Optimal Reactive Power Dispatch Considering Costs of ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 3, AUGUST 2005

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Optimal Reactive Power Dispatch Considering Costs of Adjusting the Control Devices Yong-jun Zhang and Zhen Ren, Senior Member, IEEE

Abstract—The traditional model of optimal reactive power dispatch (ORPD) for power systems is based on the principle of income maximization, which aims at minimizing active power loss of the whole networks. However, such a model may bring on excessive operations of device-control devices in real-time application. To realize reactive power dispatch, power utilities should increase equipment investment and added manpower for operation and maintenance. On the other hand, the operations would augment the fault probability of power systems. Therefore, the costs of adjusting the control devices (CACDs) are investigated, and a novel mathematical model of ORPD is presented in this paper, whose objective function is to minimize the energy loss at the current time interval and the CACD. A simulation test is presented to demonstrate that the proposed model reflects the principle of profit maximization and describes the ORPD problem with time-varying loads appropriately since it can decrease active power loss and avoid excessive controls simultaneously. Index Terms—Control action, costs of adjusting the control devices (CACDs), discrete variable, optimal reactive power dispatch (ORPD), power system, reactive power control, unit adjustment cost.

I. INTRODUCTION

O

PTIMAL reactive power dispatch (ORPD) has attracted worldwide attention for its significant influence on secure and economic operation of power systems. Reactive power dispatch has been researched extensively as a static snapshot problem, and the objective of ORPD is to minimize the active power transmission loss by means of dispatching reactive power sources while satisfying a lot of constraints, such as reactive power generation limits of generators, voltage limits of load buses, tap ratio limits, reactive power compensation limits, and power flow balance [1]–[7]. Such an objective is considered as a classic model of ORPD, or, for the sake of enhancing voltage stability, a multiobjective model that minimizes real power loss and maximizes voltage stability margin is considered [8], [9]. All of these models are based on the principle of income maximization without considering the costs of reactive power dispatch. For many applications in optimal power flow (OPF), this kind of solution is not practical because the number of control actions would be too large to be executed in actual operation, and many of the actions would be trivial [10]. Although the number of controls has little effect on the CPU time in a Newton OPF [11], the operators cannot move so many control devices

Manuscript received July 14, 2004; revised January 2, 2005. This work was supported by the Natural Science Fund of South China University of Technology. Paper no. TPWRS-00377-2004. The authors are with the Electric Power College, South China University of Technology, Guangzhou 510640, China (e-mail: 87112526 @21cn.com). Digital Object Identifier 10.1109/TPWRS.2005.851920

within a reasonable time. A curtailed number of control actions through selecting the most effective subset of controls has been investigated for a real time OPF [12]. Taylor et al. [13] present that ORPD should be seen as a time-based scheduling problem with the intention of avoiding unnecessary changes in status and output of a reactive control plant. They consider some transition constraints such as the number of control actions allowable within a time domain and the time interval required between actions performed. Paper [14] introduces the constraints of maximum allowable switching operations for on-load tap changer (OLTC) and capacitor of distribution systems. Its strategy is to minimize the power loss and improve the voltage profile for a whole day across the whole system and at the same time ensure that the number of operations is less than the maximum daily allowance. A model to minimize the energy loss over time intervals in which the transition of discrete variables is governed by the selection of time intervals is described [15]. In this manner, a near-optimal transition of discrete variables can be achieved by ensuring that they remain constant over a time interval. In [16], reactive power optimization with time-varying load demand in distribution systems is investigated. The objective is to minimize the system energy loss of 24 hours and subjected to the constraint of the maximum allowable daily operating times. In fact, the curtailed number of control actions and the maximum allowable operations for discrete control devices are presented to avoid their excessive operations, which lead to shorter life expectancy of devices and more maintenance. Hong and Liao also consider that the purpose of minimizing control actions is to reduce the depreciation cost of taps and compensators [17]. From the viewpoint of economics, the better voltage profile and the power-loss saving are the incomes of ORPD, whereas the investment of control devices and the increased operations and maintenances are the costs. It is not felicitous consequently to just consider the incomes without taking the costs into account in the objective function of ORPD. In other words, reactive power dispatch is to improve the voltage profile and decrease the active power loss at the cost of switching control devices. We, therefore, can minimize their costs of adjusting the control devices (CACDs) to realize the goal of avoiding their excessive operations while minimizing active power loss. Up to now, many numerical optimization techniques have been proposed, such as the nonlinear programming, linear programming, and interior-point methods [1]–[3], [5]. However, when these existing continuous optimization algorithms are adopted to solve the ORPD problem, discrete variables are treated as continuous ones until they are almost optimized and then rounded off to their nearest discrete values.

