Hydrothermal Scheduling Using Benders Decomposition - IEEE Xplore

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 3, AUGUST 2007

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Hydrothermal Scheduling Using Benders Decomposition: Accelerating Techniques Wilfredo S. Sifuentes, Student Member, IEEE, and Alberto Vargas, Senior Member, IEEE

Abstract—A new decomposition method is presented that includes the network through ac modeling within the hydrothermal scheduling optimization process including the losses. In short-term hydrothermal scheduling, the transmission network is typically modeled with dc power flow techniques. Such modeling, however, can lead to impractical solutions when it is verified with ac power flow. Another proposal considers in thermal systems the ac network modeling but not the optimization of losses. The approach presented here addresses issues such as congestion management and control of service quality that often arise in large and weakly meshed networks—the typical pattern of power systems in Latin America. Generalized Benders decomposition and traditional, well-known optimization techniques are used to solve this problem. The master problem stage defines the generation levels by regarding the inter-temporal constraints, whereas the subproblem stage determines both the active and the reactive economical dispatches for each time interval of the load curve. It meets the electrical constraints through a modified ac optimal power flow (OPF). Another important contribution is the inclusion of accelerating techniques aimed at reducing the number of iterations and CPU time. The methodology was proven in a real system and test systems. Results are discussed in this paper. Index Terms—Benders decomposition, hydrothermal scheduling, optimal power flow, unit commitment.

I. INTRODUCTION HORT-TERM hydrothermal scheduling (STHS) is known as one of the most challenging optimization tasks in power systems [1]. Its purpose is to minimize the total generation cost over a time period (a day or a week) involving a mix of hydro and thermal generation. This problem has been addressed with several techniques, e.g., dynamic programming [1], [2], Lagrangean decomposition [3], [4], and generalized Benders decomposition [5]. In these works, however, the transmission network was not considered or was greatly simplified. These simplifications could lead to inoperable dispatch or to the need for adjustments at the level of unit commitment when verifying operational limits with ac power flow. This problem is not only typical in Latin American countries that feature extensive and weakly-meshed networks, highly-loaded power lines, and generation plants located far from the load [7], [8] but is also present in other countries, like Spain [9]. The restructuring of electrical markets in the 1980s and 1990s for better competitiveness also turned the STHS problem into

S

Manuscript received January 27, 2006; revised November 20, 2006. This work was supported by the German Exchange Service (Deutscher Akademischer Austauschdienst/DAAD). Paper no. TPWRS-00044-2006. The authors are with the Instituto de Energía Eléctrica, Universidad Nacional de San Juan, San Juan, Argentina (e-mail: [email protected]; [email protected]. edu.ar). Digital Object Identifier 10.1109/TPWRS.2007.901751

a more complex one. Thus, when evaluating economically the operation of a power system in this stringent competitive framework, the problem incorporates additional factors, e.g., the lack of electric power to supply the demand, as measured by the non-supplied energy (NSE) factor. NSE is economically quantified by the cost of the NSE (CNSE) that must be evaluated for each period and each system bus. In spite of this deregulation trend, most countries in Latin America are still computing the short-term operating planning under some kind of centralized dispatch criteria. The most successful optimization technique applied so far to solve STHS has been Lagrangean relaxation (LR). In early applications, the optimization problem considered only thermal units [1], [3], [10]. The methodology was soon extended to consider the constraints from hydroelectric power stations [4], [12] and, subsequently, to include the transmission network modeled by dc power flow [11], [13], [14]. The success of the LR approach lies in the fact that it allows relaxing the hard constraints by defining a dual problem, which turns the original problem into a separable one. However, on account of its own nature, the solution attained for the dual problem is almost always non-practical. It requires further complex adjustments to meet every constraint of the primary problem, especially for inter-temporal constrains, such as the multi-period water balance of hydropower schemes. Other approaches of recent works [5], [6] have shown promising results using generalized Benders decomposition (GBD) [15]. With this tool and with an appropriate selection of variables, it is possible to split the problem into several small ones. This transformation allows processing the inter-temporal constraints separately from the static constraints, i.e., constraints that cover a single period. The issue of considering the transmission network within the STHS problem implies adding a very large set of nonlinear constraints that are equivalent to an OPF for each period. Since the OPF is regarded as a very complex problem [16], in STHS, the network is modeled with little detail or, else, it is greatly simplified. At present, there are just a few works that include an entire network modeling and the losses included in optimization [6], [17]. Nonetheless, they consider only purely thermal generation. Others authors include an ac network modeling and were applied in both small-scale and large-scale systems, including security and contingency constraints, but the approach is purely thermal feasibility oriented [18], [19] instead of hydrothermal optimality oriented as is presented in this paper. Doña [7] considers a complementary heuristic procedure to detect and correct reactive power deficit problems within an optimization process for a large-scale hydrothermal system, and Serrano [8] proposes an improvement of voltage levels and

