Distributed Integrated Synthetic Adaptive Multi-Objective Reactive Power Optimization

Abstract

Reactive power is the core problem of voltage stability and economical operation in power systems. Aiming at the problem that multi-objective normalization reactive power optimization function is dependent on weight, an integrated synthesis of adaptive multi-objective particle swarm optimization (ISAMOPSO) is proposed to achieve weight adaptive. Through seven test functions and three algorithm comparison experiments, it is proved that the ISAMOPSO algorithm has stronger global search capability and better convergence. Considering the optimal access position and capacity of distributed generation (DG), the ISAMOPSO algorithm is used for three-objective reactive power optimization. Finally, the results indicate that the ISAMOPSO algorithm can not only provide a variety of optimization schemes to meet different needs, but also realize dynamic reactive power optimization, which further proves that the algorithm can provide effective technical support for solving reactive power optimization problems in practical engineering.

1. Introduction

With the proposal of the smart grid, it has become a developing trend that distributed generation (DG) gradually joins and becomes an essential part of the power distribution system [1,2]. Experts in the power industry generally agree that the adoption of DG technology is the main way to save money, reduce primary energy consumption, and improve the flexibility and stability of power systems [3,4,5,6].

However, the distribution network with the DG is a multi-power structure, and the capacity and location of the DG make the grid situation complicated when it is connected. The power flow and node voltage in the line will also be changed with the change of the power output of the DG [7,8], and even reverse power flow occurs. The distribution network is at risk of voltage overruns and voltage fluctuations [9]. Therefore, it is of great significance to study the control method of distributed generation in a distribution network to improve the reliability of the distribution network.

Since the multi-objective reactive power optimization problem of distribution network with DG is a very complex non-linear problem, domestic and foreign scholars have proposed various simplified mathematical models and optimization algorithms to solve it. Many heuristics have been applied to reactive power optimization problems, such as genetic algorithms (GA) [5,10], particle swarm algorithms (PSO) [6,11,12], and deep learning algorithms [13]. Meanwhile, many scholars have improved these algorithms, which have improved the speed and accuracy of the algorithms to solve reactive power optimization problems to different degrees [14,15]. Considering that the influence of distributed generation joining the distribution network is complex and multi-layered, some mathematical models of multi-objective optimization has been proposed. Taking Reference [16] and Reference [17] as examples, the optimization objectives are to reduce the system power consumption, improve the node voltage level and increase the voltage stability margin of the distribution network. After assigning a set of weights to several functions to be optimized and transforming them into a single objective function, the GA algorithm and PSO algorithm are used to solve the problem, respectively. The most common method in dealing with multi-objective problems is the simplification of multiple objectives, such as the weighting method, the ε-dominant method, and the affiliation method. The single solution is very dependent on the selection of weights, especially in the case of being unable to subjectively judge the relationship between various objectives. It is difficult to determine the weight of each function to be optimized. In contrast, multi-objective optimization can effectively solve the multi-objective optimization problem, multiple solutions can be obtained after the iteration of the optimization algorithm, and decision-makers can flexibly select schemes according to the actual situation [18,19,20]. In Reference [21], a static reactive power optimization model with minimum network loss and voltage deviation is established and solved by the multi-objective particle swarm optimization (MOPSO) algorithm. Based on Reference [21], Reference [22] also considered the maximization of static voltage stability margin, established a three-dimensional static reactive power optimization model and solved it by non-dominated sorting genetic algorithm II (NSGA-II). However, the power output of DG varies with time, so it is more practical to consider dynamic reactive power optimization. Reference [23] established a dynamic reactive power optimization model for the distribution networks and solved it using an improved particle swarm algorithm, which was accompanied by excessive iterations and premature convergence.

Aiming at the problem that single-objective reactive power optimization can only provide a single optimization scheme, an improved particle swarm optimization (PSO) algorithm is proposed to achieve a comprehensive and adaptive three-objective optimization of distribution networks with the DG. In addition, it can provide a variety of control strategies to achieve dynamic optimization.

The contributions are highlighted as follows:

• A new optimization method (called ISAMOPSO) is proposed based on the MOPSO.
• The proposed ISAMOPSO’s performance is evaluated by testing the selected benchmark functions.
• The active power loss, voltage deviation, and reactive power compensation cost are taken as the optimization objectives, and the multi-objective reactive power optimization mathematical model of the distribution network of a distributed generation is established.
• The new proposed ISAMOPSO approach is applied to solve the reactive power optimization in the distribution networks with DG.

