Cluster voltage control method for “Whole County” distributed photovoltaics based on improved differential evolution algorithm

Abstract

China is vigorously promoting the “whole county promotion” of distributed photovoltaics (DPVs). However, the high penetration rate of DPVs has brought problems such as voltage violation and power quality degradation to the distribution network, seriously affecting the safety and reliability of the power system. The traditional centralized control method of the distribution network has the problem of low efficiency, which is not practical enough in engineering practice. To address the problems, this paper proposes a cluster voltage control method for distributed photovoltaic grid-connected distribution network. First, it partitions the distribution network into clusters, and different clusters exchange terminal voltage information through a “virtual slack bus.” Then, in each cluster, based on the control strategy of “reactive power compensation first, active power curtailment later,” it employs an improved differential evolution (IDE) algorithm based on Cauchy disturbance to control the voltage. Simulation results in two different distribution systems show that the proposed method not only greatly improves the operational efficiency of the algorithm but also effectively controls the voltage of the distribution network, and maximizes the consumption capacity of DPVs based on qualified voltage.

Graphical abstract

Keywords

distributed photovoltaics (DPVs) / cluster partitioning / improved differential evolution algorithm / voltage control / consumption capacity of distributed photovoltaics

1 Introduction

Human dependence on fossil energy such as coal, oil, and natural gas has led to the depletion of natural resources and caused serious environmental problems [1–3]. In recent decades, major countries in the world have gradually realized the importance of finding renewable alternative energy sources and have proposed their own energy strategic plans, aiming to reduce carbon emission and protect the global environment. In China, the government has attached great importance to the development of renewable energy. Solar energy, as the most common energy source in nature, has received extensive attention [4]. With the introduction of relevant policies, the Chinese government has fully launched the “whole county promotion” of distributed photovoltaics (DPVs), which marks the beginning of China’s large-scale continuous construction of DPV projects from the official level.

Solar photovoltaic power generation has the advantages of cleanliness and low carbon emission and has a development potential that cannot be ignored in the construction of low-carbon power grids [5–7]. DPV power generation advocates the principles of nearby power generation, nearby grid connection, and nearby utilization. It can not only satisfy the growing demand of residents but also bring additional revenue to installation users through the grid connection surplus power [8–10]. However, the large-scale DPV grid connection poses new challenges [11,12]. Technically speaking, first, DPVs have the characteristics of many nodes, wide coverage, and locally high-density grid connection [13], resulting in high management cost and difficulty. In addition, the traditional power grid lacks effective consumption capacity to support distributed energy, and the large-scale grid connection of DPVs will lead to bidirectional power flow in the distribution network [14], which may pose risks such as voltage violation [15,16], power fluctuation [17], and power quality degradation [18,19]. Moreover, the volatility of solar energy is also a key factor restricting the promotion of DPVs [19].

To solve the problem of voltage violation caused by DPVs integrated into the power grid, many studies are dedicated to reactive power optimization of the distribution network. Reference [15] classified the methods for solving the voltage violation caused by DPVs and expounded their effectiveness, advantages, and disadvantages respectively. Reference [20] proposed a strategy of setting the active power limit and using the energy storage device to store excessive solar energy to solve the voltage violation of the distribution network. Through the comparative analysis of the impacts on the voltage by different penetration levels of DPVs, Ref. [21] presented schemes to solve the voltage violation problems by employing automatic voltage and power factor control on PV inverters. Reference [19] proposed a strategy based on tuning transformer voltage, reactive power from PV inverters, the energy storage system, and demand response schemes for the three-phase imbalance problem caused by PVs in rural power grids, which could effectively reduce power loss and mitigate the three-phase imbalance. However, these studies were mostly based on centralized control methods, i.e., a central controller controls the reactive power optimization of the entire distribution network, and the optimal global control effect can be obtained through the unified allocation of resources [13]. Under the background of the “whole county promotion”, the characteristics of high penetration rate, many nodes, and wide coverage of DPVs have become increasingly prominent. As a result, the problems of inefficiency and high dependence on the communication system of traditional centralized control methods are further amplified. In the face of new application requirements, it is necessary to explore new control methods.

Based on the discussion of the above issues, the cluster control method emerged, which partitions the entire distribution network into multiple clusters and performs independent control in different clusters, with no interference between clusters. Due to the fact that cluster control method often ignores the overall coordination of the entire distribution network, it is generally impossible to obtain the global optimal solution, but it can obtain the local optimal solution in each cluster region, and the control efficiency can be much higher than that of centralized control methods [22]. Reference [23] proposed a network partitioning approach based on a community detection algorithm, which achieved regional voltage control in a shorter control response time using minimum reactive power compensation and active power curtailment. Reference [24] presented a novel cluster performance indicator based on electrical distance and regional voltage regulation capability and partitioned the distribution network into several clusters. On this basis, a double-layer voltage control strategy was proposed, which combined the cluster autonomous optimization with distributed inter-cluster coordination optimization at different time scales. In Ref. [25], a DPV cluster partitioning method based on spectral clustering algorithm was proposed, and an advanced particle swarm optimization algorithm was employed to achieve two-level voltage optimization in each cluster. Reference [26] proposed a multi-agent deep reinforcement learning (MADRL) framework, which used unsupervised clustering to partition the distribution network into multiple clusters and employed the improved MADRL algorithm for cluster control. However, most existing studies were based on the distribution network with low penetration PVs that were difficult to adapt to the new situation under the background of the “whole county promotion” of DPVs. Therefore, more in-depth exploration is needed.