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This not only introduces some numerical approximations but also causes violations of the inequality constraints and/or an unnecessary increase in the objective function. This serious deficiency limits their practical application [3]. The artificial intelligence methods, such as simulated annealing algorithm, genetic algorithm (GA), evolutionary programming, and evolutionary strategy [4], [6], [18], have been developed for the discrete ORPD problem. GA is the focus of researchers’ attention in this area. GA uses multiple-point search instead of single-point search, and the only information required is the objective function, which makes the search simple and strong for the ORPD problem. Moreover, without special demand for a start point, GA is good at detecting and handling the infeasible solutions. This paper, thus, presents a new model for optimal reactive power dispatch whose objective function is to minimize the total costs, including energy loss of transmission network and CACD, in Section II. Then, the model of CACD and the simplified model of ORPD are described in Section III. Section IV gives a solving algorithm, namely, the cataclysmic genetic algorithm, for ORPD problems and presents a method to handle the constraints for ensuring no voltage violations, even with time-varying load demand. A test is presented in Section V. II. PROBLEM FORMULATION In order to improve the voltage profile and decrease the active power losses along the transmission lines under various operating conditions, power system operators can select control measures as little as possible, such as switching reactive power compensators, changing generator voltages, and adjusting transformer tap settings. If the cost of each discrete control device for an ORPD operation, namely, unit adjustment cost (UAC), is specified, the costs of adjusting the discrete control devices for the reactive power dispatch can be attained when multiplying the devices’ UACs by their operating times. Here, the operating times of an OLTC is defined as the difference of tap positions when implementing reactive power dispatch, and that of a shunt reactive compensator is defined as the difference of the bank number switched in. For example, if the current tap position of an OLTC transformer is 4 and the optimal value is 6, it is needed to adjust OLTC two times from 4 to 6. Then, we name the operating times of this transformer as 2. Since continuous control variables such as generator voltages can be adjusted more frequently by the excitation circuit, the costs of adjusting them can be neglected. From the viewpoint of [10], “Many of the controls whose actions need to be limited in number are adjustable only in discrete steps.”

where active power loss of the network at current time interval (kW); electricity price ($/kWh); time interval, usually 1 h; row vector that consists of unit adjustment costs of dis; crete control variables column vector that consists of deviations of discrete control variables, namely, operating times . and are defined as follows: (2) (3) (4) (5) where unit adjustment cost of adjusting the OLTC tap , ($/times); unit adjustment cost of switching the compensator , ($/times); column vector that consists of operating times of trans(times); former taps column vector that consists of operating times of com(times); pensator banks column vector that consists of the current tap positions ; before optimization column vector that consists of the optimal (or regarded ; as optimal) tap positions column vector that consists of the current numbers of compensator banks switched in before optimization ; column vector that consists of the optimal (or regarded as optimal) numbers of compensator banks switched in . It is assumed that the system loads and the status of discrete control variables are constant during a time interval (usually an hour). The assumption is reasonable if the time interval is short enough so that there is no great change in operation modes and load flows. B. Constraints The above functions are subjected to power-flow equality constraints and the physical and operating limits as follows:

(6)

A. Objective Function For an -bus power system with compensation buses, transformers, and generators, the proposed objective of optimal reactive power dispatch is to minimize the total cost, including the cost of energy loss of the network and the costs of adjusting the discrete control devices, which can be formulated as follows: (1)

where control variable ; state variable ; power-flow equations; column vector that consists of the angles of all buses ; except slack bus