The remaining paper is structured as follows: Section 2 presents the model description of the distribution networks with DG. Section 3 proposes the ISAMOPSO and tests its validation and performance. Section 4 presents two case simulations on the IEEE-33 node system with DG to verify the effectiveness of the proposed ISAMOPSO. Section 5 presents the conclusions.

2. Multi-Objective Reactive Power Optimization Model

2.1. Objective Function

The three most important objective functions in a power system are selected as the objective functions for reactive power optimization, which are active power loss, voltage deviation, and reactive power compensation cost.

2.1.1. Active Power Loss

The expression for active power loss is as follows:

$$\min f_{Ploss}={\displaystyle \sum _{k=1}^{N_{bra}}{G}_{k\left(i,j\right)}\left[V_i^2+V_j^2-2V_i{V}_j\cos \left({\theta }_i-{\theta }_j\right)\right]}$$

(1)

where $f_{Ploss}$ is active power loss, $N_{bra}$ is the total branch number, $G_{k\left(i,j\right)}$ is the conductance of branch $k$ connecting nodes $i$ and $j$, voltage amplitude and voltage phase of the corresponding nodes $V$ and $\theta$, respectively.

2.1.2. Voltage Deviation

The expression for voltage deviation is as follows:

$$\{\begin{array}{l}\min f_{\mathsf{\Delta }V}={\displaystyle \sum _{i=1}^{N_L}\left(\frac{V_i-V_i^{spec}}{\mathsf{\Delta }{V}_i^{\max }}\right)}\hfill \\ \mathsf{\Delta }{V}_i^{\max }=V_i^{\max }-V_i^{\min }\hfill \\ V_i^{spec}=\frac{V_i^{\max }+V_i^{\min }}{2}\hfill \end{array}$$

(2)

where $N_L$ is the number of load nodes, $\mathsf{\Delta }{V}_i^{\max }$ is the maximum allowable voltage deviation of node $i$, and $V_i^{spec}$ is the expected voltage amplitude of node $i$.

2.1.3. Reactive Power Compensation Cost

The expression for reactive power compensation cost is as follows:

$$\min f_{\cos t}=f_{\cos tC}+f_{\cos tDG}={\displaystyle \sum _{s\in N_C}{C}_{CAPs}\left|Q_{qs}\right|}+S_{DG}\left(\frac{r{\left(1+r\right)}^n}{{\left(1+r\right)}^n-1}{C}_1+C_2\right)$$

(3)

where $s\in N_C$ is the adjustable point of reactive power compensation equipment, $Q_{qs}$ and $C_{CAPs}$ are the actual input capacity and unit capacity investment of reactive power compensation equipment of node $S$, respectively. When $C_{CAPs}=1$, the total investment of reactive power compensation equipment is to minimize the investment of reactive power compensation, $S_{DG}$ is the capacity of distributed power, $r$ is the discount rate, n is the service life of distributed power, $C_1$ is the investment cost of distributed power, and $C_2$ is the operation cost of distributed power.

2.2. Constraint Condition

2.2.1. Equality Constraint

The mathematical expression of the power flow balance equation of active power and reactive power after DG access is shown below.

$$\{\begin{array}{l}{P}_{Gi}+P_{DGi}-P_{Li}=V_i{\displaystyle \sum _{j=1}^{Nb}{V}_j\left(G_{ij}\cos {\delta }_{ij}+B_{ij}\sin {\delta }_{ij}\right)}\hfill \\ Q_{Gi}+Q_{DGi}+Q_{Ci}-Q_{Li}=V_i{\displaystyle \sum _{j=1}^{Nb}{V}_j\left(G_{ij}\sin {\delta }_{ij}-B_{ij}\cos {\delta }_{ij}\right)}\hfill \end{array}$$

(4)

where $P_G$ and $Q_G$ are the active power and reactive power injected into the generator, $P_L$ and $Q_L$ are the active power and reactive power consumed by the load, $B$ is the admittance, $G$ is the conductance, $P_G$ is the active output of DG, and $Q_{DG}$ and $Q_C$ are the reactive power capacity of DG and the reactive power capacity of reactive power compensation equipment.