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losses reduction process by considering the smallest deviation of the previously computed economic dispatch. A heuristic decomposition is presented in [9] applied to the Spanish market to correct both voltage levels and congestion issues. As a conclusion on the state of the art, at present, there is not any strong mathematical formulation available applied to STHS with an ac modeling. The present work offers a contribution on this sense. It combines the approach of Alguacil [5] to treat the inter-temporary constraints and the approach of Ma [20] for the electrical constraints, though with an extension to consider a complete power flow modeling. This compounded approach also allows representing cascaded hydro power plants, an issue that is hard to address with other approaches. Two drawbacks (always present in Benders decomposition) are its slow final convergence, also called tailing-off effect, and the high computational burden for the master problem [21]. The former is minimized by introducing accelerating techniques in a specific stage of the procedure and the latter by using a specialized optimization tool. The solution of hydrothermal systems with hydro predominance is significantly more complex than the thermal system. This problem is extremely difficult to deal with using LR approaches but relatively straightforward using the MIP technique, as is recognized in [22]. This paper is organized as follows. The generic hydrothermal coordination is formulated in Section II. Section III describes the proposed decomposition and the addition of heuristics to reduce the number of iterations. Section IV gives the numerical results and discusses the application to an actual system. Section V gives concluding remarks. II. PROBLEM FORMULATION The following notation is introduced. Sets: Time horizon of scheduling. Number of thermal units. Number of hydro-power plants. Number of buses in the system. Number of lines in the system. Constants:

Minimum active and reactive power output of thermal unit . Maximum active and reactive power output of thermal unit . Ramp rate of thermal unit . Minimum up time of thermal unit . Maximum active and reactive power outputs of hydro-power plant . Minimum active and reactive power outputs of hydro-power plant . Minimum and maximum limits active power of the line or branch l. Module and angle of the admittance matrix elements. Minimum and maximum voltage limits of bus k. Penalty costs due to active and reactive deficits. Rx

Penalty costs due to reactive excess. Minimum and maximum water volume of the dam associated to the hydro-power plant .

Main Variables: Active and reactive power output of thermal unit for period . Active and reactive power output of hydro-power plant for period . Operation state of thermal unit .

for period

Operation state of hydro-power plant period . Startup of thermal unit Startup.

for

for period

Water volume and flow rate of the dam associated to hydro-power plant for period . Active and reactive power deficits for period on bus .

Free, linear, and quadratic terms of the costs curve of thermal unit . Startup cost of thermal unit . Active and reactive power load for period on bus . Spinning reserve required for period .

Excess values of active and reactive power for period on bus . Voltage value for period on bus . Voltage angle for period on bus . The STHS can be formulated as shown in (1) at the bottom of the page.

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SIFUENTES AND VARGAS: HYDROTHERMAL SCHEDULING USING BENDERS DECOMPOSITION

The first group represents the production and startup costs, and the second group represents the total amount of penalty costs. The production costs are those incurred by the committed thermal units. They can be modeled either linearly or in quadratic form, in relation to the generated power output . , and meet two imporThe introduction of variables tant objectives in the formulation: First, they prevent the occurrence of non-feasibilities in the optimization approach, which allows closing the nodal balance (active and/or reactive balance) for any condition. Second, the penalty magnitudes correspond to the costs resulting from being unable to supply active or reactive power to the system (an extension of the concept of fault machine). These variables will not be zero when the proposed generating schedule and the transmission network cannot satisfy the nodal balance. Problem (1) is subject to the following constrains: 1) Auxiliary constraint to detect start-up.

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8) Nodal balance on bus riod .