ZHANG AND REN: OPTIMAL REACTIVE POWER DISPATCH

column vector that consists of the reactive power out; puts of generators column vector that consists of nodal voltage magnitudes ; column vector that consists of generator voltage ; magnitudes of slack bus and PV buses column vector that consists of nodal voltage magnitudes of PQ buses ; column vector that consists of transformer tap posi; tions column vector that consists of numbers of reactive . The corcompensator banks switched in responding reactive power injection vector , where is the row vector that consists of stan. dard sizes of a single compensator, In the objective function (1), minimizations of the cost of energy loss and the CACD represent the income of ORPD maximization and the expenditure minimization, respectively, and indicates to maximize the energy saving minimization of after the CACDs are recouped. Therefore, we consider that the proposed mathematical model is based on the principle of profit maximization as compared with the traditional model that only minimizes power loss. III. COSTS OF ADJUSTING THE CONTROL DEVICES A. Unit Adjustment Cost When implementing reactive power dispatch, reactive power control devices are operated under high voltage, and impact currents will be generated into the control devices. If they are switched too frequently, their insulating intensity must be reduced, and consequently, some hidden damages of power systems will emerge. On the other hand, too frequent operations overburden operators and are likely to lead to some mistakes. From these viewpoints, each operation of reactive power dispatch has its cost. These above factors and the investment of control devices should be introduced into the ORPD model as the costs of adjusting the discrete control devices, which involve the five aspects as follows. 1) The investment of discrete control devices: Such as those of the on-load tap changers and the switched reactive compensators with breakers. 2) The extra investment of transformers due to reactive power control causing their service duration to be shorter. The investments of transformers are not involved entirely because they serve for transmission but not for reactive power dispatch only. However, a transformer’s life expectation must be reduced by the regular adjustment of OLTC. 3) The overhaul costs related to reactive power control only, including those of OLTCs and compensators. 4) The system loss due to the fault of reactive power control devices. If the relay protection installations can work and the damage of transformers or capacitors is avoided, this part is mainly the loss of power cut. This aspect can be ” operation ignored if the power system can satisfy “ mode.

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5) The pay for operators related to reactive power control. To divide the sum of the above costs during the service durations of the control devices by their total allowable adjustment times, the unit adjustment cost of discrete control devices can, therefore, be formulated as follows:

(7) (8) where design value of the total allowable adjustment times of ; the OLTC of transformer (times); design value of the total allowable adjustment times ; of the compensation bus (times); life expectancy of the transformer when the tap never adjusted (year); life expectancy of the transformer after the tap times (year), ; changed life expectancy of the compensation bus after times (year); switched overhaul periods of the OLTC of transformer (year); overhaul periods of the compensator (year); investment of the OLTC of transformer ($); investment of transformer ($); unit overhaul cost of OLTC of transformer ($/times); system loss due to the fault of the OLTC of transyears ($); former during investment of the breakers and their operating equipments of compensation bus ($); investment of the shunt capacitors/reactors of compensation bus ($); unit overhaul cost of the compensator ($/times); system loss due to the fault of the compensator during years ($); unit cost for operators’ salaries related to reactive power control ($/times). If operators are added for reactive power control and each operator’s annual is formulated as salary is ($/year), (9)

It is assumed that each control device has adjusted its total allowable adjustment times during its whole service term. The UAC indicates the investment, maintenance cost, and latent system loss for each control of reactive power dispatch. If a switching action cannot reduce the amount of power loss more than its UAC or no help to eliminate the deviation of state variables, we would rather stop this movement. Although different devices have different UAC, from the above-mentioned formulae, it can be concluded that the UAC of each control device is a constant represented by its average cost of adjustment at any time. It is valuable for ORPD application because the constant UAC means there is no coupling between time intervals mentioned in the models in [16] and