2.2.2. Inequality Constraint

Where $V_{Gi\min }$ and $V_{Gi\max }$ are the minimum and maximum of the terminal voltage, and $N_G$ is the number of generator nodes. $K_{j\min }$ and $K_{j\max }$ are the lower and upper limits of the $j$ adjustable transformer ratio, and $N_K$ is the number of adjustable transformers. $C_{k\min }$ and $C_{k\max }$ are the lower limit and upper limit of $k$ compensation capacity of reactive power compensator, and $N_C$ is the number of nodes of reactive power compensation equipment. $Q_{DGg\min }$ and $Q_{DGg\max }$ are the lower and upper limits of the reactive power capacity of DG, and $N_{DG}$ is the number of nodes compensated by DG. $Q_{Gi\min }$ and $Q_{Gi\max }$ are the lower and upper limits of generator output reactive power, and $N_G$ is the number of generator nodes. $V_{l\min }$ and $V_{l\max }$ are the lower and upper limits of the voltage amplitude of the load node, and $N_B$ is the number of load nodes.

$$V_{Gi\min }\le V_{Gi}\le V_{Gi\max },\left(i=1,2,\cdots ,N_G\right)$$

(5)

$$K_{j\min }\le K_j\le K_{j\max },\left(j=1,2,\cdots ,N_K\right)$$

(6)

$$C_{k\min }\le C_k\le C_{k\max },\left(k=1,2,\cdots ,N_C\right)$$

(7)

$$Q_{DGg\min }\le Q_{DGg}\le Q_{DGg\max },\left(g=1,2,\cdots ,N_{DG}\right)$$

(8)

$$Q_{Gi\min }\le Q_{Gi}\le Q_{Gi\max },\left(i=1,2,\cdots ,N_G\right)$$

(9)

$$V_{l\min }\le V_l\le V_{l\max },\left(l=1,2,\cdots ,N_B\right)$$

(10)

3. ISAMOPSO Algorithm

To improve the global searching ability of PSO, the learning factor and inertia weight of PSO are improved adaptively.

3.1. The Basic Principle of the PSO Algorithm

The core operation of the PSO algorithm is the nearest individual away from the particle to avoid a collision and guides particles to move toward the center of the target and group. The core principle of PSO mainly includes the update of speed and position. The update process is expressed as:

$$v_i^{k+1}=\omega v_i^k+c_1{r}_1(x_{pi}^k-x_i^k)+c_2{r}_2(x_{gi}^k-x_i^k)$$

(11)

$$x_i^{k+1}=x_i^k+v_i^{k+1}$$

(12)

where $k$ is the number of iterations, $i$ is the particle number, $c_1$ and $c_2$ are learning factors, and $r_1$ and $r_2$ are independent pseudorandom numbers, obeying uniform distribution on $[0,1]$. $\omega$ is the inertia weight. $v_i^k$ is the search speed for a particle $i$ at the kth iteration in the range $\left[-V_{\max },V_{\max }\right]$. $x_i^k$ is the position of a particle $i$ at the kth iteration, and its range is $\left[-x_{\max },x_{\max }\right]$. $x_{pi}^k$ is the optimal position searched so far by a particle $i$ in the kth iteration. $x_{gi}^k$ is the optimal position in the kth iteration of the entire particle swarm.

3.2. Introducing Random Black Hole Strategy

The basic principle of introducing the black hole mechanism into PSO in [24] is to regard the particle population in the solution space as a planet in space, the fitness value of the particle as an attraction, and the real optimal solution as a black hole, and update the velocity and position of the particle according to the black hole mechanism. The principle is shown in Figure 1.

During the iteration process, for each dimension d of each particle, a black hole is randomly generated in the region with the dimension $p_{gd}$ corresponding to the global optimal solution as the center and r is the radius. A constant threshold $p$ is set as the ability of black to capture the corresponding dimension $x_{id}$ of the particle threshold $p\in [0,1]$. Set a random number $l\in [0,1]$ for $x_{id}$. If the random number $l\le p$, $x_{id}$ is captured by a black hole; if the random number $l>p$, after introducing the random black hole strategy, the particle update formula is:

$$x_{id}=\{\begin{array}{l}{x}_{id}+v_{id},l>p\hfill \\ p_{gd}+2r\left(r_3-0.5\right),l\le p\hfill \end{array}$$

(13)

where $r_3$, $l$ and $p$ are random numbers with uniform distribution on [0,1]. $l$ is the probability of a particle $x_{id}$ appearing on its corresponding dimension, $r$ is the radius of the black hole.

