Abstract: Reactive power optimization and compensation in power systems are effective means to improve system operating voltage, reduce network losses, and enhance system stability. This paper summarizes the current domestic and international practices in reactive power optimization and compensation, and discusses and studies the existing problems in reactive power compensation and optimization. Keywords: Reactive power optimization, reactive power compensation, nonlinearity, network loss, voltage quality 1 Introduction With the rapid development of the national economy and the increase in electricity consumption, the economic operation of the power grid has received increasing attention. Reducing network losses and improving the transmission efficiency and economic efficiency of the power system are practical problems faced by power system operators and are also one of the main directions of power system research. In particular, with the implementation of the electricity market, transmission companies (power grid companies) can bring higher benefits and profits by effectively reducing network losses and improving the economic efficiency of system operation. Reactive power optimization and compensation in power systems are an important part of the research on the safe and economical operation of power systems. Through the rational allocation of reactive power sources and the optimal compensation of reactive loads in the power system, not only can the voltage level be maintained and the stability of power system operation be improved, but also active power losses and reactive power losses can be reduced, enabling the power system to operate safely and economically. Reactive power optimization calculation, given the system network structure and system load, aims to minimize network losses under various constraints by adjusting control variables (reactive power output and terminal voltage levels of generators, installation and switching of capacitor banks, and adjustment of transformer taps). Reactive power optimization not only ensures the entire network voltage operates near its rated value but also achieves considerable economic benefits, perfectly combining power quality, system operational safety, and economy. Therefore, the prospects for reactive power optimization are very broad. Reactive power compensation can be considered a sub-part of reactive power optimization; it minimizes network losses under various constraints by adjusting the installation location and capacity of capacitors. 2. Principles and Types of Reactive Power Optimization and Compensation 2.1 Principles of Reactive Power Optimization and Compensation In reactive power optimization and compensation, the first step is to determine suitable compensation points. Reactive power load compensation points are generally determined according to the following principles: 1) Based on the characteristics of the network structure, select several central points to control the voltage of other nodes; 2) Based on the principle of local reactive power balance, select nodes with larger reactive power loads. 3) Reactive power stratification and balancing, i.e., avoiding the mutual flow of reactive power between different voltage levels, to improve the economic efficiency of system operation. 4) The reactive power compensation degree in the network should not be lower than the 0.7 standard stipulated by the Ministry. 2.2 Types of Reactive Power Optimization and Compensation Reactive power compensation in power systems includes not only capacitive reactive power compensation but also inductive reactive power compensation. In ultra-high voltage transmission lines (500kV and above), due to the large capacitive charging power of the lines, it is estimated that the capacitive charging power per kilometer at 500kV reaches 1.2 Mvar/km. Therefore, it is necessary to compensate for inductive reactive power in the system to offset the capacitive power of the lines. In practice, inductive reactive power compensation is carried out in 500kV substations of the power grid, and high-voltage reactors and low-voltage reactors are connected in parallel to balance reactive power in the 500kV power grid. 3. Reactive Power Optimization of Transmission and Distribution Networks (Closed-loop Networks) Reactive power compensation in power systems can be discussed from two aspects: the optimization and compensation of reactive power in transmission and distribution networks (closed-loop networks) and distribution lines and users (open-loop networks). 3.1 Objective Function of Reactive Power Optimization The well-known law of equal network loss increment rate in reference [3] states that network loss is minimized when the network loss increment rate is equal. The reactive power compensation point should be set at a point with a small network loss increment rate (reactive power compensation is usually performed when the network loss increment rate is negative). This allows for iterative solving in conjunction with the optimal network loss increment rate to obtain the optimal point. However, this method does not consider the adjustment effect of other control variables, and in actual operation, it is impossible to make the network loss increment rate equal through repeated iterations, as this would involve too much computation and be time-consuming. Meanwhile, domestic