genetic operators, recent GA implementations usually employ the following additional operators. 1) Fitness Scaling: In order to avoid early domination of extraordinary individuals and to encourage a healthy competition among equals, a scaling of the fitness of the population is necessary. In our approach, the genotype fitness is scaled by a linear transformation [28]. 2) Elitism: Elitism ensures that the best solution found thus far is never lost when moving from one generation to another. The best solution of each generation replaces a randomly selected individual in the new generation. 3) Hill Climbing: A hill-climbing operator, based on the concept of phenotype-mutation [18], is applied only to the best individual to increase the GA speed at smooth areas of the search space. 4) Elite Self-Fertilization: A local search process of “selffertilization” is adopted for the elite solution. At the end of each generation, the best chromosome breeds with itself for a predefined small number (5)–(10) of generations, creating improved perturbations of its initial genetic material.

A. Parameter Set Selection and Encoding Method The implementation of a GA solution to an optimization problem begins with the selection of the decision variables and their encoding scheme, which are crucial for the overall efficiency of the algorithm. The parameter set must be selected in such a way that the resulting model fully describes the physical system and the encoding method must guarantee the effective transfer of information between chromosome strings. Until now, due to the fact that the number of hydro units is greater than the number of hydroplants, the majority of conventional and GA solutions to the hydroscheduling subproblem adopted modeling at the hydroplant level. According to that approach, the water discharge of the plant is allowed to continuously vary between zero and maximum discharge limits. In order to model minimum discharge limits, prohibited operating zones and discrete pumping operation, additional integer variables and/or penalty terms that distort the search space should be introduced. In our approach, an efficient unit-wise modeling, taking into consideration the hydro unit on/off and pumping operating states, is introduced. The proposed method models nonzero minimum discharge limits and discrete pumping operation within the chromosome encoding/decoding process and is capable of handling prohibited operating zones with minor modifications. The parameter set used in our approach consists of the hydro unit discharge rate and the unit on/off or pumping status. For notation simplicity, the encoding method is given for one hydroplant in Fig. 1. The chromosome substring corresponding to genes representing the real part hydroplant contains (water discharge) and the integer part (unit status). The length (precision) of the real part encoded parameters depends on the desired accuracy for the turbine discharge. The length of the integer part depends on the number of installed (identical) units at each hydroplant. The entire chromosome bit-string is obtained by concatenating all of the substrings representing the parameter set of each hydroplant over the scheduling horizon. B. Parameter Set Decoding In order to evaluate the encoded information stored in chromosome genes, a decoding mechanism is necessary. 1) Unit Status Decoding: Genes, representing the hydro unit status for each time interval, are decoded as unsigned integers for nonpumping hydroplants and as signed integers in the case of pumping hydroplants. In both cases, the integer represents the number of generating/pumping units. The decoded unit state affects the variation limits of the real part in the case of generating mode while in the pumping mode, defines the constant value of the pumping flow.

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2) Unit Discharge Decoding: With the unit on/off or pumping status fixed by the decoding of the integer part, genes representing discharge rates are decoded as unsigned integers and normalized in the range of the discharge variation (9). C. Fitness Function When GAs are applied for the solution of an optimization problem, the “fitness function” (FF) that assigns a quality value to every member of the population must be defined. This quality value is used as a comparative measure of each solution against other members of the population and its formulation is a key problem for the application of GAs to practical optimization problems. In some cases, the FF is identical to the objective function of the optimization problem. In many cases, however, the FF must be constructed taking into account the objective, the constraints imposed, and the difficulty of the resulting search space. In constrained optimization problems, an estimation of the distance of the current solution from the feasible region must be calculated. This is not an easy task since most real-world problems have complex constraints that distort the search space. Several methods have been proposed in the literature for the treatment of constraints [29]. Among them, the penalties approach is the most popular in the engineering community. In our approach, the FF is of the form (19) where fitness value assigned to the member (operation schedule) of the population; cost of serving the load with the thermal units only; cost (objective) of the current operation schedule; penalty term assigned to the operation schedule penalizing the violation of constraints. component is the thermal system In our approach, the , the following operating cost in (1). For the penalty term expression was adopted: (20) where: amount of violation of constraint ; function of ; coefficient indicating violation of constraint ; constant weighting factor indicating the relative im; portance of constraint . stringency factor for the violation of constraint ; number of constraints. Among the constraints defined in the problem formulation, those concerning (4)–(8), and (9) are satisfied within the thermal subproblem solution and the proposed encoding method, respectively. For the remaining constraints, (2), (3), (12), and (15), the following violation-to-cost transformation functions were adopted. is 1) Load Balance: The constraint violation transformed into cost using the function (21)