3.3. Adaptive Adjustment of Inertia Weight and Learning Factor

The ordinary MOPSO algorithm can easily fall into the local optimum. By changing the values of $\omega$, $c_1$ and $c_2$ in the iterative process of the algorithm, it can effectively avoid the premature end of the iterative process without finding the optimal solution. The calculation method is:

$$\omega ={\omega }_0+\frac{r_4}{2}{\left(1-{\omega }_0\right)}^{\frac{t}{T_{\max }}}$$

(14)

$$\{\begin{array}{l}{c}_1=2{\cos }^2\left[\frac{\pi t}{2T_{\max }}\right]\hfill \\ c_2=2{\cos }^2\left[\frac{\pi \left(T_{\max }-t\right)}{2T_{\max }}\right]\hfill \end{array}$$

(15)

where ${\omega }_0$ takes the constant on $[0,0.5]$, $r_4$ takes the random number on $[0,1]$ which obeys uniform distribution, and $T_{\max }$ is the maximum iteration number. Additionally, $t$ is the current iteration number.

3.4. The Diverse Selection of Leader Particles

The swarm induction strategy is introduced into PSO, and a disturbance is imposed on the population to increase the diversity of swarms. These disturbed particles as leading particles may lead the population to jump out of the local optimum to enhance its development ability. The individuals in the new sensing population are compared with the non-dominated individuals of the current generation, and the high-quality particles are selected for the next generation of particle swarm, and the formula is:

$$X_{tur}=X_{new}+\beta V_{\max }\cdot sign\left(2\left(rand\left(1,N\right)-0.5\right)\right)$$

(16)

where $X_{tur}$ is the particle position of the new sensing population, $X_{new}$ is the particle position of the next generation population, $\beta$ is the disturbance amplitude obeying uniform distribution $[0,1]$, $V_{\max }$ is the maximum flight velocity of the particle, and the size of the perceived group is $N$.

The particle with the largest fitness value was selected as the lead particle to guide the evolution of the next generation of the population. When a leading particle is selected, all particles in the non-dominant solution set are dynamically calculated according to the equations, which are expressed as:

$$\{\begin{array}{l}fitness=\frac{1}{{\displaystyle \sum _{i=1}^M{w}_i{f}_i}}\hfill \\ w_i=\frac{{\lambda }_i}{{\displaystyle \sum _{i=1}^M{\lambda }_i}},{\lambda }_i\in \left(0,1\right)\hfill \end{array}$$

(17)

The number of objective functions is $M$, and the value of the ith objective function is $f_i$, and the ${\lambda }_i$ obeys the uniformly distributed random number $[0,1]$.

3.5. Cyclic Elimination Strategy of Crowding Distance

Before sorting the crowding distance, the R_t population is sorted non-inferiorly and divided into different dominating levels. By calculating the crowding distance instead of one-time deletion, the crowding distance is gradually deleted from small to large using the idea of a dynamic cycle. The schematic diagram of the operation process is shown in Figure 2.

$R_t$ is divided into different classes according to the dominance relationship, where the $F_1$ class is the highest, and the class decreases in the following order. If the number of members of $F_1$ is less than $N$, then all the members of $F_1$ are selected for the population P_t+1, and the remaining members are selected from $F_2$, $F_3$... until the size of $N$. The number of members in $F_3$ is greater than the number of remaining members in $P_{t+1}$, so we need to perform crowding distance sorting at the same level of $F_3$ and select the number of members into $P_{t+1}$. The crowded distance ranking is used to calculate the density of other solutions around a solution, which is calculated as shown in (18).

$$P_i={\displaystyle \sum _{m=1}^M\frac{P_{(i+1)·f_m}-P_{\left(i-1\right)·f_m}}{f_{\max ·m}-f_{\min ·m}}}$$

(18)

Only one solution is deleted at each time, and the upper limit of deletion is determined immediately after deletion until the scale of $P_{t+1}$ is $N$. $N_{delete}$ is the number of eliminations, which is expressed as:

$$N_{delete}=N_1+N_2+\cdots +N_{\mathrm{k}}-N$$

(19)

where $N_{\mathrm{k}}$ is the number of k-level non-inferior solutions, $N$ is the size of the population $P_{t+1}$, and the sum of the current number of k-level non-inferior solutions is just greater than $N$ to achieve the elimination condition.