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[17]. Then, the proposed decoupling ORPD model is more simple and easier to solve than the models of prior works, and it is more suitable for the real-time application that emphasizes the solving speed more than the actual optimal solution [12]. B. Normalization of Unit Adjustment Cost If defining (10) as the unit adjustment cost vector based on power (kW/times), then function (1) can be transformed as follows: (11) whose objective is to minimize the sum of the active power loss of the whole network and the costs of adjusting control devices represented in power. Such objective function is consistent with (1) in substance. Although the UAC of discrete control devices can be calculated theoretically by (7) and (8), it is difficult and unpractical to determine an exact value of the UAC for each control device since the life of device is a probability expectation, and the system loss due to the fault of reactive control devices is an estimate, too. Usually, if ignoring the loss of control device fault that is difficult to count due to lack of statistic, the estimate value of UAC of an OLTC is varying from 3–10 kW/times, and that of a compensation bus is varying from 2–6 kW/times. In terms of the definition of UAC, the transformers, especially those at the same voltage level, have similar values of UAC in a power system. The UAC of shunt compensators has such a characteristic, too. It is, therefore, practicable to normalize the UAC of the same kind of control devices by their average value in the real-time engineering application. Then, the average unit and that of all compensators adjustment cost of all OLTCs can be defined as (12)

IV. SOLVING METHOD A. Cataclysmic Genetic Algorithm (CGA) CGA [18] is adopted to solve ORPD problems. This algorithm is one of the modified genetic algorithms with the following characteristics. 1) The cataclysm operator is adopted to improve the diversity of the small size populations and to avoid the premature convergence. Mimicking the cataclysm of the evolution course of ecosystems, in which most species are extinct but very few survive, the cataclysm operator updates all individuals randomly, except for the current optimum after tens of generations. 2) An adjacent mutation operator integrating with grouping integer-encoding strategy is used to elude the void solutions and the Hamming cliffs. These techniques redound to improve the search efficiency and local search capability of genetic algorithms. 3) Without special demand for a start point, it can detect and handle the infeasible solutions and is fit for solving the nonlinear ORPD problem involving multiobjective, multirestriction, multiextremum, and discreteness. 4) It has fine performance of global optimization and high search efficiency. B. Handling Constraints When adopting genetic algorithms to solve the nonlinear optimization problems, the constraints are considered into the objective function using penalty terms. The control variables, including generator voltages, tap positions of transformers, and numbers of compensator banks switched in (directly proportional to the reactive power outputs of compensators), are self-constrained when encoding them as chromosomes. In other words, the chromosomes are generated just within the limits of all control variables so that the constraints of the control variables are always satisfied. The state variables, including voltages of PQ-buses, ( , where is the set of number of PQ buses), and reactive power generation, ( , where is the set of number of generator buses), however, are constrained as penalty terms into the integrative objective function as follows:

(13) (15) in (11). So, the objective and replace the elements of vector function can be depicted as follows: (14) Although such a model is not so exact as compared with (1) or (11), its feasibility and rationality are testified by the application in a real network in China, Hainan Power Systems.

where is a penalty factor proportional to a generation (iteration) number when using GA to solve ORPD problems, and is the maximum agent of constraints, i.e., (16), shown at the , , function bottom of the page, where is to find the maximum of its elements, and is defined as (17)

(16)


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If ( ) and ( ) are all within their limits, . So, function (15) each element of (16) equals zero and just consists of power loss and the CACD, actually. However, , and any deviation of state variable makes consequently. Also, then, the objective function of the infeasible solution is enlarged with such a positive penalty term. Moreover, CGA uses the objective function (15) as the fitness function directly and adopts a tournament selection operhave high fitness. ator [18]. Individuals with a small value of As we know, the optimization result of such an objective function usually makes some state variables close to the boundaries of restrictions. Under real-time circumstances, these variables are likely to exceed their restrictions when the system’s operation mode shifts remarkably at the time interval between two optimization computations that run hourly in a general way. Thus, a menace to the security of power systems comes into being. Moreover, the difference between ORPD computational results and the actual values of power flow also cause some state variables to be over their limits. Therefore, it is necessary to reduce the restriction bandwidths of state variables , which involve nodal voltage magnitudes of PQ buses and reactive power outputs of generators, as follows:

Fig. 1.