To address the problems caused by the “whole county” DPVs, this paper partitions the distribution network into multiple clusters, and different clusters exchange terminal voltage information through a “virtual slack bus.” In each cluster, based on the control strategy of “reactive power compensation first, active power curtailment later,” the residual capacity reactive power compensation of PV inverters and DPV active power curtailment are taken as independent variables, and an improved differential evolution algorithm based on Cauchy disturbance is employed to control the voltage. The method proposed in this paper not only greatly improves the operational efficiency of the algorithm but also effectively controls the voltage of the distribution network, improves the power quality, and maximizes the consumption capacity of DPVs based on qualified voltage. Compared with existing works, this paper makes the following contributions:

1) Aiming at the limitation of existing cluster partitioning methods that rarely consider the influence of active power-voltage sensitivity, the proposed cluster partitioning index takes into account the influence of active and reactive power, based on which, a normalized distribution network edge weight matrix is established. At the same time, compared with the clustering methods employed in most papers, the method used in this paper not only performs adaptive clustering for distribution network, but also has a faster speed.

2) The differential evolution algorithm is improved and applied to the distribution network cluster control. Compared with the basic differential evolution algorithm, the improved algorithm shows a better optimization accuracy and stability of the calculation results and can effectively solve the premature convergence problem of the traditional optimization algorithm.

3) A cluster voltage control method based on virtual slack bus is proposed. Different clusters will exchange voltage information through terminal nodes, which can avoid excessive calculation errors due to independent control ignoring the global coordination to a certain extent.

2 Cluster partitioning method

2.1 Quantification of electrical distance

In the distribution network, the electrical distance between nodes represents the degree of electrical coupling, which is often obtained through the sensitivity relationship between voltage amplitude and power. Taking reactive power–voltage sensitivity as an example, it is expressed as

(1)S_{QV}^{ij}=\frac{\mathrm{\Delta }{V}_i}{\mathrm{\Delta }{Q}_j}.

This equation represents the influence of node j on the voltage amplitude change of node i after unity reactive power is injected and reflects the coupling degree between the two nodes to some extent. Active power–voltage sensitivity S_{PV}^{ij} is defined similarly. The power–voltage sensitivity matrix S_{SV}^{ij} is jointly determined by the reactive power–voltage sensitivity matrix and the active power–voltage sensitivity matrix and is given by

(2)S_{SV}^{ij}=\left(S_{PV}^{ij}+S_{QV}^{ij}\right)/2.

Based on the definition of power–voltage sensitivity, the electrical distance is further defined as

(3)d_{ij}=\lg \frac{S_{SV}^{jj}}{S_{SV}^{ij}}= \lg \frac{\mathrm{\Delta }{V}_j}{\mathrm{\Delta }{V}_i},

where d_ij represents the common logarithm of the rate of the voltage change value of node j to the voltage change value of node i after the power change of node j. In other words, as d_ij increases, the influence of node j on node i decreases, and the electrical distance between the two nodes increases. Therefore, d_ij can be used as the basis for the partitioning of the distribution network.

Based on this, the electrical distance is normalized to establish the edge weight matrix, which can be represented as

(4)a_{ij}=\{\begin{array}{cc}{\displaystyle \frac{d_{max}-d_{ij}}{d_{max}-d_{min}}},& d_{ij}\ne 0,\\ 0,& d_{ij}=0,\end{array}

where d_max and d_min are the maximum and minimum elements in the electrical distance matrix, respectively. This approach can limit the elements in the edge weight matrix to the range of [0, 1].

2.2 Cluster partitioning method based on modularity optimization

To realize coordination and complementarity between clusters, it is necessary to satisfy the principle of electricity balance in the distribution network and the demand for electrical connection between nodes. In this case, the electrical connections between nodes in the same cluster are close, while the connections between clusters are loose, so as to facilitate the effective operation of cluster voltage control. Therefore, based on the edge weight index of electrical distance quantification, a cluster partitioning method based on modularity optimization is proposed, whose process is shown in Fig.1. Different from k-means and its derivative algorithm adopted in most relevant studies [27,28], the method proposed in this paper can perform adaptive cluster partitioning according to the attributes of the distribution network without specifying the number of clusters in advance, and has a strong expansibility.

The objective function of cluster partitioning is

(5)max\rho =\frac{1}{2A}\sum _{i,j}\left[a_{ij}-\frac{k_i{k}_j}{2A}\right]\psi (i,j),

where \rho is modularity; a_ij represents each element of edge weight matrix, i.e., the weight of connected edges between nodes i and j; A={\displaystyle \frac{1}{2}\sum _{i,j}{a}_{ij}} represents the sum of all elements of the edge weight matrix; k_i={\displaystyle \sum _j{a}_{ij}} is the sum of the weights of all edges connected to node i; when nodes i and j belong to the same cluster, and \psi (i,j)=1, otherwise \psi (i,j)=0.

The value of modularity usually ranges from 0 to 1. As the value increases, the clustering effect and the accuracy of cluster partitioning increase [29,30]. Some relevant studies have adopted similar cluster partitioning methods based on modularity [23,24].

3 Improved differential evolution algorithm

Cluster partitioning serves the purpose of distribution network control. In this paper, differential evolution (DE) algorithm is employed to achieve voltage control in the distribution network. Its process includes initialization, mutation, crossover, and selection. This algorithm requires very few control variables and has the advantages of fast convergence speed, good robustness, and strong optimization ability. However, when dealing with multi-parameter collaborative optimization in the distribution network, DE might also suffer the problems of weak optimization ability, premature convergence, and poor stability. This paper refers to the Cauchy disturbance method proposed in Ref. [31] and adjusts it, based on which the mutation process of the basic DE is improved.