3.6. ISAMOPSO Algorithm Performance Verification and Analysis

The ISAMOPSO algorithm is used for the complex problem which is non-linear, multivariable, and under multi-constraint conditions. Therefore, the performance of the algorithm is evaluated by a comprehensive evaluation index that reflects the convergence and distribution uniformity of the algorithm, including the Inverted Generational Distance (IGD) and the Hypervolume (HV).

IGD is expressed as:

$$IGD=\frac{{\displaystyle \sum _{x^{*}\in P^{*}}d\left(x^{*},P\right)}}{n}$$

(20)

where $P$ is the real Pareto optimal solution set, $P^{*}$ is the non-dominated solution set solved by the algorithm, $d\left(x^{*},P\right)$ represents the minimum Euclidean distance between the solution on the Pareto optimal surface and the individual in the non-dominated solution set solved by the algorithm, and $n$ is the cardinal number of $P^{*}$. The smaller the IGD value, the closer the non-dominated solution set is to the actual Pareto optimal solution set.

HV is expressed as:

$$HV=\lambda {\cup }_{\mathrm{i}=1}^{\left|S\right|}{v}_i$$

(21)

where $\lambda$ represents the Lebeg measure, $v_i$ represents the hypervolume composed of reference points and non-dominated individuals, and $S$ represents the non-dominated solution set. The larger the HV value, the stronger the dominance of a single solution, and the closer it is to the Pareto boundary.

During the experiment, the performance of ISAMOPSO is compared with those of NSGA-II [25], MOPSO [26], and speed-constrained multi-objective particle swarm optimization (SMPSO) [27]; the swarm size is set to 100; the Maximum iteration is set to 10,000; and DTLZ1-DTLZ7 as test functions are run independently 30 times on the algorithm. The IGD performance index values and the mean values and standard deviations are listed in

Table 1. The result of the mean and standard deviation of IGD using the test function in different algorithms.

Test Function	Index	Algorithm
		NSGAII	MOPSO	SMPSO	ISAMOPSO
DTLZ1	Mean	2.6664 × 10−2	3.1767 × 10−2	3.0328 × 10−2	2.6595 × 10−2
	Std	1.28 × 10−3	1.75 × 10−2	1.20 × 10−2	1.09 × 10−3
DTLZ2	Mean	6.8392 × 10−2	7.7528 × 10−2	6.6816 × 10−2	6.3912 × 10−2
	Std	3.38 × 10−3	8.42 × 10−3	2.30 × 10−3	1.79 × 10−3
DTLZ3	Mean	6.8458 × 10−2	1.2056 × 10−1	6.8460 × 10−2	9.9426 × 10−2
	Std	2.19 × 10−3	7.13 × 10−2	2.63 × 10−3	4.33 × 10−2
DTLZ4	Mean	6.6723 × 10−2	1.7982 × 10−1	1.1460 × 10−1	1.5361 × 10−1
	Std	3.02 × 10−3	1.15 × 10−1	1.06 × 10−1	1.64 × 10−2
DTLZ5	Mean	5.7555 × 10−3	7.7760 × 10−3	5.3310 × 10−3	4.8635 × 10−3
	Std	3.61 × 10−4	6.66 × 10−4	4.20 × 10−4	1.35 × 10−4
DTLZ6	Mean	5.7767 × 10−3	8.5702 × 10−3	5.3689 × 10−3	4.6755 × 10−3
	Std	2.82 × 10−4	9.20 × 10−4	2.09 × 10−4	9.00 × 10−5
DTLZ7	Mean	7.9676 × 10−2	1.0997 × 10−1	1.0327 × 10−1	8.2313 × 10−2
	Std	6.02 × 10−3	9.98 × 10−2	3.81 × 10−2	6.18 × 10−3

. It can be found that the mean value of IGD calculated by the ISAMOPSO algorithm is the smallest compared with the result of the other algorithm in the test functions DTLZ1, DTLZ2, DTLZ5, and DTLZ6, and the mean value of IGD calculated by the ISAMOPSO algorithm is slightly worse than that of SMPSO in the test function DTLZ3, and test functions, DTLZ4 and DTLZ7 are slightly worse than that of NSGAII. To more intuitively show the convergence and distribution of the optimal solution set obtained by various algorithms. The result with the best IGD value is selected to show the comparison between the approximate Pareto optimal frontier obtained by the four algorithms on all test functions and the real Pareto optimal front, as shown in

Table 2. Pareto front comparison results of the test functions in different algorithms.