and foreign scholars have conducted extensive research on reactive power optimization, proposing numerous optimization algorithms for mathematical models of reactive power optimization. There are two main mathematical models for reactive power optimization. The first model ignores the cost of reactive power compensation equipment, aiming primarily at minimizing system network losses. The objective function for reactive power optimization in the optimized state can be expressed as: The second model aims for optimal system operation, taking into account the reduced network losses due to compensation and the cost of adding compensation equipment. It can be expressed as: Where β is the price per kilowatt-hour, τmax is the annual maximum load loss hours, α and γ represent the annual depreciation and maintenance rate and investment recovery rate of the reactive power compensation equipment, respectively, KC is the price per unit of reactive power compensation equipment, and QC∑ is the total reactive power compensation capacity. Model two considers investment and can be considered a more ideal model. Especially with the implementation of the electricity market, various sectors pursue economic benefits, making consideration of reactive power investment more reasonable. 3.2 Optimization Algorithm Due to the nonlinearity of power systems, the diversity of constraints, the mixture of continuous and discrete variables, and the large computational scale, reactive power optimization in power systems presents certain challenges. Linearizing nonlinear reactive power optimization models is the starting point for many algorithms, such as reactive power flow optimization based on sensitivity analysis, linear programming interior-point methods for reactive power comprehensive optimization, and reactive power flow optimization with penalty terms and interior-point methods. All of these methods obtain the optimal solution by expanding the nonlinear programming model using Taylor series, ignoring terms of second order and above, and establishing a linearized model. However, because these methods ignore terms of second order and above during linearization, their convergence cannot be guaranteed. To improve the convergence of optimization calculations, the idea of ​​penalty functions has been introduced into linear programming, resulting in reactive power flow optimization models and algorithms with penalty terms, which minimize or eliminate the out-of-bounds behavior of dependent variables. However, this cannot fundamentally solve the non-convergence problem after linearization. To address the shortcomings of linear algorithms, some nonlinear algorithms have been proposed, such as mixed integer programming, the constrained polyhedron method, and nonlinear primordial-dual algorithms. Although these methods can theoretically find the optimal solution, the inherent characteristics of reactive power optimization make the calculations complex and time-consuming, and reliable convergence cannot be guaranteed. To improve the convergence and nonlinearity of handling discrete variables (transformer tap adjustment, capacitor bank switching) in reactive power optimization, new artificial intelligence-based methods have been proposed, including genetic algorithms, Tabu search, heuristic algorithms, improved genetic algorithms, distributed computation genetic algorithms, and simulated annealing algorithms. These algorithms have improved the convergence and computational speed of reactive power optimization to a certain extent, and some methods have been put into practical application with good results. However, the following problems still need to be solved in reactive power optimization: 1) Since reactive power optimization is a nonlinear problem, and nonlinear programming often converges to local optima, how to find its global optimum still needs further research and discussion. 2) Since the objective function for minimizing network loss is itself a function of the square of voltage, the final solution obtained in reactive power optimization may have many bus voltages close to the upper limit of voltage, while actual operating departments do not want the voltage to be close to the upper limit. If the voltage constraint range is reduced, it may cause the reactive power optimization to fail to converge or require repeated corrections and iterations to find a solution (requiring manual changes to local constraint conditions). How to unify voltage quality and economic operation indicators still needs further research. 3) Real-time problem of reactive power optimization. With the improvement of the automation level of the power system, high requirements are placed on the real-time performance of reactive power optimization. How to avoid non-convergence and find the optimal solution in a short time still needs further research. 