3.1 Algorithm improvement strategy based on Cauchy disturbance

The difference between improved DE (IDE) and the basic DE is that the Cauchy-disturbance-based mutation and the central-solution-based crossover operation are incorporated. The Cauchy disturbance is based on the Cauchy distribution function, which is given by

(6)f(x; x_0,\gamma )=\frac{1}{\pi }\arctan \left(\frac{x-x_0}{\gamma }\right) +\frac{1}{2}.

When Cauchy disturbance is performed, it is necessary to first select the best bNP solutions in X, denoting them as bX, and then disturb these solutions based on Eq. (6). The disturbance equation is expressed as

(7)dX=bX+\eta \tan (\pi \times (\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}(bNP,n)-0.5)),

where \eta is the Cauchy disturbance coefficient, n is the dimension of the solution set, which can be understood as the number of nodes in the distribution network in this paper, and rand(bNP ,n) is a random number matrix of bNP rows and n columns. Considering that more nodes are involved when centralized control is employed, and fewer nodes are involved in a single cluster when cluster control is employed, the expression of bNP is defined as

(8)bNP=\{\begin{array}{ll}30-\text{round}(15\times \delta ),& \text{Centralized control},\\ 10-\text{round}(5\times \delta ),& \text{Cluster control}.\end{array}

The Cauchy disturbance coefficient can be represented as

(9)\eta =0.005\times (gen/G+\delta ),

where gen is the current number of iterations, and G is the maximum number of iterations. Moreover, \delta in Eqs. (8) and (9) is the adjustment parameter, defined as

(10)\delta =\frac{F_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}-F_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}+\zeta }{F_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}-F_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}+\zeta },

where F_worst is the worst objective function value, F_best is the best objective function value, F_mean is the average value of the objective function, and \zeta is a small disturbance parameter.

Through the above operations, the Cauchy disturbance set dX for the best solution set bX is generated, which can be understood as some solutions near the best solutions in the solution space. The purpose is to search for better solutions near the best solutions in the solution space.

Furthermore, based on bX, dX, and the worst set w{X}^{(-1)} eliminated in the previous iteration, the Cauchy mutation set mX is generated, which is represented as

(11)mX=( X-bX)\cup dX \cup w{X}^{(-1)}.

In the basic DE, the following mutation strategy is generally employed.

(12)m_i=X_{r1} +M\left(X_{r2}-X_{r3}\right),

where M is the mutation coefficient. In this strategy, the three parameters related to the mutation are all derived from the original solution set X. The mutation process is not directional and has a great randomness, which results in a slower convergence speed and a higher probability of falling into local optimum. Therefore, after generating the Cauchy disturbance set dX and the Cauchy mutation set mX, the following strategy is employed for the mutation operation.

(13)m_i=X_i+ M\left(X_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}-X_i+X_{r1}-m{X}_{r2}\right),

where X_best is the best quality solution in X or (X-bX)\cup dX (the best of the two sets); X_{r1} and m{X}_{r2} are derived from solution sets X and mX, respectively. X_best has a certain directionality, which guides the entire population to gradually concentrate toward the current optimal solution and improves the convergence speed and the accuracy of the algorithm. However, due to the greedy selection characteristics of the convergence process, some potential solution information may be discarded, but mX helps to alleviate this problem. In the iterative process, the algorithm restores the discarded solutions with a certain probability, which increases the population diversity [31]. Therefore, the mutation operation is not only related to the initial solution set X, but also to the Cauchy disturbance set dX and the inferior solutions eliminated in the previous iteration. This strategy changes the optimization direction of the algorithm and to some extent prevents the differential evolution process from getting trapped into local optimum prematurely.

3.2 Constraints of differential evolution population size

The population size NP of the DE reflects the size of the population information in the algorithm. As NP increases, the population information increases, which will increase the calculation workload; as NP decreases, the calculation speed increases, which will lead to limited population diversity, and is not conducive to the algorithm to obtain the global optimal solution. Moreover, the setting of the NP value should be relevant to the number of distribution network nodes n, which can reflect the size of the distribution network.

When the DPV active power curtailment is not considered, the optimization independent variable is the reactive power compensation input of the DPV inverter at each node, and the expression of NP is

(14)

When considering the DPV active power curtailment, the optimization independent variables are the reactive power compensation input of the DPV inverter and the active power curtailment of DPVs. There is a collaborative relationship between these two independent variables. Therefore, a larger differential evolution population size must be employed to ensure the collaborative optimization ability of the algorithm. NP is represented as

(15)

When the cluster partitioning is not considered, n represents the total number of nodes in the distribution network; otherwise, n represents the number of nodes in each cluster.

4 Cluster voltage control model

4.1 Optimization strategies of distribution network

4.1.1 Objective function

In this paper, the control objective is to minimize the weighted sum of node voltage deviation, network active power loss, and DPV active power curtailment, which is given by

(16)minF= {\alpha }_1{F}_{\mathrm{d}\mathrm{v}}+{\alpha }_2{F}_{\mathrm{P}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}+ {\alpha }_3{F}_{\mathrm{P}\mathrm{c}\mathrm{u}}+ F_{\mathrm{p}\mathrm{t}},

(17)\{\begin{array}{l}{F}_{\mathrm{d}\mathrm{v}}={\displaystyle \sum _i(U_i-1{)}^2,}\\ F_{\mathrm{P}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}={\displaystyle \sum _{ij}{P}_{{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{ij}},}\\ F_{\mathrm{P}\mathrm{c}\mathrm{u}}={\displaystyle \sum _i{P}_{{\mathrm{c}\mathrm{u}}^i},}\end{array}