(active and reactive) for every pe-

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(11)

(2)

9) Capacity limits of transmission lines or branches for every period . Due to regulation in some Latin American countries, the active power limit is shown. However, constraint (12) can be replaced for the apparent power limit or current thermal limits for the methodology presented here.

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2) Minimum up-time operation.

10) Power output limits (active and reactive) of thermal units.

3) Ramping. (4)

(13) (14)

4) Minimum and maximum active and reactive power output of hydraulic unit generation. 11) Bus voltage

limits.

(5) (6) 5) Hydraulic balance considering continuity and time-delay of inflow. The constraints from hydroelectric power plants are frequently modeled linearly. Nevertheless, there exists some exception when a nonlinear modeling must be considered [23]—a case not regarded in this work. A particular case is the consideration of time delay in cascaded hydroelectric power plants; these constraints are easily handled through a linear representation. (7) 6) Minimum and maximum volume of each reservoir. (8) 7) Spinning reserve requirement (SRR). In this work, SRR is considered, and it has a fixed value previously computed through studies comparing reliability and operation cost.

(9) Main electrical operation constrains are the following ones:

(15) Constraints (6) and (10)–(15) correspond to the formulation of various ac OPFs, though with a difference in that they contain variables that link these constraints through several time intervals, which—in turn—prevents their individual treatment. Constraints (2)–(5) and (7)–(9) correspond to a mixed integer problem, whereas constraints (6) and (10)–(15) correspond to a nonlinear problem with continuous variables (provided some variables are fixed equal to a given value). III. PROPOSED DECOMPOSITION The original GBD [15] technique was developed with the purpose of manipulating continuous and discrete variables separately. The master problem—a mixed integer optimization problem—deals with the discrete variables and with only one continuous variable, . The continuous variable together with the Benders cuts represent the subproblem into the master problem. The subproblem is a nonlinear optimization problem. The Benders cuts are linear approximations to the subproblem about the solution point found in the master problem. Therefore, the master problem must be solved successively (by accumulating the Benders cuts), and the subproblem as well, until reaching the convergence criterion. The proposed decomposition is slightly different from the original GBD approach. In the present work, there are more continuous variables— —in the master problem, which allow

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transforming the subproblem into a set of independent optimization problems as well as greatly reducing its computational burden. For sake of simplicity, an intuitive decomposition is presented first. Subproblem: The subproblem is defined as

(16) Objective function (16) is subject to constraints (6) and (10)–(15). This problem is nonlinear and separable. This way, the subproblem becomes a set of independent optimal power flows. Then, (16) will actually be as follows:

3) It is highly advisable to fix the bounds of variables. The solution gets reduced to an operative and feasible region to degrees). (e.g., voltage angles bounds: It is very important to remark that, with this formulation, the solution for the subproblem solution will always be feasible. Hence, the nonlinear solver must guarantee this is reached in order to attain the Lagrangean multipliers. The main advantage of this formulation is that it prevents the need of building feasibility cuts, because in nonlinear programming, it is not always possible to know when the optimization problem is not feasible or that the solver cannot reach at the right solution. Master Problem: The master problem includes the minimization of the fixed costs (free term of the cost curve) and the startup costs of the thermal units subject to the operative inter-temporal constrains (minimum operation times and minimum out-of-service time). It also contains the complete set of constraints of the hydroelectric power plants

(19) (17) The original variables, computed in the master problem as input data to the OPF, must be included as a constraint explicitly, because it is necessary to know their dual value once the solution of the subproblem has been attained. Each dual value is used to construct the Benders cuts, which are added to the master problem in each iteration to improve the proposed dispatch. Considering the case for variable for a specific period and a thermal unit , the additional constraint that should be added to the OPF would be

The objective function (19) is subject to constraints (2)–(5), (7), and (8). In Latin America, a common practice is to assign the spinning reserve only to hydraulic plants. Therefore, the master problem can also process constraint (9). It is then necessary to add the Benders cuts when computing each iteration. The added Benders cuts represent the subproblem in a linear way. The formulation of the master problem is of mixed integer lineal programming (MIP) type. For each single period , the Benders cut is