Function	NSGAⅡ	MOPSO	SMPSO	ISAMOPSO
DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7

. The results show that the non-dominated solution obtained by the ISAMOPSO algorithm is closer to the real Pareto optimal solution set than that obtained by other algorithms.

The HV performance index values and the mean values and standard deviations are listed in

Table 3. The result of the mean and standard deviation of HV using the test function in different algorithms.

Test Function	Index	Algorithm
		NSGAII	MOPSO	SMPSO	ISAMOPSO
DTLZ1	Mean	8.2429 × 10−1	7.9694 × 10−1	8.1700 × 10−1	8.1891 × 10−1
	Std	3.74 × 10−3	4.61 × 10−2	5.11 × 10−3	3.13 × 10−2
DTLZ2	Mean	5.3293 × 10−1	5.0980 × 10−1	5.3286 × 10−1	5.3832 × 10−1
	Std	5.53 × 10−3	1.29 × 10−2	3.90 × 10−3	3.56 × 10−3
DTLZ3	Mean	5.0795 × 10−1	4.6962 × 10−1	5.2880 × 10−1	5.3488 × 10−1
	Std	5.30 × 10−2	6.38 × 10−2	6.18 × 10−3	4.64 × 10−3
DTLZ4	Mean	5.4192 × 10−1	4.9007 × 10−1	5.2322 × 10−1	5.2573 × 10−1
	Std	3.29 × 10−3	4.11 × 10−2	3.85 × 10−2	6.46 × 10−3
DTLZ5	Mean	1.9930 × 10−1	1.9691 × 10−1	1.9944 × 10−1	1.9965 × 10−1
	Std	1.70 × 10−4	2.61 × 10−3	2.09 × 10−4	1.25 × 10−4
DTLZ6	Mean	1.9958 × 10−1	1.9781 × 10−1	1.9962 × 10−1	2.0015 × 10−1
	Std	1.30 × 10−4	4.58 × 10−4	1.74 × 10−4	3.57 × 10−5
DTLZ7	Mean	2.7128 × 10−1	2.6964 × 10−1	2.6817 × 10−1	2.7167 × 10−1
	Std	1.68 × 10−3	1.17 × 10−2	2.67 × 10−3	2.57 × 10−3

. The HV means calculated by the ISAMOPSO algorithm on the test function DTLZ1-DTLZ7 are greater than or approximately equal to the HV mean calculated by other algorithms, indicating that the dominant nature of the ISAMOPSO algorithm is stronger than that of its three algorithms, while the individual dominance of NSGA-II is stronger than the dominance of each individual obtained by the other two algorithms. Moreover, the Pareto optimal solution set obtained by the ISAMOPSO algorithm is closer to the real Pareto optimal front, indicating that the convergence and diversity of ISAMOPSO algorithm are better.

4. Applied ISAMOPSO for Reactive Power Optimization

To find the optimal capacity and location of the DG in the power distribution system and solve the multi-objective reactive power optimization problem of distribution networks with DG, this paper takes the IEEE-33 node distribution system as an application example [28]. There are 33 branches in the unimproved IEEE-33 node system. The active load is 3715 kW, the reactive load is 2300 kvar, the system reference voltage is 102.66 kV, and the reference capacity is 10 MVA. The calculation accuracy in the test is 10⁻⁴. The IEEE-33 node distribution system is shown in Figure 3.

4.1. Influence of Distributed Generation on Distribution Network

After the DGs are connected, the whole distribution network becomes a complex structure of multiple power sources, which is not only used to transmit power but also can generate power. The random power emitted by DG will cause the change in power flow, and some DGs with larger capacity will even cause the voltage instability at each node of the grid and the increase in network loss, which is closely related to the connection location and capacity of DGs. Therefore, it is necessary to consider the location and capacity of DGs connected to the distribution network.