(18)F_{\mathrm{p}\mathrm{t}}= {\alpha }_4\left[\sum _i\left(U_{\mathrm{v}\mathrm{u},i}-U_{\mathrm{u}\mathrm{p}}\right)+\sum _j\left(U_{\mathrm{v}\mathrm{l},j}-U_{\mathrm{l}\mathrm{o}\mathrm{w}}\right)\right],

where F_dv is the sum of the squares of the voltage deviation of each node of the distribution network, and U_i is the voltage of each node; F_Ploss is the sum of the active power loss of each branch of the distribution network, and P_{{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{ij}} is the active power loss of each branch; F_Pcu is the sum of DPV active power curtailment, and P_{{\mathrm{c}\mathrm{u}}^i} is the active power curtailment of each DPV node; F_pt is the voltage violation penalty term, and U_{\mathrm{v}\mathrm{u},i} and U_{\mathrm{v}\mathrm{l},j} are the node voltage values where the voltage exceeds the upper threshold and the lower threshold, respectively; U_up and U_low are the set voltage upper threshold and voltage lower threshold; {\alpha }_1, {\alpha }_2, and {\alpha }_3 are the weight coefficients of node voltage deviation, active power loss, and DPVs active power curtailment, respectively; and {\alpha }_4 is the penalty factor for voltage violation.

The function of the voltage violation penalty term is to avoid voltage violation, whose control ability is related to the value of {\alpha }_4. When no voltage violation occurs, F_pt = 0. Set {\alpha }_4\gg {\alpha }_1,{\alpha }_2,{\alpha }_3. On this basis, once the voltage violation occurs, the value of the optimization objective function will be affected greatly, and the occurrence of the node voltage violation can therefore be avoided to a certain extent. In this paper, let {\alpha }_4= 100.

4.1.2 Newton–Raphson power flow

The power balance equation of the Newton–Raphson power flow algorithm in Cartesian coordinates is represented as [32,33]

(19)\left(P_{{\mathrm{G}}^i}-P_{{\mathrm{c}\mathrm{u}}^i}\right)-P_{{\mathrm{L}}^i}= e_i\sum _{j\in i}\left(G_{ij}{e}_j-B_{ij}{f}_j\right)+ f_i\sum _{j\in i}\left(G_{ij}{f}_j+B_{ij}{e}_j\right),

(20)Q_{{\mathrm{G}}^i}-Q_{{\mathrm{L}}^i}=f_i\sum _{j\in i}\left(G_{ij}{e}_j-B_{ij}{f}_j\right)- e_i\sum _{j\in i}\left(G_{ij}{f}_j+B_{ij}{e}_j\right),

(21)U_i=e_i+ \mathrm{j}{f}_i,

where e_i and f_i are the real and imaginary parts of the voltage at node i; P_{{\mathrm{G}}^i} is the maximum active power output value of DPV at node i, P_{{\mathrm{G}}^i}-P_{{\mathrm{c}\mathrm{u}}^i} is the actual power generation of DPV at node i, and Q_{{\mathrm{G}}^i} is the reactive power compensation of DPV inverter output at node i; P_{{\mathrm{L}}^i} and Q_{{\mathrm{L}}^i} are the active power and reactive power of the load at node i; and G_ij and B_ij are the real and imaginary parts of each element Y_ij of the admittance matrix, i.e., conductance and susceptance.

Based on the calculation results of Newton–Raphson power flow, the active power loss P_{{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{ij}} of branch ij can be derived as

(22)P_{{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{ij}}=G_{ij}\left(U_i^2+U_j^2-2U_i{U}_j\cos {\theta }_{ij}\right),

where U_i and U_j are the voltages at both ends of the branch, and {\theta }_{ij} is the phase angle difference between U_i and U_j.

4.1.3 Operation constraints of DPVs

When performing active power curtailment for DPV power generation and invoking the remaining capacity of DPV inverters for reactive power compensation, the following constraints should be considered.

(23)\{\begin{array}{l}0⩽P_{{\mathrm{c}\mathrm{u}}^i}⩽P_{{\mathrm{G}}^i},\\ Q_{{\mathrm{G}}^i}^2⩽S_i^2-{\left(P_{{\mathrm{G}}^i}-P_{{\mathrm{c}\mathrm{u}}^i}\right)}^2,\\ Q_{{\mathrm{G}}^i}=\left(P_{{\mathrm{G}}^i}-P_{{\mathrm{c}\mathrm{u}}^i}\right)\tan \phi ,\\ 0.95⩽\cos \phi ⩽1,\end{array}

where S_i is the capacity of the PV inverter at node i, and \phi is the power factor angle of the inverter.

4.1.4 PV power curtailment rate

In this paper, PV power curtailment rate (PCR) is employed as the consumption capacity index of DPVs. It refers to the rate of the DPV power generation which is curtailed because the distribution network cannot consume it with the current maximum total power generation of DPVs, given by

(24)\mathrm{P}\mathrm{C}\mathrm{R}=\frac{{\displaystyle \sum _i{P}_{{\mathrm{c}\mathrm{u}}^i}}}{{\displaystyle \sum _j{P}_{{\mathrm{G}}^j}}}.

From the above equation, as the value of PCR decreases, the DPVs consumption capacity of the distribution network increases.

4.2 Cluster voltage control method

4.2.1 Voltage control method based on virtual slack bus

In the cluster voltage control model adopted in this paper, different clusters will exchange voltage information through end nodes, which effectively avoids excessive calculation errors due to independent control ignoring the global coordination.

