Abstract : Capacitor optimization configuration and switching is an important aspect of distribution network optimization. This paper reviews the research history and current development status of capacitor optimization configuration and switching, focusing on a detailed review of various algorithms for capacitor optimization switching, analyzing the characteristics and existing problems of each algorithm, in order to promote further development in this research field. Keywords : Distribution network, capacitor configuration, switching algorithm 1 Introduction Capacitors, as important reactive power compensation equipment in distribution networks, are widely used in distribution systems. By rationally configuring and controlling capacitors in the distribution system, the voltage quality of the distribution system can be improved, the power factor can be improved, network losses can be reduced, and system capacity can be increased. The capacitor optimization problem in distribution networks is divided into two main categories: planning and operation. The planning problem mainly determines the installation location, type, and rated capacity of capacitors to minimize investment costs while meeting voltage constraints. The planning problem is also known as the capacitor optimization configuration problem. The operation problem, based on the existing reactive power equipment configuration (the location and maximum capacity of capacitors are determined), determines the switching scheme of switchable capacitor banks according to actual load changes to minimize network losses (energy consumption) or operating costs. The operation problem is also known as the capacitor optimization switching problem. Since the 1950s, the efficiency of parallel capacitors has been a focus of scientific and technological workers, with numerous related documents [1,2]. However, most of these studies focus on planning (i.e., capacitor optimization configuration), while fewer studies focus on operational optimization switching. A review of the research and development of capacitor optimization configuration has been published [3-6]. This paper focuses on summarizing the algorithms for capacitor optimization switching. 2. Capacitor Optimization Configuration The capacitor optimization configuration problem aims to determine the location, size, and number of capacitors in a power distribution system to maximize benefits while satisfying all equality and inequality constraints under various load levels. This is a mixed-integer nonlinear combinatorial optimization problem, where the objective function is non-differentiable. References [3-6] describe the research and development of the capacitor optimization configuration problem. Due to the lack of efficient computational tools, analytical methods were commonly used in the early stages, requiring the objective function to be continuously differentiable. To obtain such an objective function, some simplifications that do not conform to reality are required, such as assuming that the location and size of the capacitors are continuous variables, that the feeder cross-sections are equal, and that the load at each point changes according to a uniform pattern. Despite these assumptions, the objective function obtained in most cases remains quite complex. The main drawback of this type of method is that the optimization results do not match reality. With the development of computer technology, various mathematical programming methods have been applied to solve capacitor optimization problems. Some of these methods can treat the location and size of capacitors as discrete variables, which is a significant improvement over analytical methods. Although these methods can obtain optimal solutions, they are computationally intensive and inefficient. Since the 1990s, heuristic methods, artificial neural networks (ANN)/expert system methods, and methods based on randomized optimization techniques (including genetic algorithms (GA), simulated annealing (SA), and tabu search (TS)) have been applied to capacitor optimization problems in distribution networks. Compared to analytical and mathematical programming methods, heuristic and expert system methods are intuitive, easy to understand, and simple to implement, but they cannot guarantee the optimal solution. Applying artificial neural network methods to capacitor optimization configuration for different distribution system load states requires frequent training samples. For a distribution network of a certain scale, obtaining all possible load patterns is difficult, and training samples also requires a significant amount of time. Methods based on randomized optimization techniques are particularly suitable for solving combinatorial optimization problems because they can effectively handle non-differentiable objective functions. Practice has shown that these randomized optimization methods have better global optimization capabilities than traditional optimization methods, but their convergence and computational speed need further improvement. 3. Capacitor Optimization Switching Distribution network capacitor optimization switching is a control measure used to determine the switching strategy of installed capacitor banks in the distribution system under different load states (for adjustable capacitor banks, the number of banks to be switched must also be determined) to reduce system power or energy loss during operation. Based on the different optimization methods used, it can be divided into the following categories. 3.1 Traditional Mathematical Programming Algorithms 1) Nonlinear Programming In 1982, Grainger et al. first used nonlinear programming to solve the capacitor optimization problem. They used a constant current model to simulate the load and capacitor, constructed a corresponding mathematical model, and carried out a series of research works [7-10]. Since the constructed model [7-9] could not consider the voltage static characteristics of the components, it had certain limitations. In 1985, Grainger [11] pushed the research to a deeper level, introduced the concept of standardized equivalent feeders, and solved the reactive voltage control problem of a more complex distribution network with side branches. He decomposed it into two sub-problems: capacitors and voltage regulators, and solved them using nonlinear programming. In the capacitor sub-problem, planning and operation were considered at the same time. It was determined where to install capacitors of what capacity and how to control these fixed and controllable capacitors to minimize the annual comprehensive cost. That is, based on the consideration of capacitor installation costs, the energy was saved to the maximum extent by optimizing the switching of capacitors. Of the above literature, only literature [9] considers the integer constraint of the capacitor and uses the branch and bound method to solve it; other literatures [7-8, 10-11] treat the position and size of the capacitor as continuous variables, which is inconsistent with the actual situation. 