4 Reactive power compensation on distribution lines and reactive power compensation for users 4.1 Reactive power compensation on distribution lines Since the resistance of 35kV, 10kV and some low-voltage distribution lines is relatively large, the power loss and voltage loss caused by reactive power flow on the lines are large. Therefore, the reactive power compensation theory is based on this. The classical line compensation theory believes that the location of the capacitor installation can be seen in the table below. The principle can be briefly described as follows: When the reactive power transmitted by the line is Q, the line length is L, and the compensation distance of each group is x, the compensation capacity of each group is Qx. Qx = Qx / L. When it is assumed that the capacitor is installed in the center of the compensation interval, the reduction in line loss is the greatest. The reactive power flow diagram is shown in Figure 1: When the distance of the installation location of the i-th group of capacitors from the end is: For any group of capacitors, the distance from the end is: xi = L(2i-1)/(2n+1) Its optimal compensation capacity is: nQx = 2nQ/(2n+1) Thus, the data in Table 1 can be obtained. Reactive power compensation for distribution lines can effectively reduce network losses, but its effect is not as good as compensation on the low-voltage side. This conclusion assumes that the reactive power flow is uniformly distributed. If the reactive power flow on the line is non-uniformly distributed, the conclusion will be different. At the same time, when installing capacitor banks on the line, their maintenance and operation are relatively inconvenient, and the investment in compensation equipment is not considered. Therefore, it is recommended to adopt the following method. 4.2 Reactive Power Compensation for Users For enterprises and high-load power users, reactive power compensation is divided into high-voltage centralized compensation, low-voltage centralized compensation, and low-voltage local compensation according to the type. Reference [8] points out that under the condition of equal compensation capacity, low-voltage local compensation reduces line loss the most, and therefore has the best economic benefits. This is understandable. Since low-voltage local compensation compensates for the inductive part of the load, the reactive current flowing through the line and transformer is greatly reduced, and obviously this method achieves the best economic benefits. However, the above does not indicate what the optimal compensation capacity should be, nor does it take into account the investment in reactive power equipment. Reference [6] points out the optimal compensation capacity for open networks. The three common open networks are shown in Figure 2. 4.2.1 Optimal Reactive Power Compensation for Radial Open Networks For users or open networks with distribution transformers, the optimal reactive power compensation capacity for the connection of open networks is derived in detail with reference [6]. Its objective function adopts the second type of objective function. For analysis, a simple derivation is given below: For a radial network, the relationship between the annual network calculation expenditure and reactive power compensation can be expressed as: Since the main research focuses on the impact of reactive power on active power network loss, the impact of active power on network loss can be ignored. Equation (4) can be simplified to the following equation: The compensation QCn,op at the remaining nodes is the same as the above equation. 4.2.2 Optimal reactive power compensation for trunk and chain open networks For trunk and chain open networks, reactive power compensation is set at point i=1. Its QC1,op is the same as that of the radial open network. If reactive power compensation is set at i=1,2, see Figure 2(b) and (c). At this time, the annual expenditure can be expressed by the following formula: Similarly, the expression for QC2,op can be obtained as (for simplification, the voltage of node 2 can be considered to be approximately equal to the voltage of node 1): In the formula, R∑ is the sum of the line resistances of the trunk or chain-connected open network, where R∑=R1+R2. Extending to the network with i nodes and m trunk or chain-connected open network segments, the general formula for calculating the optimal reactive load compensation capacity QCi,op at each point of the open network can be obtained as: The above formula is simple and clear, and combines the well-known equal network loss incremental rate and the optimal network loss incremental rate. The optimal compensation capacity can be obtained in one calculation by using the formula, avoiding the iterative calculation process. For a specific example, see Example 6-2 in reference [3]. In Example 6-2, the optimal compensation capacity is obtained by solving 5 sets of equations and 6 iterations, while the above derivation formula can be used to calculate it in one step. 5 Conclusion Reactive power optimization and compensation in power systems require relatively accurate load data, generator data, transformer parameters, etc. Meanwhile, in the actual operation of power systems, the state of the power system changes continuously; therefore, reactive power optimization and compensation should be applied flexibly according to the actual situation. With the further realization of dispatch automation, distribution network automation, and unmanned substations, algorithms with fast calculation and good convergence are needed. Simultaneously, with the implementation of the electricity market and the gradual maturation of reactive power pricing theory, the theory of reactive power optimization will also change and be further improved accordingly. 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