In each individual cluster, independent voltage control will be performed based on Eqs. (16)–(18). Based on the radial structure of the distribution network, the power flow calculation is generally performed from the front-end to the back-end, and there must be a branch connection between two adjacent clusters. Therefore, after the power flow calculation and voltage control are completed in the front-end cluster, the node connected to the back-end cluster will be found from the front-end cluster, and the node and its voltage information will be copied to the back-end cluster as the slack bus of the back-end cluster. Similar to the definition in Ref. [24], this node is named as the “virtual slack bus” for the back-end cluster. The power flow calculation and voltage control of the back-end cluster will be performed based on the voltage of the virtual slack bus. A schematic diagram of this operation is shown in Fig.2. After the cluster is partitioned, the slack bus of cluster b is denoted as b₀, whose voltage can be expressed as

(25)U_{b0}=U_{aj},

where aj is the upstream node connected to cluster b in cluster a.

In addition, the power flow between clusters is ignored, thereby simplifying the calculation process of inter-cluster coupling. In this way, the information exchange between clusters can be largely reduced, which to a certain extent compensates for the decline of global coordination caused by cluster control.

Through the above calculation, the reactive power compensation and active power curtailment of each DPV node are obtained. These data are input into the global power flow model for a global calculation, and the difference between cluster control and centralized control is compared to verify the effectiveness of the cluster voltage control method.

4.2.2 Control method of “compensation first and curtailment later”

When voltage control is executed, the residual reactive power capacity of each DPV inverter is first invoked for reactive power compensation. When the computer program finds that the maximum reactive power compensation cannot solve the voltage violation problem, further active power curtailment is executed until the voltage of each node is limited within the allowable range.

4.2.3 Voltage threshold setting of two control methods

Under global conditions, the allowable voltage deviation is set to 0.05 p.u. When the cluster voltage control strategy is adopted, considering that it ignores the global coordination to a certain extent, the deviation may increase when the results of cluster control are placed under the global condition for checking calculation. As a consequence, when the voltage control is executed within a single cluster, the allowable voltage deviation must be smaller. In this paper, the allowable voltage deviation is set as 0.025 p.u., taking half of that of the centralized control.

5 Case study

In this paper, the cluster partitioning method and the IDE-based voltage control method are analyzed and discussed based on both the IEEE 33-bus system and a county distribution network in Guangdong province, China. The programs involved in this paper are all executed in MATLAB.

5.1 Case I: IEEE 33-bus system

5.1.1 Cluster partitioning

The result of cluster partitioning of the IEEE 33-bus system containing DPVs is shown in Fig.3. When DPVs are low-penetration, \rho = 0.716, and when the penetration is high, \rho = 0.7202, both cases have a good cluster partitioning effect. The black nodes in Fig.3 represent the absence of DPV, the blue nodes represent the DPV nodes under the condition of low penetration of DPVs grid connection, and the red nodes represent the increased DPV nodes under the condition of high penetration of DPVs grid connection compared with the low penetration. When high penetration DPVs are integrated into the power grid, not only are there more DPV nodes, but the inverter capacities of some nodes also increase, which poses greater pressure to the normal operation of the distribution network.

5.1.2 Verification of “compensation first and curtailment later”

Fig.4 shows the convergence path of the objective function value F under low penetration and high penetration conditions of DPVs when using IDE and the centralized control methods, respectively. The black curve corresponds to the low penetration condition. The algorithm converges after 137 iterations, and the horizontal part of the latter part of the black curve is the process of determining whether the convergence condition is satisfied. On the other hand, the red curve corresponds to the high penetration condition. The algorithm converges after a total of 313 iterations, and the front part of the curve corresponds to the case of only invoking the inverter capacity for reactive power compensation. However, even the maximum compensation cannot solve the voltage violation problem in the distribution network, active power curtailment will be further executed, and the voltage of each node can be adjusted back to the threshold range. The sharp drop in the middle of the curve indicates that the computer program determines that only reactive power compensation cannot achieve the control goal. Therefore, active power curtailment needs to be executed. The change of the curve also confirms the necessity of the voltage violation penalty term F_pt, i.e., when the voltage violation occurs, the value of F_pt is very large, resulting in a large F, which is contrary to the goal of Eq. (16). Therefore, in order to obtain a smaller F, the algorithm must be optimized in the direction where there is no voltage violation.

5.1.3 Performance comparison between IDE and other algorithms

The performance of IDE, the basic DE, and two classical optimization algorithms (particle swarm optimization (PSO) and ant colony optimization (ACO)) are compared by employing centralized voltage control mode and cluster voltage control mode, respectively. To obtain enough samples, the four algorithms are independently calculated 10 times under different mode conditions. Counting the number of mode occurrences in each set of results, the results are averaged, and the standard deviation and coefficient of variation (the ratio of standard deviation and average value) are further calculated for each group of objective function values. The weight coefficients values of the objective function are set to the default value: {\alpha }_1={\alpha }_2={\alpha }_3=1. In addition, due to the fact that only reactive power compensation is executed under the low penetration of DPVs grid connection, the calculation process is simple, and the results of different algorithms are not significantly different. Therefore, the DPVs integrated into the distribution network are set as high penetration.

First, the performance of the four algorithms in the centralized voltage control mode are compared, and the results are shown in Tab.1.

According to Tab.1, in the centralized control mode, the ACO has the slowest speed and poor optimization ability, and each calculation reaches the set maximum number of iterations (1000 times). Although PSO is fast, its optimization ability is poor, robustness is low, and the possibility of falling into local optimal is high. The basic DE has a higher optimization accuracy, but the calculation speed is slow. Compared with the first three, IDE has significant performance advantages. It can not only obtain more accurate function values, but also more stable results, which has a good algorithm robustness. In addition, it has a stronger optimization ability, fewer iterations, and a shorter time consumption.

Furthermore, the performance differences of the four algorithms in the cluster voltage control mode are compared, and the results are shown in Tab.2.