2) Spinning Reserve: The constraint violation transformed into cost using the function

is (22)

3) Reservoir Target Volumes and Operating Limits: The hydraulic coupling of cascaded reservoirs makes the target volume constraints by far the hardest to satisfy. However, due to the stochastic nature of natural inflows, the target volumes can be treated as soft constraints that can be violated or relaxed within a predefined tolerance. In our approach, a tolerance of a 1000 m was adopted for each reservoir. In contrast to the target volume constraints, the reservoir operating limits are always treated as hard constraints. The reservoir target volume and operating limit violations are transformed into cost using the following function: (23) In every case, is the incremental cost of the most expensive thermal unit, VOLL is the value of lost load in /MWh, is the minimum hydro-unit generating efficiency (inand verse specific consumption) in megawatt-hours per m . Given a candidate solution to the HTC problem (in the form of a chromosome), the FF is calculated as follows. a) The number of online hydro units and the discharge rate of each hydroplant are determined by decoding the chromosome. b) Using (13), the volume trajectory of each reservoir is computed. Hydro-unit efficiencies are derived by interpolating the data in the operating tables of each hydroplant. c) The power output and reserve contribution of each hydroplant are computed from (10) and (11). For each hour, the total power and reserve contribution of the hydro system are subtracted from the required load and reserve. d) The remaining load and reserve requirements are covered from the thermal system. A priority list [6] thermal UC combined with a -iteration dispatch is used for the thermal system scheduling. The computed schedule minimizes the thermal production cost while satisfying all of the local thermal unit constraints (4)–(8). e) The hydro and thermal system schedules computed above may violate some problem constraints. Due to thermal unit ramping constraints, the power balance (2) may be occasionally violated. Spinning reserve constraints and target volume as well as reservoir operating limits may also be violated by the candidate solution. All violated constraints are added as penalty terms (21)–(23) to the thermal system cost computed in Step d for the FF calculation (19). V. PROBLEM-SPECIFIC GENETIC OPERATORS It is widely recognized that the simple GA (SGA) scheme is capable of locating the neighborhood of the optimal or near-optimal solutions but, in general, requires a large number of generations to converge. This problem becomes more intense for large-scale optimization problems with difficult search spaces where the possibility for the SGA to get trapped in local optima

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Fig. 2. Genotype swap operator. Fig. 4. Genotype inverse operator.

Fig. 3. Genotype cross-swap operator.

increases and the convergence speed decreases. At this point, a suitable combination of the basic with other problem-specific genetic operators must be introduced in order to enhance the performance of the GA. Problem-specific genetic operators usually combine local search techniques and expertise on the problem. In this paper, six problem-specific operators that increase the convergence speed and improve the solution quality are introduced. The first three operators enhance the genetic search in optimization problems invoking real-valued decision variables. The last three operators are motivated by the desirable operation of hydroplants. From the implementation point of view, the first five operators introduce random modification to all chromosomes of a new generation and they are applied with a probability of 0.2 to both the real and the integer part of the chromosome. If the modified chromosome proves to have better fitness, it replaces the original one in the new population. Otherwise, the original chromosome is retained. The Genetic Swap-Shift Operator (GSSO) is always applied to the elite individual of each generation only.

Fig. 5.

Genotype copy operator.