(18) is the previous value obtained in the master problem, where and is the Lagrange multiplier of the constraint; it gives a linear approximation of the rate-of-change of the subproblem costs caused by a unitary change of . The observed sensitivity arises from the fact that controls the limits of constrains (13) and (14). These constrains have variables that participate in the objective function of the subproblem. As in the previous case, it is necessary to add specific constrains of and variables into the subproblem. The strict application of GBD requires that the subproblem be convex; else, the linearized Benders cut may cut off feasible solutions. The non-convexities in the subproblem result from the presence of sine and cosine functions in the nodal balance (active and reactive) equations. However, some conditions can prevent or reduce much of these occurrences. 1) It is necessary to reach large bus angle-differences in order to change the convexity of these functions. Luckily, this situation is seldom found in practice. 2) The penalized variables altogether with the magnitude of the penalization significantly reduce the non-convex zones and prevent the subproblem from becoming non-feasible. Moreover, the first tool used in order to optimize the subproblem is sequential quadratic programming; then, the region is approximated by quadratic functions.

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The upper line means a fixed value for the variable computed in the previous iteration in the master problem and the remaining values in the subproblem . Hence, represents the subproblem within the master problem. Master Problem versus Subproblem: As a clarifying example, let us consider the case of a given OPF, with its active power totally supplied (as ensured by unit commitment computed in the master problem), though incapable of reaching the imposed voltage level for a given bus. This condition will be compensated for by activating a reactive power deficit machine on this bus. In this scenario, the dual value of variable will depend on the electrical proximity of the generator to the reactive-deficit bus. If the generator is located on a bus that has reactive power reserve electrically far from the bus showing the voltage-problem, the dual value of will be zero or very small, because an increase of reactive power injection in that bus will contribute very little or nothing to raise the voltage level. Therefore, the generators (thermal and hydro units) nearest to the reactive-deficit bus will have greater values associated to the dual value of the variable that controls its operation state . With all this information, the master problem determines a


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variables in the subproblem are the penalized variables, angles, and voltages. For each period , the Benders cut are defined as

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Fig. 1. Proposed algorithm structure.

new generation schedule. A similar analysis can be done when congestion appears. Fig. 1 shows the algorithm structure. Decomposition Strategy: In the actual decomposition, the linear part of the cost-curve of thermal units was moved to the master problem (including its active power limits constraint). This consideration allows processing inter-temporary constraints—e.g., (9) or (4)—without making simplifications, and it obtains a stronger lower bound, though the computational burden is increased as well. The master problem is redefined as

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The authors analyzed several scenarios and got as a conclusion that it is necessary to implement a particular decomposition strategy appropriate for the specific system, e.g., in hydrothermal or thermal systems, the quadratic term can also be moved to the master problem when the presented decomposition presents a slow convergence but the master problem needs to be solved using a QMIP solver. We adopt to include the quadratic term into the subproblem and then testing this decomposition provided that actual Latin-Americans hydrothermal systems present both hydro and thermal units very different characteristics and the quadratic terms are not significant (in fact, in some countries, only a linear cost curve representation is used). Then in our case, the subproblem is redefined as

(22) It is necessary to add the constraint to consider the influence of the quadratic part of the cost curve in the master problem. Under this decomposition framework, the state

Accelerating Techniques (ACCT): The inclusion of ACCT is aimed at reducing the computing time for the entire problem. It is described briefly below. 1) When Benders decomposition is used in linear programming, it is only necessary to keep the active Benders cuts to the next iterations. In mixed integer programming, this approach cannot be used in the same way. The problem arises from the fact that a Benders cut cannot be active but that cut is actually forcing to an integer solution of some variable. The distance of the cut can be evaluated as the difference between zero and the constraint evaluated at the solution point. The selected distance in order to delete a cut is chosen by trial and error, just to prevent the master problem giving the same solutions in the next iterations. 2) The first iterations are performed with relaxed mixed integer programming (RMIP). In the first iterations, the master problem has little information, and the proposed schedule does not have to be exactly computed. Therefore, the next better point (schedule) is calculated with RMIP. 3) The upper and lower bounds of the solution for hydraulic power plants in the last iterations are tightened. After a careful observation to understand why GBD shows a long tail (slow final convergence), it was realized that, in specific periods, the hydraulic generation undergoes a hydro generation complementary effect. This means, for example, that the total amount of active generation of two power plants change slightly between two iterations but change much more between them with a very little improvement of the lower and upper bounds. This problem is brought up by the linear representation of the entire problem in the master problem. Controlling this behavior will lead to a faster convergence, with slightly worsening of the solution. 4) The stopping criteria adopted with GBD, and GBD with ACCT, is that the difference between the upper and lower bounds should be smaller than the adopted tolerance. Results Validation: It is not possible to show the contrast with other methodology applied to this problem (STHS) with ac modeling. However, in order to guarantee the optimality of the solution, the following scheme is proposed. 1) Solve the original problem with the proposed methodology. 2) By fixing all binary variables and using the remaining results found in step one as initial variable values, solve again the original problem without considering any decomposition (totally coupled problem). 3) Contrast the results found in steps 1 and 2. This scheme is founded on the fact that GBD converges in one , where iteration [24] if the subproblem is solved at point