According to Tab.2, in the cluster control mode, it is difficult for ACO to obtain the optimal result. ACO has a low robustness, a long optimization time, and the worst performance. PSO is relatively better, but it still has some problems such as low optimization accuracy and robustness. Compared with the basic DE, IDE still has a better optimization accuracy and algorithm robustness, and the obtained results are more stable. However, IDE no longer possesses an efficiency advantage over the basic DE, and the calculation process lasts even longer than the basic DE. The reason for this is that in the cluster control mode, the number of nodes contained in each cluster is greatly reduced compared to the centralized control mode, thus greatly reducing the complexity of the optimization process. As a result, even the basic DE can reach the optimum through fewer iterations. At the same time, IDE includes the Cauchy disturbance operation, which slightly increases the time cost of each iteration.

In summary, IDE has a better algorithm performance than PSO, ACO, and the basic DE, manifested in a better optimization accuracy and stability of calculation results. Although the time of IDE spent in the cluster control mode is slightly longer than that of the basic DE, the time difference is relatively small compared to the overall situation, increasing by only 9.8%. Therefore, the subsequent discussion will be based on IDE.

5.1.4 Comparison of cluster voltage control and centralized voltage control

First, the results of different voltage control methods under low penetration of DPVs grid connection are compared and shown in Fig.5. It can be seen that before the voltage control is executed, the upper limit of the voltage is exceeded only in nodes 16 and 17 in the distribution network, and both the centralized control method and the cluster control method can adjust the voltage back to the setting range. The difference is that the voltage obtained by the cluster control method is lower than that obtained by the centralized control method. One reason for this is that the low penetration of DPVs grid connection has relatively no obvious effect on the overall voltage rise of the distribution network. The other reason for this is that the cluster control ignores the line voltage drop of the entire distribution network, which leads to a tendency of inputting more inductive reactive power to reduce the line voltage when each cluster is individually controlled. Fig.6 shows the power factor of each node, which satisfies the constraints of Eq. (23).

Second, the control results of different voltage control methods under the condition of high penetration of DPVs grid connection are compared. A comprehensive comparison of Tab.1 and Tab.2 indicates that the cluster voltage control method sacrifices the accuracy of the optimization results to a certain extent, and the obtained objective function value F is increased by 14.3% compared with that of the centralized control method. Nevertheless, it greatly improves the computing efficiency conversely, and the time spent is only 4% of the centralized control. Moreover, the node voltage distribution obtained by the two control methods is compared and shown in Fig.7. Before the voltage control is executed, there are many node voltage violations in the distribution network, and therefore the distribution network cannot operate safely and stably. Both voltage control methods can adjust the voltage of each node back to the threshold range, between which, the centralized voltage control method controls the voltage of node 16 to 1.05. For the cluster control method, the node voltage deviation in each cluster is limited within a deviation of 0.025. After considering the global coordination, all node voltages can also be limited within the threshold range, among which, the voltage of the node 16 with the highest voltage is 1.047, which indicates that the cluster control method is effective. In addition, the result shown in Fig.8 suggests that the power factor of each node satisfies the constraints.

Finally, the solution parameters of the two control methods under the condition of high penetration of DPVs grid connection are compared, which include the sum of the square of node voltage deviation F_dv, the sum of the active power loss of the distribution network F_Ploss, and the sum of the active power curtailment F_Pcu. The results are shown in Fig.9. Compared with the centralized control, cluster control will increase the amount of active power curtailment and correspondingly reduce the values of node voltage deviation and distribution network active power loss slightly. Thereinto, PCR is about 5.98% under centralized control but about 7.55% under cluster control.

To sum up, the cluster control method can effectively control the voltage of each node within the threshold range, avoid the occurrence of the voltage violation, and improve the power quality. However, the cluster control method only considers the interaction within a single cluster and ignores the global coordination. Compared with the centralized control method, it will lead to an increase in the amount of active power curtailment and reduce the ability of the distribution network to consume DPVs. Nevertheless, the cluster control method has a short solution time and is much more computationally efficient than centralized control. Therefore, it has a better application potential under the premise that a large number of DPVs with randomness and volatility are connected to the distribution network.

5.1.5 Influence of weight coefficients on consumption capacity of DPVs

Based on the background of the high penetration of DPVs, the IDE and the cluster voltage control method are employed to discuss the effect of changing the DPVs active power curtailment item weight coefficient {\alpha }_3 on the PV consumption capacity of the distribution network. In addition, in order to obtain a lower voltage deviation and active power loss, set {\alpha }_1= {\alpha }_2=1.

Fig.10 shows the distribution of PCR and the voltage violation penalty term F_pt of the distribution network when the value of {\alpha }_3 varies. The data are the average values obtained after excluding the deviation values after 10 independent calculations. As can be seen from Fig.10, when {\alpha }_3=0, PCR is about 48.26%, and nearly half of the DPV power generation is discarded, which indicates that the distribution network has a weak ability to consume DPVs. In the interval of , with the increase of {\alpha }_3 value, PCR decreases. When {\alpha }_3>0.1, PCR remains unchanged, which is about 7.55%. In addition, when {\alpha }_3 takes a value in the [0, 1] interval, F_pt is always 0, which manifests that no voltage violation occurs.

Fig.11 shows the node voltage distribution of the distribution network when {\alpha }_3 is 0, 0.01, 0.1, and 1, respectively. As {\alpha }_3 increases, the voltage of each node gradually increases at first and then tends to be constant. The two curves corresponding to {\alpha }_3=0.1 and {\alpha }_3=1 in Fig.11 almost completely overlap.

Based on the above results, it can be seen that {\alpha }_3 is very important to DPVs consumption capacity of the distribution network. Its role is reflected in that the increase of {\alpha }_3 leads to an increase in the weight value of the active power curtailment of DPVs in the control objective function, thus resulting in the gradual decrease in PCR. However, the influence of {\alpha }_3 is limited. After its value reaches a certain threshold, the control ability of the objective function does not change any more.