Fig. 6. Genotype max-min operator.

new areas of the search space far away from the current solution, and preserves the diversity of the population. D. Genotype Copy Operator (GCO) The GCO randomly selects one parameter in the chromosome and copies its genotype to the immediately preceding or following position, with equal probability (Fig. 5). This operator has been introduced in order to force continuous operation of hydroplants at zero output during low-load periods and at maximum output during high-load periods in combination with the GMmO described next. E. Genotype Max-Min Operator (GMmO)

The GSO selects two parameters in the chromosome at random and swaps their genotypes (Fig. 2). It searches for improved modifications of the current solution by exchanging production schedules between the same or different hydroplants.

The concept of the GMmO is also inspired by the desirable operation of hydroplants. In most cases, hydro-units operate at full load during peak load hours and at zero output during low load hours. The GMmO selects one parameter in the chromosome at random and, with a probability dependent on the hour of the day, fills its genotype with 1s or 0s resulting in the introduction of the max or min power output level (Fig. 6).

B. Genotype Cross-Swap Operator (GCSO)

F. Genotype Swap-Shift Operator (GSSO)

Crossover is the main operator responsible for information exchange between highly-fitted individuals. However, useful parts of information may also lie in low-fitted individuals excluded by selection. The GCSO exchanges production schedules even between low-fitted individuals. It randomly selects two different individuals in the population and two parameters in their chromosomes and swaps their genotypes (Fig. 3).

The GSSO is always applied to the best individual of each generation. It randomly selects one river basin and a hydroplant and creates perturbations of generation scenarios within a randomly selected, small set of consecutive parameters of the selected plant while the scheduling of the remaining hydro production system is kept unchanged. Fig. 7 demonstrates the operation of the forward GSSO when applied to a set of three consecutive parameters (hourly discharges). Beginning with the first parameter in the set, GSSO swaps its value (genotype) with the value of the immediately following parameter. The new chromosome is then tested for improved quality and, if successful, replaces the old one. The swapping position then moves to the next parameter and the same procedure is repeated until the end of the selected parameter set. In case of a nonsuccessful swap, the swapping position remains the same and the following to the next parameter is chosen as a

A. Genotype Swap Operator (GSO)

C. Genotype Inverse Operator (GIO) The importance of the mutation operator is well recognized. However, because of its random nature and small application rate, the possibility of producing better solutions is low. The GIO acts like a sophisticated mutation. It randomly selects one parameter (as opposed to 1 b) in the chromosome and inverses its bit values from 1 to 0 and vice versa (Fig. 4). The GIO searches for bit-structures of improved performance, exploits

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TABLE II HYDRO SYSTEM DATA

Fig. 7. Genotype swap-shift operator.

TABLE I SYSTEM NET-LOAD TABLE III TEST PARAMETERS AND RESULTS

swapping candidate. The application of the GSSO to the current river basin ends when all of the installed hydro plants have gone through the swap and shift procedure. The backward GSSO is implemented in a similar fashion. In our test experiments, a maximum number of six parameters was used. It is obvious from this description that the GSSO searches for improved solutions by shifting hydroplant discharges in time, while keeping the total volume of water discharge over the scheduling period fixed. VI. TEST RESULTS The developed EGA method was applied to the HTC of the Greek power system. The Greek power system consists of 13 hydroplants (two of them are pumped-storage) and 28 thermal units. The total installed thermal and hydro capacity is 6864 MW and 2960 MW, respectively. The system peak load during the summer of 2002 was about 8.5 GW. The system net load curve of June 10, 2002 is given in Table I. System imports, small, run-of-the-river hydro plants production, and wind production have been subtracted from the actual load curve. Hydro system data are presented in Table II. The structure of the hydro system in Greece is quite simple. There are six river basins, each having at most three reservoirs/hydro plants in cascade. There is no time delay (compared to the hourly time step of our model) between the discharge of a hydroplant and the resulting inflows of the downstream reservoir. Thus, time have been set equal to zero for each hydroplant in delays (13). However, our computer code has been designed to model a complex network of rivers with time delays between reservoirs. The proposed solution approach was applied to the HTC of the Greek Power System using two scenarios of increasing difficulty. In the first “relaxed” scenario, the reserve requirement along with the start-up, shut-down, no-load costs, ramp-rate, and time constraints of the thermal units are neglected. In the