2) Linear programming Deng Youman [12] studied the problem of the number of capacitors to be switched from the perspective of real-time control, derived its successive linear integer programming model, and proposed a dual relaxation solution and successive integration method suitable for the characteristics of capacitor switching in the distribution network. The obtained model is simple, the solution process has no oscillation phenomenon, the convergence is fast, and the amount of calculation is small. However, the optimization result depends on the initial state of the network. For the same system, when the initial value of the capacitor is different, the optimization result is different. At the same time, the error caused by successive integration depends on the single capacity of the capacitor. In literature [13], the author further considered the uncertainty of the predicted load value using the trapezoidal fuzzy number in the fuzzy set, and used successive linear integer programming to optimize the capacitor switching of the three-phase unbalanced system. In [14,15], Teng JH considered how to use common linear programming techniques to achieve real-time optimization control of capacitors in unbalanced and balanced power distribution systems. 3) Quadratic programming Wang JC [16] considered the capacitor optimization problem in asymmetrical power distribution networks, established its mathematical model, decomposed the problem into two sub-problems: capacitor configuration problem and real-time switching problem, and solved it using quadratic integer programming. 4) Dynamic programming Hsu YY et al. [17] proposed a dynamic programming method to determine the optimal switching strategy of feeder capacitors in the next 24 hours. Its goal is to minimize feeder line loss while ensuring voltage quality. The constraints include restrictions on the number of capacitor switching operations. If the switching state of the capacitor is taken as a state variable, dynamic programming will have a dimensionality curse when there are many capacitors. To overcome the dimensionality curse that may occur when using dynamic programming, the authors define the state variable at stage n as the total number of capacitor switching times from stage 0 to n. This method significantly reduces the dimension of online calculation of dynamic programming and speeds up the convergence speed. However, the amount of calculation still increases exponentially with the number of capacitors. When there are many capacitors, it is still not ideal. The shortcoming is that the load is treated as a constant current. Using traditional mathematical optimization theory, the method is mature and has good convergence. The global optimal solution can be obtained theoretically. However, when applying these methods, in order to satisfy the continuous differentiability of the objective function, some simplification assumptions are often required. This will make the optimization results inconsistent with the actual situation, thus limiting its application range to a certain extent. 3.2 Artificial Neural Network Algorithm The biggest feature of the ANN method is that the nonlinear relationship between input and output can be stored in the weights of neurons through sample training. Santoso NI [18] used a two-stage ANN to realize the real-time control of capacitor switching. The first-stage ANN uses the measured value of the bus (power and voltage) and the current tap value of the capacitor as input to predict the load level. The second-stage ANN determines the control strategy according to the load level. Das et al. proposed a method based on artificial neural networks to address the problem that traditional optimization methods are time-consuming and unsuitable for online applications. The research results show that the method is more than 100 times faster than traditional optimization methods [19]. 3.3 Methods based on randomization optimization In recent years, many scholars have applied methods based on randomization optimization techniques (including SA, GA and TS) to the research and production practice of power systems, including capacitor optimization switching in distribution networks. 1) Simulated annealing algorithm (SA) In 1990, Chiang HD [20] used the SA algorithm to determine the installation location, type, capacity and switching scheme of capacitors under different load levels. It considered the actual situation of capacitors, load constraints and operating constraints under various load levels, and performed calculations using a 69-node system as an example. Subsequently, the authors extended the capacitor optimization problem from three-phase symmetrical systems to asymmetrical systems [21]. The difference from the previous paper is that it considers the voltage static characteristics of the load and the capacitor replacement problem, and still uses the simulated annealing method to solve it. Wang Shouxiang et al. [22] also applied the simulated annealing algorithm to solve the three-phase phase switching problem of distribution capacitors. The algorithm considers the actual three-phase imbalance of the distribution system, the daily load change curve of the system, and the actual number of capacitor operations. 2) Genetic Algorithm (GA) Literature [23] believes that the number of capacitor operations is very important to both users and power companies. The number of operations should be considered as a separate objective rather than a constraint. At this time, capacitor optimization switching includes two important objectives: minimizing the daily operating loss of the feeder and minimizing the number of operations. These two objectives have different dimensions and conflict with each other. To solve this problem, the authors proposed an interactive compromise optimal method combined with the genetic algorithm. The significant feature of this method is that it can provide a set of flexible compromise optimal solutions to help system operators determine the best capacitor switching scheme. Liu Li et al. [24] used the genetic algorithm to solve the distribution network capacitor optimization switching problem, overcoming the shortcomings of traditional optimization methods in handling integer constraints of capacitors. It has no requirements for initial conditions and has global optimization capabilities. 