Tested network.

Fig. 2.

Hourly total load curves.

(18) where and are initial upper/lower limits of state variable , which are set from the viewpoint of power system, and are adjusted upper/lower limits of state variable in terms of the requirement of real-time ORPD, and is an adjustand ment coefficient varying from 3%-8%. Because , the state variables optimized get more margins from the initial limit boundaries, and the security margin can be augmented with the varying loads under real-time conditions. In addition, there are two measures to avoid the deviations of nodal voltage magnitudes from their limits when the system loads varying remarkably, i.e., to regulate voltages of generators and to shorten the optimization time interval. V. SIMULATION STUDY A real power system is used to evaluate the proposed model, which is shown in Fig. 1. The hourly loads of the tested network are shown in Fig. 2. The tap ratio of transformer T1 is 220( ), and those of other transformers are all 110( ). There are six load buses, at which some shunt capacitor banks are installed. From Bus 1 to Bus 6, the compensation sizes are 4 4.008 Mvar, 2 3 Mvar, 2 3 Mvar, 1 4.8 Mvar, 1 3.3 Mvar, 2 2.016 Mvar, respectively. Three schemes with different UACs and the restriction ranges of nodal voltage magnitudes are listed in Table I. The UACs of scheme #1 are all zero, so its objective function is only the active power loss as a traditional model, i.e., (19) In Table I, and are the upper and lower limits of nodal voltages of PQ buses, respectively. The initial upper and lower limits of nodal voltages are adopted in schemes #1 and #2 directly and, contrastively, the

TABLE I PARAMETERS OF 3 TESTED SCHEMES

reduced limit bandwidths in scheme #3, in which . Furthermore, an ordinary voltage and reactive power control, scheme #4, is compared with aforementioned schemes, whose objective is to minimize the number of control actions as (20) and subjected to the restrictions (6). The upper and lower limits of nodal voltage magnitudes of scheme #4 are 1.07 and 1.00, respectively. Due to limitations, only some dispatch curves of discrete control devices are illustrated here, as shown in Figs. 3–6. Obviously, the dispatch curves of scheme #1 without taking CACD into account are more zigzag than those of the two schemes considering CACD, i.e., schemes #2 and #3. The total operating times of all four schemes within 24 hours of all transformer taps and capacitor banks are shown in Tables II and III, respectively. In scheme #1, the intraday operating times of Tap 1 and Tap 6 are all 14, which exceed 10, the maximum allowable switching operation. However, with

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TABLE II TOTAL OPERATING TIMES OF TRANSFORMER TAPS

TABLE III TOTAL OPERATING TIMES OF CAPACITOR BANKS Fig. 3.

Dispatch curves of Tap 1.

Fig. 4.

Dispatch curves of Tap 2.

Fig. 7. Response curves of voltage at Bus 1.

Fig. 5. Dispatch curves of Capacitor 1.

Fig. 6. Dispatch curves of Capacitor 2.

consideration of CACD, schemes #2 and #3 can curtail the number of control actions to a different extent and eliminate all violations of allowable switching times. In the four schemes, the total operating times within 24 hours of all discrete control

variables are 72, 24, 23, and 32, respectively. It is apparent that the introduction of CACD can avoid excessive operations of control devices when their numbers of control actions are curtailed over 60%. Also, the total numbers of controls corresponding ORPD considering CACD are even less than that corresponding scheme #4, whose objective function is to minimize the operating times. On account of different UACs, the dispatch curves of schemes #2 and #3 are not in step entirely; nevertheless, almost all of intraday operating times of the control devices are equal except that of capacitor banks 1: The number in scheme #2 is 9 whereas is 8 in scheme #3. In the mass, the larger the UACs, the less operations can be anticipated. From Tables II and III, another conclusion is that the different UACs would not change the intraday operating times remarkably if they near their average like in schemes #2 and #3. The response curves of voltages at Bus 1 are shown in Fig. 7. It is many times that voltages come close to or reach the upper limit 1.07 in scheme #1 and scheme #2. However, due to the reduced limit bandwidths, the voltages are always not as near their secure upper limits 1.07 in scheme #3. The lowest/highest voltages of these six load buses in the 24 time intervals are listed represent the lowest and highest in Table IV, where and voltages of these buses in the whole day. Those of scheme #1 are too high, and those of scheme #3 are seemly.