second “fully-constrained” scenario, all of the previously neglected constraints are included. In both scenarios, the parameter set consists of 624 control variables divided into two subsets. The first subset of 312 continuous variables represents the hydroplants water discharge rate and the second one the discrete hydro units operating states. The parameter precision chosen for the continuous and discrete control variables were 12 b and 3 b, respectively, resulting in a chromosome length of 4680 b. The GA population size was taken equal to 50 and a number of 300 maximum generations was selected as the stopping criteria for the “relaxed” case. The population size and maximum number of generations were increased to 75 and 400, respectively, for the “fully-constrained” scenario. Two sets of 20 genetic runs were performed for both scenarios on an AMD Athlon 1.8-GHz PC. The first set was conducted with the proposed EGA solution approach while the second one with a standard SGA featuring the same modeling specifications except for the advanced and problem-specific operators of the EGA. In order to establish a common base of comparison between the EGA and SGA approaches, the SGA was allowed to perform at least the same number of FF evaluations as the one of the best EGA run. All of the conducted EGA runs were feasible; that is, all of the system-wide thermal and hydro constraints were satisfied at the final solution. In contrast to the EGA, none of the SGA experiments provided a feasible final solution. Table III summarizes the results obtained for each scenario and solution approach. The results of the EGA and SGA approaches are compared in terms of their maximum FF value

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Fig. 8.

FF evolution comparison for the EGA and SGA.

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Fig. 10. Peak-shaved load curve of the best run for the “fully constrained” scenario with SGA.

TABLE IV TEST PARAMETERS AND RESULTS FOR SYSTEMS OF GREATER SIZE

Fig. 9. Peak-shaved load curve of the best run for the “fully constrained” scenario with EGA.

since all infeasible SGA runs achieved smaller operating costs. The operating costs of the best and worst EGA runs for the “reand , (0.216% diflaxed” scenario was ference). The corresponding operating costs for the “fully conand , (1.439% strained” scenario were difference). The increase in the percentage difference between the worst and best run of the “fully constrained” scenario is justified by the resulting distortion of the search space when the constraints (3) and (5)–(8) are included. Fig. 8, illustrates the improvement achieved in the convergence speed with the inclusion of the advanced and problemspecific operators. The EGA is capable of locating the area of the final solution (99.4% of the best FF value) in 100 generations with a computational time of 207 s. As opposed to the EGA scheme, the SGA took only 13 s to evaluate 100 generations but the corresponding FF value is far away from the best FF value (52.3%). Fig. 9 shows the original system load curve and the resulting thermal generation along with the hydro generation of the best run computed by our EGA-HTC algorithm for the “fully constrained” scenario. As expected, the hydro plants generate electricity during peak load hours and pump water during low load hours. Fig. 10 depicts the corresponding “peak-shaved” load curve achieved by the best run of the SGA. Even after 7500 generations and 525 000 FF evaluations, the quality of the solution provided by the SGA is very poor. In order to investigate the scalability of our method with respect to both system size and study horizon, two additional cases of the “fully constrained” HTC problem were investigated. In the first case, the study horizon is doubled (48 h) and in the second case, the size of the Greek system is doubled by duplicating all generating units. A set of ten (10) random runs was