3) Tabu Search (TS) Zhang Xuesong [25] proposed to use TS to solve the problem of switching the entire (0-1) optimization plan of the distribution network capacitor group, and compared the efficiency of the same problem with the Tabu search method and the genetic algorithm respectively, and proved that TS has a fast convergence speed. Deng Jixiang et al. [26] made some improvements to the practical application of the Tabu search method based on the above, and used the optimization coding technology in the improved genetic algorithm to process the problem of switching the compensation capacitor in the segment. The disadvantage of TS is that it has a strong dependence on the initial solution. The SA algorithm can generally obtain the global optimal or global suboptimal solution, but the method has a strong dependence on the parameters and annealing scheme and a large amount of computation; GA has the advantages of simple use, strong robustness and easy parallelization, but it still faces many problems, such as slow calculation speed and different optimization results when different gene strings are selected; the TS method can effectively obtain the global optimal solution or local optimal solution, but the depth control of the Tabu table is still the main difficulty in its application. 3.4 Hybrid Algorithm In recent years, some researchers have also devoted themselves to the research of combining traditional mathematical programming method and intelligent method. Reference [27] also uses a similar method of combining ANN with dynamic programming to solve the capacitor switching problem. This method is implemented in three steps. First, historical load data is collected and the optimal decision is determined offline using dynamic programming. Second, the load curve is classified using two classification algorithms of ANN. In each category, the optimal decision obtained in the previous step is averaged to obtain the pre-schedule table. Finally, the decision with high credibility in the pre-schedule table is fixed, and the decision with low credibility is optimized using dynamic programming to obtain the final control scheme. The first two steps are carried out offline, and the last step is applied online. When applied online, the calculation speed is greatly accelerated because the number of state variables is greatly reduced. The characteristic of ANN technology for control is that it is fast in online calculation and is particularly suitable for real-time control. The long offline training time of ANN is not the main disadvantage. The key problem is the acquisition of training samples. Since the load of the distribution network changes frequently, a large number of samples are needed to train the ANN for each load mode, which limits its practicality. Miu KN and Chiang HD [28] studied the application of GA in the capacitor optimization configuration and control problem of three-phase unbalanced distribution network and constructed a two-level optimization model. The first-level optimization uses a genetic algorithm to determine a feasible solution space, and the second-level optimization uses a heuristic algorithm based on sensitivity analysis, using the feasible solution space obtained in the previous level as the initial value for the search to continue the optimization. This method takes less time than simply using GA, but the accuracy of the solution is reduced. Reference [29] proposed an improved GA/TS hybrid algorithm and used it for real-time switching of capacitors in the distribution network. The initial value was solved using GA, and the optimal solution was obtained using TS. Chen Xingying et al. [30] established a reactive power optimization control problem for the distribution network with the goal of minimizing network loss from the perspective of economic operation. The mathematical constraints mainly emphasized the limitation of the number of capacitor switching, and the fuzzy dynamic programming method was used to calculate the optimal switching problem of capacitors in the distribution network. The hybrid algorithm can give full play to the advantages of each method and avoid its disadvantages. From this point of view, this type of algorithm has great development potential. However, how to "take the best from each algorithm" and give full play to their respective advantages still needs further research. 4 Conclusion The problem of optimal configuration and switching of capacitors in the distribution network is a large-scale nonlinear combinatorial optimization problem. Traditional mathematical optimization theories offer mature methods with good convergence, theoretically yielding global optimal solutions. However, most of these methods require the objective function to be continuously differentiable, necessitating simplification assumptions that can lead to discrepancies between the optimization results and actual conditions, thus limiting their applicability. ANN-based algorithms can produce results quickly, but their accuracy depends on the sample size, which is difficult to obtain and requires significant training time. SA-based algorithms can theoretically achieve global optimal solutions, but suffer from parameter dependence and high computational cost, limiting their application to capacitor optimization and planned switching where computational speed is less critical. GA, while possessing global optimization capabilities, suffers from convergence and computational limitations. TS-based algorithms face challenges in determining the list size; lists that are too large or too small negatively impact global optimization capabilities. Fuzzy mathematics and expert system methods rely on the development of other technologies. In summary, each method has its advantages and disadvantages. For the problem of capacitor optimization and planned switching in distribution networks, the requirements for computational speed are not very demanding, and computational speed can be sacrificed to obtain a high-precision solution. 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