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means the entire set of binary variables, and are their optimal values. If the status (0/1) of both thermal and hydro operation units were correctly computed, then the original problem can be transformed into a nonlinear optimization problem by fixing the binary variables, with the possibility of having some further cost reductions due to an improvement in water allocation. The reason for this is that, in GBD, the nonlinear problem was reconstructed by hyper-planes, and the minimum may be slightly different from the actual one. This nonlinear problem, in spite of having fewer variables and having as initial variable values the results obtained by the proposed decomposition, takes a long time to converge. The methodology was implemented with Dash Optimization Suite® [25], using Xpress—MIP to solve the master problem, and Xpress-SLP for the subproblem. It was tested on an Athlon XP 2.0-GHz PC with 1 GBytes of RAM memory.

TABLE I IMPACT ON OPERATIVE COSTS DUE TO NETWORK MODELING CONSTRAINTS

TABLE II BUS VOLTAGE (IN P.U.) COMPUTED FOR THE FIRST SIMULATION. OUT-OF-LIMIT VOLTAGES IN BOLD

IV. RESULTS Test Example: The methodology was tested in the NorthCenter Interconnected System of Peru (SICN), characterized by an annual energy demand of 11085 GWh, and a peak of 2200 MW. The system was modeled considering 12 hydraulic power plants associated to nine river basins, some of them in cascade, 22 thermal plants, four SVCs. The electric network was modeled with 61 buses and 103 lines. Active and reactive generations of five noncontrollable hydro plants were directly discarded in the nodal balance as a negative load. An entire survey day was split into 48 periods. The voltage limits were set to 1.1 and 0.95 p.u. in all buses (excepting the lower limit in “SICN-17” bus, which was set in 1.0 p.u. due to operational requirements). The quadratic cost curve term for thermal units was neglected because it was not available. However, the proposed algorithm has not presented any problem in various tests done with estimated quadratic costs. The penalty costs used were UM 1750 for active deficit and UM 1650 for reactive deficit. During a typical day with a high hydropower share (89%), four simulations were made to observe the impact of ac modeling. In the first and second simulations, the reactive nodal balance and other constraints containing reactive variables were disabled, and the voltage was fixed in 1.0 p.u. in all buses in order to reproduce a dc modeling, though maintaining a nonlinear losses representation. Besides, in the first and the third simulation, the line limit capacity constraints were not set. Finally, the two last simulations were made with a full ac modeling. Due to the large number of results (voltages, angles, generation levels, etc.), only a few of them are shown. All results are referred to the proposed method, including the above-described accelerating techniques, unless stated otherwise. Table I shows the total costs for the four simulations. It is possible to see the impact over the cost of the activation of both groups of constraints: voltage limits and/or capacity limits. The CPU time solution corresponds 55% to the master problem and 45% to the subproblem. These values are only representative for this specific problem. For larger problems, the execution time increases quickly due to the MIP nature of the master problem. Table II also shows the impact of not considering the voltage constraints.