conducted for each case. The results are presented in Table IV. In both cases, the resulting peak-shaved load curve (not shown) is similar to the one of Fig. 9. However, the solution quality seems to have deteriorated since the best run operating cost of the “double system size” case is a little higher than twice the best operating cost of the original system, while it should be lower. VII. CONCLUSION A method for the solution of the HTC problem using an EGA has been presented. The solution method has been applied to a real power system, the Greek power system, comprising 13 hydroplants and 28 thermal units, with satisfactory results. Our test results demonstrate the superiority of the EGA, incorporating problem-specific operators, over the SGA, which failed to solve the HTC problem of the test system. By incorporating the number of online units of each hydroplant in the GA chromosome, constraints such as nonzero minimum discharge limits and discrete pumping operation were modeled without the need of additional FF penalty terms. The major advantage of the GA solution to the HTC problem is its modeling flexibility: constraints such as variable-head hydroplant efficiencies, discrete pumping operation, nonzero minimum discharge limits, and dead zones can be easily incorporated either by judicious chromosome selection or as penalties in the FF. The execution time explosion with the problem size and the lack of formal proof for convergence to the global optimum solution (due to their stochastic nature two different GA runs converge to different solutions) are the main disadvantages of the GA solution to the HTC problem. With the advent of parallel and distributed computing, the implementation of the GA optimization scheme to the HTC problem on parallel machines exploiting GA’s inherently parallel nature should be investigated.

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ACKNOWLEDGMENT The authors wish to thank A. Tasoulis, J. Blanas, K. Bokis, and T. Striggos of the Transmission Exploitation Department of D.E.I., currently with the Hellenic T.S.O., for providing the system data and their valuable experience. REFERENCES [1] M. E. El-Hawary, Hydro-thermal scheduling of electrical power systems, in IEEE Tutorial Course 90EH0328-5-PWR. Application of Optimization Methods of Economy/Security Functions in Power System Operations. [2] R. A. Duncan, G. E. Seymore, D. L. Streiffert, and D. J. Engberg, “Optimal hydrothermal coordination for multiple reservoir river systems,” IEEE Trans. Power App. Syst., vol. PAS-104, pp. 1154–1161, May 1985. [3] H. Brannlund, J. A. Budenko, D. Sjelvgren, and N. Andersons, “Optimal short term operation planning of a large hydrothermal power system based on a nonlinear network flow concept,” IEEE Trans. Power Syst., vol. PWRS-1, pp. 75–82, Nov. 1986. [4] H. Habibollahzadeh and J. A. Bubenko, “Application of decomposition technique to Short—Term operations planning of Hydro—Thermal power systems,” IEEE Trans. Power Syst., vol. PWRS-1, pp. 41–47, Feb. 1986. [5] L. F. B. Baptistella and J. C. Geromel, “A decomposition approach to the problem of unit commitment schedule for hydro thermal systems,” in Proc. Inst. Elect. Eng. D, vol. 127, Nov. 1980, pp. 250–258. [6] A. J. Wood and B. F. Wollenberg, Power Generation, Operation and Control. New York: Wiley, 1984. [7] J. Yang and N. Chen, “Short term hydrothermal coordination using multi-pass dynamic programming,” IEEE Trans. Power Syst., vol. 4, pp. 1050–1056, Aug. 1989. [8] B. G. Gorenstin, N. M. Campodonico, J. P. Costa, and M. V. F. Pereira, “Stochastic optimization of a hydro-thermal system including network constraints,” IEEE Trans. Power Syst., vol. 7, pp. 791–797, May 1992. [9] F. Zhuang and F. D. Galiana, “Toward a more rigorous and practical unit commitment by Lagrangian relaxation,” IEEE Trans. Power Syst., vol. 3, pp. 763–773, May 1988. [10] S. Virmani, E. C. Adrian, K. Imhof, and S. Mukherjee, “Implementation of a Lagrangian relaxation based unit commitment problem,” IEEE Trans. Power Syst., vol. 4, pp. 1373–1380, Nov. 1989. [11] H. Yan, P. B. Luh, X. Guan, and P. M. Rogan, “Scheduling of hydrothermal power systems,” IEEE Trans. Power Syst., vol. 8, pp. 1358–1365, Aug. 1993. [12] , “Optimization-based scheduling of hydrothermal power systems with pumpedstorage units,” IEEE Trans. Power Syst., vol. 9, pp. 1023–1031, May 1994. [13] S. Ruzic, N. Rajakovic, and A. Vuckovic, “A flexible approach to short-term hydrothermal coordination. Part I: Problem formulation and general solution procedure,” IEEE Trans. Power Syst., vol. 11, pp. 1564–1571, Aug. 1996. [14] , “A flexible approach to short-term hydrothermal coordination. Part II: Dual problem solution procedure,” IEEE Trans. Power Syst., vol. 11, pp. 1572–1578, Aug. 1996. [15] , “Short-term hydrothermal coordination by Lagrangian relaxation: Solution of the dual problem,” IEEE Trans. Power Syst., vol. 14, pp. 89–95, Feb. 1999. [16] A. Merlin and P. Sandrin, “A new method for unit commitment at electricite de france,” IEEE Trans. Power App. Syst., vol. PAS-102, pp. 1218–1225, May 1983. [17] K. T. Aoki, K. Satoh, and M. Itoh, “Unit commitment in a large-scale power system including fuel constrained thermal and pumped storage hydro,” IEEE Trans. Power Syst., vol. PWRS-2, pp. 1077–1084, Nov. 1987. [18] A. Bakirtzis, V. Petridis, and S. Kazarlis, “A genetic algorithm solution to the economic dispatch problem,” in Proc. Inst. Elect. Eng., Gen., Transm. Dist., vol. 141, July 1994, pp. 377–382. [19] G. B. Sheble, T. T. Maifeld, K. Brittig, G. Fahd, and S. Fukurozaki-Coppinger, “Unit commitment by genetic algorithm with penalty methods and a comparison of Lagrangian search and genetic algorithm-economic dispatch example,” Elect. Power Energy Syst., vol. 18, no. 6, pp. 339–346, 1996. [20] S. A. Kazarlis, A. G. Bakirtzis, and V. Petridis, “A genetic algorithm solution to the unit commitment problem,” IEEE Trans. Power Syst., vol. 11, pp. 83–92, Feb. 1996.