5.2 Case II: A 10 kV county distribution network

5.2.1 Cluster partitioning

To further test the application of the IDE and cluster control method in the field of reactive power optimization of the actual distribution network, an example analysis is conducted based on a 10 kV county distribution network system, as shown in Fig.12. The distribution network has a large scale and complex structure, involving a large number of power distribution areas and lines. It contains 109 nodes in total, and the red nodes in Fig.12 are the DPV nodes.

The centralized voltage control is also attempted for the system. However, due to the large number of nodes and variables involved in the system, the calculation process is complicated, and it takes several days for only one calculation, which is not applicable in engineering calculation.

The result of cluster partitioning of the distribution network is shown in Fig.12, where each stained area represents a cluster. The system is partitioned into 13 clusters. The modularity \rho = 0.858, the result of cluster partitioning is reasonable.

5.2.2 Influence of weight coefficients on consumption capacity of DPVs

For this distribution network, Fig.13 shows the distribution curve of PCR and the voltage violation penalty term F_pt of the distribution network wheen the value of {\alpha }_3 varies. The data are the average values obtained after excluding the deviation values after 10 independent calculations. It can be seen that the value of PCR decreases gradually with the increase of {\alpha }_3. When and {\alpha }_3>0.11, voltage violation occurs. The former is caused by excessive PV curtailment, and the latter is caused by excessive emphasis on maximizing the consumption capacity of DPVs, resulting in the fact that the voltage exceeds the upper limit. In the interval of 0.04⩽ {\alpha }_3⩽0.11, there is no voltage violation, and PCR gradually decreases with the increase of {\alpha }_3. When {\alpha }_3=0.11, PCR is about 3.17%. Therefore, {\alpha }_3=0.11 is the most suitable weight coefficient of the optimization objective function for the system under the current DPVs-connected situation. In addition, it is obvious that the value of F_pt tends to increase sharply when voltage violation occurs, and the setting of Eq. (16) can avoid this trend as much as possible.

Fig.14 shows the node voltage distribution without voltage control, where {\alpha }_3 is 0.01, 0.04, 0.08, 0.11, and 0.18, respectively. As shown in Fig.14, a large number of voltage violations occurs in the distribution network without voltage control. With the increase of {\alpha }_3, the voltage of each node gradually increases. If {\alpha }_3 is too small, it may cause excessive PV curtailment and result in a low voltage. On the contrary, if {\alpha }_3 is too large, the excessive pursuit of maximum consumption capacity of DPVs will lead to voltage violation, thus reducing the security and stability of the distribution network.

Overall, the more nodes in the distribution network, the more DPVs are connected, the more complex the coupling relationship is. Therefore, the reactive power optimization of the whole county complex distribution network containing high penetration DPVs may need to sacrifice part of the consumption capacity to ensure the safe and stable operation of the distribution network.

6 Conclusions

Based on the above case studies, the following conclusions can be drawn:

1) The distribution network voltage control method based on the idea of “compensate first and curtailment later” can adjust the algorithm process according to the situation of the voltage violation in the distribution network, and control the voltage of each node of the distribution network within the threshold range.

2) Compared with PSO, ACO, and the basic DE, the IDE based on Cauchy disturbance adopted in this paper has a better algorithm performance, and a better optimization accuracy and stability in calculation results.

3) The cluster control method can obtain a higher computing efficiency by sacrificing only part of the optimization accuracy under the premise that a large number of DPVs with randomness and volatility are connected to the distribution network. Therefore, it has a wide applicability. This advantage of the cluster control method becomes increasingly significant as the complexity of the distribution network increases.

4) The reactive power optimization of the complex distribution network with high penetration DPVs may need to sacrifice part of the consumption capacity to ensure the safe and stable operation of the distribution network.

In the early stage of the “whole county promotion” of DPVs in China, inadequate management, imperfect supporting facilities, and chaotic DPVs construction were widespread, which seriously affected the operation of the distribution network. The method proposed in this paper can effectively control the voltage violation problem caused by a large number of DPVs connected to the distribution network, and therefore improve the power quality and ensure the safe and reliable operation of the power system.

In future work, the proposed method still needs to be expanded and improved. First, MDE consumes more time than the basic DE in the cluster control mode. Therefore, it is necessary to further reduce unnecessary redundant steps in the algorithm and improve the computational efficiency and optimization stability of the algorithm. Next, this paper only considers the reactive power regulation capacity of the inverter, which is far from enough for the comprehensive promotion of the construction of DPVs in the future. Therefore, it is also necessary to further explore the collaborative control of PV inverters, energy storage equipment, step-down transformers and other devices to further reduce the occurrence of PV power curtailment. Finally, the cluster control method should be further improved, such as combining the voltage control method with the soft open point (SOP), aiming to design a more suitable control method for complementary coordination among DPV clusters.