TABLE III BUS VOLTAGE (IN P.U.) COMPUTED TO THE FOURTH SIMULATION. ALL VOLTAGES ARE AT THEIR OPERATIVE LIMITS

These voltages were obtained after fixing the commitment status (0/1) of both thermal and hydro units and performing the proposed algorithm with the voltage constraints.1 The fourth simulation had all its constraints active. Table III shows the voltage levels for the fourth simulation. Table IV shows the unit commitment status for dc (first simulation) and ac results (fourth simulation). The voltage buses with problems were resolved with a new unit commitment. At peak hour, the thermal unit U12 startup and stay online lasted one hour because of reactive requirements. Fig. 2 shows the load diagram. The very flat thermal generation dispatched suggests a good solution for the proposed decomposition. Impact From the ACCT: Table V shows the impact from ACCT aimed at reducing the total CPU time. The CPU time considerably decreases (five times), with no reduction in solution quality. In various tests performed, the ACCT was able to diminish the solution quality, and it reached 0.4% of tolerance. The key reason for this reduction on CPU time lies on the less iteration needed to achieve convergence. This fact is very important because the master problem grows with each iteration. Then, its solution takes more CPU time. Fig. 3 shows the convergence with the addition of ACCT (13 iterations) and without ACCT (25 iterations). Results Validation: As previously stated, once the discrete variables were computed, the original problem was turned into a large-scale nonlinear (totally coupled) one. 1To check a dc dispatch, normally an OPF is performed in each period, and the active power difference is taken by the thermal units. But in the proposed case, a more generic “checking” is performed by allowing further moves of both thermal and hydro generation.


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TABLE IV UNIT COMMITMENT STATUS “A” MEANS ONLY PRESENT IN AC DISPATCH, “AD” PRESENT IN BOTH AC AND DC DISPATCH AND “D” ONLY PRESENT IN DC DISPATCH

Fig. 3. Number of iterations required to achieve convergence—lower and upper bound evolution (fourth simulation). TABLE VI TOTAL COST CONSIDERING THE PROPOSED METHOD INCLUDING ACCT, AND THE FULLY COUPLED NONLINEAR PROBLEM

Fig. 2. Dispatch for full ac modeling: fourth simulation. Noncontrollable hydro units are not included. TABLE V TOTAL COST CONSIDERING PURE GBD AND GBD PLUS ACCT: FOURTH SIMULATION

The additional cost savings noted in Table VI are caused by an improved water allocation resulting from the nonlinear representation of the total problem. It may be argued that 1.41% (the largest difference) could be non-negligible, but it is important to say that this full operative solution was achieved in reasonable computing CPU time, while avoiding any post-heuristic schedule correction. Discussion: In the proposed master problem of decomposition, the active power balance constraint is not explicitly considered; rather, it is actually present as signal cost. The main drawback to include this constraint is the fact that the active power balance is actually the entire set of active nodal balance made up of completely nonlinear constraints. Even though if they were linearized and added to the master problem, it would become so large and would take extremely long CPU times to reach the solution. In fact, there is an optimization technique for nonlinear mixed integer problems called outer approximation [21] that solves MINL problems by adding linearized constraints to the master problem, converging in fewer iterations than with Benders decomposition. Nevertheless, it is applicable only to small-scale problems because of the above-mentioned reasons. Another possibility is to add feasibility cuts, as in [5]; though, actually, these are not mathematically deducible. In some cases, they could cause the master problem to converge to a wrong solution. For instance, if it includes the next constraint into the master problem (losses computed in the previous iteration) losses

(24)

Several tests showed that this constraint can reduce the number of iterations, with a much worse solution than with the proposed method (see Table VII).

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TABLE VII TOTAL COST, CPU TIME, AND ITERATIONS CONSIDERING THE ADDITIONAL CONSTRAINT (24)

TABLE VIII TOTAL MISMATCHES DEFICIT OR EXCESS [MWH OR MV ARH]: FOURTH SIMULATION

The implementation of constraint (24) in our proposed decomposition without any other ACCT does not help to reduce computation time. The same tailing-off effect has been noted, though with a difference: The gap between the upper and lower bounds in the first iterations is smaller than with the proposed method. This fact tells us that the proposed dispatch is better, though the water movement between sub-periods is again the greatest problem that prevents achieving a faster convergence. In addition, it is strictly necessary to be sure that the generation is always able to cover both the demand and the losses. If congestion appears, and the only solution to meet the nodal balance is to shed load [18], [19] (equivalent to dispatching a fault machine in that bus), this heuristic cut will lead to an incorrect solution. For actual systems, the authors have not noted any numerical difficulty as regards unit status. Most 0/1 definitive status of thermal units are defined in the first iterations. However, when reactive problems are present, this status could change for certain units (near to the voltage problem bus) until the reactive problem is overcome. The main observed reason for slow convergence lies in the variations of hydro generation between iterations in the last iterations. Due to the linear representation of the subproblem into the master problem, the hydro generation can change abruptly in some units between two successive iterations. This behavior causes power flow variations, with the consequent variations of losses. In this condition, the proposed generation does not exactly match the active power balance in the subproblem, so a smaller amount of penalized imbalance will be present. This imbalance, in spite of being small or very small, contributed largely to the total cost due to its penalization. From the tenth iteration, % of its prehydro generation is allowed to change only vious value. Fewer percentages lead to faster convergence but also could lead to a non-feasible master problem. The last iterations of the process are used to reduce the deficit very close to zero. In this instance, the deficit is very small (see Table VIII), but it has a high effect over the cost. This is possible