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TABLE IV NODAL VOLTAGE DISTRIBUTIONS

Fig. 9.

Relationship of objective functions with the number of controls.

TABLE V CPU TIME OF MAIN ORPD PROCEDURE

Fig. 8. Hourly active power losses of the network.

Fig. 8 describes the curves of the hourly active power losses of the network under different control schemes. Although the curve of scheme #1 is the lowest one, the hourly active power losses of schemes #1, #2, and #3 are very closed and are less than those of scheme #4 at each time interval. So, the proposed model has very little side effects as far as the capability of power-loss saving is concerned. Tinney et al. [10] give a hypothetical curve to describe the objective function as a function of , the number of controls used to solve an OPF problem, which is shown in Fig. 9. On this that curve, point A corresponds to the smallest set of actions can produce a feasible solution. As more controls are used, the power loss of the system decreases. At point D, all of the controls are used, and the power loss is the absolute minimum, but the number of controls is the largest. Point D is the solution obtained by scheme #1, and point A is the solution by scheme #4. They are not so suitable for real-time ORPD application. When the cost of adjusting control devices is considered, as represents the sum of power loss and shown in Fig. 9, CACD. Here, it is assumed that each control has the same UAC, so the curve CACD is directly proportional to . Then, point B is the minimum of , which is corresponding the solution of schemes #2 or #3. It is obvious that the number of controls at is less than , and the corresponding power loss point B (at point C) is only a little higher than that obtained by (at point D) and is much lower than the soscheme #1 (at point A). That is why the curves lution of scheme #4

of power loss of schemes #1, #2, and #3 are very closed and are less than that of scheme #4. The tests are performed on an Intel PIV 2.0 GHz. The average CPU time listed in Table V is only for the main ORPD procedure, which does not include the input or output time. The solution speed of the proposed algorithm is fast enough.

VI. CONCLUSION Traditional ORPD usually minimizes the active power losses of transmission networks, which causes excessive operations that lead to lower life expectation of devices and more maintenance under real-time circumstances. This paper presents a new mathematical model for ORPD, in which the objective function is to minimize the sum of the active power loss of the whole network and the costs of adjusting control devices represented in power. Simulation tests demonstrate that the solution of the proposed model can 1) decrease active power losses with fewer operations; 2) avoid excessive operations of control devices; 3) keep good voltage profiles of power systems under realtime circumstances. As different unit adjustment costs would not influence the daily operating times remarkably if they near their average, it is, therefore, practicable to normalize the UAC of the same kind of control devices by using their average value, and the rationality of the simplified objective function (14) is validated. Under the circumstances of the electricity market, some market factors make the definition of UAC more complicated. It is a challenge to establish an exact model for ORPD considering various market factors.

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Yong-jun Zhang was born in Guangdong Province, China, on January 19, 1973. He received the B. S. and Ph.D. degrees in electrical engineering from South China University of Technology, Guangzhou, China, in 1995 and 2004, respectively. He is currently a Lecturer with the Electric Power College, South China University of Technology. His research interest is mainly in the application of artificial intelligence technique to reactive power optimization.

Zhen Ren (SM’84) was born in Jiangsu Province, China, on March 16, 1938. He graduated from the Department of Electric Machines, Zhejiang University, Zhejiang Province, China. He is now a Professor and a Doctoral Tutor with the Electric Power College, South China University of Technology, Guangzhou China. His special fields of interest include wavelet theory and its application in power systems, dc transmission, and reliability of power systems.