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Christoforos E. Zoumas (S’98) received the Dipl. Elect. Eng. degree from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1996, where he is currently pursuing the Ph.D. degree. His research interests include power system operation with emphasis in hydroelectric production optimization and computer applications in power systems.

Anastasios G. Bakirtzis (S’77–M’79–SM’95) received the Dipl. Mech. and Elect. Eng. degree from the National Technical University of Athens, Athens, Greece, in 1979, and the M.S.E.E. and Ph.D. degrees from the Georgia Institute of Technology, Atlanta, in 1981 and 1984, respectively. Currently, he is a Professor with the Electrical Engineering Department at the Aristotle University of Thessaloniki, Thessaloniki, Greece, where he has been since 1986. In 1984, he was a Consultant to Southern Company, Atlanta, GA. His research interests include power system operation and control, reliability analysis, and alternative energy sources.

John B. Theocharis (M’82) received the Dipl.Eng. and Ph.D. degrees from the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1980 and 1985, respectively. Currently, he is an Associate Professor with he Aristotle University of Thessaloniki. His research interests include adaptive control and modeling, neural networks, and fuzzy systems with applications to the identification and control of complex nonlinear systems.

Vasilios Petridis (M’77) received the diploma in electrical engineering from the National Technical University, Athens, Greece, in 1969, and the M.Sc. and Ph.D. degrees in electronics and systems from King’s College, University of London, London, U.K., in 1970 and 1974, respectively. Currently, he is a Professor in the Department of Electronics and Computer Engineering with the Aristotle University of Thessaloniki, Greece. He was a Consultant to the Naval Research Center in Greece, and Director of the Department of Electronics and Computer Engineering and Vice-Chairman of the Faculty of Electrical and Computer Engineering at the Aristotle University of Thessaloniki. He was co-author of the monograph “Predictive Modular Neural Networks: Application to Time Series” in 1998 and the author of four books on control and measurement systems and co-author of many research papers. His research interests include control systems, machine learning, intelligent and autonomous systems, artificial neural networks, evolutionary algorithms, fuzzy systems, modeling and identification, robotics, and industrial automation.