to observe in the computed lower bound. This criterion requires more CPU time than the nodal mismatch criteria adopted in [18] and [19], but the quality of solution is better and a near global optimum solution is assured. In our proposal, the quadratic term is located in the subproblem with the goal to alleviate the computational burden of the master problem and to obtain faster solutions, because as is observed, this step requires the most computational effort. Our premise was based on the fact and experience about power systems where the operative characteristics of unit generators are significantly different. This fact causes that commonly appears one or two marginal units (in pure thermal systems) or the uniform operation of thermal units in hydrothermal systems with hydro predominance. The master problem, after several iterations, begins to delimitate the final solution region of the problem and exist few alternatives (thermal dispatch) with similar cost. In this manner, the last iterations are necessary to adjust the hydro generation units for reducing the network losses (due to its linear representation reconstructed by the Benders cuts). We also test a purely thermal system (IEEE 118 bus) with many similar (some of them equal) cost curves and important quadratic terms. The proposed method showed a slow convergence because the linear representation2 of the cost curves into the master problem, a thermal unit will always try to absorb most of the deficit. Then we proceed to consider the quadratic term in the master problem transforming this in a type of mixed integer quadratic problem. Under these test conditions, the problem was solved in few iterations. As a conclusion, the type of modeling (the quadratic term in the subproblem or in the master problem) is problem-dependent like was showed. There is not one methodological solution that fits all cases in unit commitment and short-term hydrothermal coordination. V. CONCLUSION The proposed methodology is innovative from the viewpoint that it allows modeling the transmission network using an ac OPF in the STHS optimization. It has fundamental importance for largely expanded and weakly meshed systems, such as those of Latin American countries or systems with a lack of reactive support. The presented decomposition scheme is simple, pure, and robust, even for dominant hydraulic power systems. It is also easy to implement, because it uses well-known, optimized, fast techniques, such as MIP and ac OPF algorithms. Its main disadvantage is that it may require a large number of iterations. The large iteration numbers required to achieve convergence are reduced (and CPU time) with the addition of simple rules. Important constraints like transmission congestion and voltage control are considered in the proposed methodology, without needing any post-schedule correction. This ensures the near optimal global solution, while the quality service requirements are satisfied. The chosen decomposition of the problem allows considering the network’s entire modeling, with little impact on computing CPU time. 2In spite of having the Benders cuts that transfer the impact of the quadratic cost term, the representation still have been linear.


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Wilfredo S. Sifuentes (S’92) was born in Peru in 1969. He received the Mechanical-Electrical degree from Universidad Nacional de Ingenieria (UNI), Lima, Perú, in 1992. In 2001, he was awarded a DAAD scholarship to carry out his Ph.D. studies at the Instituto de Energía Eléctrica–Universidad Nacional de San Juan (UNSJ), San Juan, Argentina. He worked at COES-SINAC as a dispatcher and real-time system operator of the Peruvian interconnected system. His research interests include optimization theory, planning, and economics of electric energy systems.

Alberto Vargas (M’97–SM’02) was born in Argentina. He received the Electromechanical Engineer degree in 1975 from the Universidad Nacional de Cuyo, Mendoza, Argentina, and the Ph.D. degree in electrical engineering in 2001 from the National University of San Juan (UNSJ), San Juan, Argentina. At present, he is a Professor of postgraduate studies at the Institute of Electrical Energy (IEE)—UNSJ. Since 1985, he has been a Chief Researcher of the team of Electrical and Competitive Market, at IEE.