References

[1] Bonthagorla P K, Mikkili S, Optimal P V. Array configuration for extracting maximum power under partial shading conditions by mitigating mismatching power losses. CSEE Journal of Power and Energy Systems, 2022, 8(2): 499–510

[2] Swenson R. The solar evolution: Much more with way less, right now—The disruptive shift to renewables. Energies, 2016, 9(9): 676

[3] Martins F, Felgueiras C, Smitkova M. . Analysis of fossil fuel energy consumption and environmental impacts in European countries. Energies, 2019, 12(6): 964

[4] Wang S C. Current status of PV in China and its future forecast. CSEE Journal of Power and Energy Systems, 2020, 6(1): 72–82

[5] Ahadi A, Miryousefi Aval S M, Hayati H. Generating capacity adequacy evaluation of large-scale, grid-connected photovoltaic systems. Frontiers in Energy, 2016, 10(3): 308–318

[6] Shahsavari A, Akbari M. Potential of solar energy in developing countries for reducing energy-related emissions. Renewable & Sustainable Energy Reviews, 2018, 90: 275–291

[7] Singla P, Duhan M, Saroha S. A comprehensive review and analysis of solar forecasting techniques. Frontiers in Energy, 2022, 16(2): 187–223

[8] Lu Q, Yu H, Zhao K L. . Residential demand response considering distributed PV consumption: A model based on China’s PV policy. Energy, 2019, 172: 443–456

[9] Liu S Y, Bie Z H, Liu F. . Policy implication on distributed generation PV trading in China. Energy Procedia, 2019, 159: 436–441

[10] Wu Y N, Xu M J, Tao Y. . A critical barrier analysis framework to the development of rural distributed PV in China. Energy, 2022, 245: 123277

[11] Gandhi O, Kumar D S, Rodríguez-Gallegos C D. . Review of power system impacts at high PV penetration Part I: Factors limiting PV penetration. Solar Energy, 2020, 210: 181–201

[12] Chen J L, Xu X Y, Yan Z. . Data-driven distribution network topology identification considering correlated generation power of distributed energy resource. Frontiers in Energy, 2022, 16(1): 121–129

[13] Zhang Y J, Qiao Y, Lu Z X. . Voltage control for partially visible distribution networks with high DG penetration. Power System Technology, 2019, 43(5): 1528–1535

[14] Camilo F M, Almeida M E, Castro R. . Multi-conductor line models for harmonic load-flow calculations in LV networks with high penetration of PV generation. Journal of Modern Power Systems and Clean Energy, 2022, 10(5): 1288–1301

[15] Hashemi S, Ostergaard J. Methods and strategies for overvoltage prevention in low voltage distribution systems with PV. IET Renewable Power Generation, 2017, 11(2): 205–214

[16] Nour A M M, Hatata A Y, Helal A A. . Review on voltage-violation mitigation techniques of distribution networks with distributed rooftop PV systems. IET Generation, Transmission & Distribution, 2020, 14(3): 349–361

[17] Xu J, Fu H B, Liao S Y. . Demand-side management based on model predictive control in distribution network for smoothing distributed photovoltaic power fluctuations. Journal of Modern Power Systems and Clean Energy, 2022, 10(5): 1326–1336

[18] Jamal T, Urmee T, Calais M. . Technical challenges of PV deployment into remote Australian electricity networks: A review. Renewable & Sustainable Energy Reviews, 2017, 77: 1309–1325

[19] Yang A Q, Cai Y X, Chen X P. . An adaptive control for supporting village power grid integrating residential PV power generation. Energy Reports, 2022, 8: 3350–3359

[20] LiQ RZhang J C. Solutions of voltage beyond limits in distribution network with distributed photovoltaic generators. Automation of Electric Power Systems, 2015, 39(22): 117–123 (in Chinese)

[21] WangYWen F SZhaoB, . Analysis and countermeasures of voltage violation problems caused by high-density distributed photovoltaics. Proceedings of CSEE, 2016, 36(5): 1200–1206 (in Chinese)

[22] Wang C L, Tao Y G. Locally and globally optimal solutions of global optimisation for max-plus linear systems. IET Control Theory & Applications, 2022, 16(2): 219–228

[23] Zhao B, Xu Z C, Xu C. . Network partition based zonal voltage control for distribution networks with distributed PV systems. IEEE Transactions on Smart Grid, 2018, 9(5): 4087–4098

[24] Chai Y Y, Guo L, Wang C S. . Network partition and voltage coordination control for distribution networks with high penetration of distributed PV units. IEEE Transactions on Power Systems, 2018, 33(3): 3396–3407

[25] Ding J J, Zhang Q, Hu S J. . Clusters partition and zonal voltage regulation for distribution networks with high penetration of PVs. IET Generation, Transmission & Distribution, 2018, 12(22): 6041–6051

[26] Cao D, Zhao J B, Hu W H. . Attention enabled multi-agent DRL for decentralized Volt-VAR control of active distribution system using PV inverters and SVCs. IEEE Transactions on Sustainable Energy, 2021, 12(3): 1582–1592

[27] Liu L J, Zhang Y, Da C. . Optimal allocation of distributed generation and electric vehicle charging stations based on intelligent algorithm and bi-level programming. International Transactions on Electrical Energy Systems, 2020, 30(6): e12366

[28] Sun J, Xu J, Ke D P. . Cluster partition for distributed energy resources in Regional Integrated Energy System. Energy Reports, 2023, 9: 613–619

[29] Newman M E J. Fast algorithm for detecting community structure in networks. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2004, 69(6): 066133

[30] Newman M E J. Modularity and community structure in networks. Proceedings of the National Academy of Sciences of the United States of America, 2006, 103(23): 8577–8582

[31] Deng L B, Sun H L, Zhang L L. . η code: A differential evolution with η Cauchy operator for global numerical optimization. IEEE Access: Practical Innovations, Open Solutions, 2019, 7: 88517–88533

[32] Qian J, Wang P, Chen G G. Improved gravitational search algorithm and novel power flow prediction network for multi-objective optimal active dispatching problems. Expert Systems with Applications, 2023, 223: 119863

[33] WagleRSharma PSharmaC, . Optimal power flow based coordinated reactive and active power control to mitigate voltage violations in smart inverter enriched distribution network. International Journal of Green Energy, 2023: 1–17

RIGHTS & PERMISSIONS

Higher Education Press 2023