4.1.1. Influence of Distributed Generation Capacity on Distribution Network

According to the structural characteristics of the IEEE-33 node distribution system, 14 groups of independent experiments were conducted on 14 representative nodes. The DG was incorporated into the 3rd, 6th, 9th, 12th, 15th, 18th, 19th, 21st, 23rd, 25th, 26th, 28th, 31st, and 33rd nodes to test the influence of the voltage and network loss of the distribution network system. The total load of the IEEE-33 node system is 3715 kW + j2300 kvar. The capacities of DG are 20%, 40%, 50%, 70%, 90% and 100% of the total load. The results of the experiments are shown below.

The per-unit values of the network loss when the different capacities of DG are incorporated into different 14 nodes are shown in

Table 4. The per-unit values of the network loss when the different capacities of DG are incorporated into different 14 nodes.

Node	0	20%	40%	50%	70%	90%	100%
3	0.0293	0.0228	0.022	0.0216	0.0211	0.0208	0.0207
6	0.0239	0.0159	0.011	0.0097	0.0089	0.0104	0.012
9	0.0239	0.0141	0.0107	0.011	0.0148	0.0225	0.0276
12	0.0239	0.0134	0.0118	0.0135	0.021	0.0331	0.0405
15	0.0239	0.0134	0.015	0.0189	0.0315	0.0491	0.0595
18	0.0239	0.0147	0.0198	0.0258	0.043	0.0655	0.0785
19	0.0239	0.0232	0.0229	0.0229	0.0231	0.0237	0.0241
21	0.0239	0.0237	0.0264	0.0286	0.035	0.0434	0.0484
23	0.0239	0.0224	0.0217	0.0217	0.0223	0.0236	0.0246
25	0.0239	0.0216	0.0226	0.0243	0.0296	0.0373	0.042
26	0.0239	0.0155	0.0107	0.0094	0.009	0.0112	0.0132
28	0.0239	0.0135	0.009	0.0087	0.0111	0.0173	0.0216
31	0.0239	0.0118	0.0095	0.011	0.0185	0.0306	0.0382
33	0.0239	0.0121	0.0108	0.0131	0.022	0.0359	0.0444

.

Through the analysis of Figure 4a,g,i, it can be concluded when the DG is close to the power supply side, it has no obvious effect on the voltage rise no matter how much the capacity of the DG. Meanwhile, the voltage increases significantly and even exceeds the limit value when the DG is incorporated into the 15th, 18th, 31st, and 33rd nodes, which are far away from the power supply side. The higher the DG capacity, the higher the node voltage of the system, as can be seen in Figure 4e,f,l,m.

From Table 4, the network loss decreases with the increase in the capacity of the DG. The capacity of the DG should be between 20% and 40% of the total load. If the DG is incorporated near the power supply side, the capacity of the DG should be selected at 20% of the total load to reduce system power loss. Otherwise, the capacity should be 40% of the total load to increase the power flow transmission capacity and minimize the network loss so that the voltage distribution is more stable.

4.1.2. Influence of Distributed Generation Position on Distribution Network

Using the control variables method, when the capacity of the DG is certain, the location of the DG in the IEEE-33 node distribution system should be changed to study the influence of the node voltage distribution. To improve the experimental reliability and summarize the general rule, six groups of independent experiments with different capacities were set up. The capacities of DG were 20%, 40%, 50%, 70%, 90% and 100% of the total load. The voltage amplitudes of 14 nodes in each group of experiments are shown in the following (Figure 5):

The results of experiments show that no matter how much capacity of DG is incorporated into the 3rd, 6th, 19th, and 23rd nodes of the system, the voltage amplitude of 33 nodes is the same as the value there is no DG in the system. When the DG is incorporated into the 15th and 18th nodes, the voltage amplitude of 33 nodes is higher the higher the capacity of DG, but when the DG is incorporated into the 28th and 31st nodes, the voltage amplitude of 33 nodes is nearly no change. When the DG is incorporated into the 21st and 35th nodes, only the voltage of nearby nodes is affected. The closer the access location of DG supply is to the end of the line and the load center, the more obvious its effect on voltage lifting, so it should be chosen to merge into the middle and rear sections of the mainline. From the above experiments, it is most appropriate to select the 12th, 15th, and 18th nodes for the integration point, and the capacity of DG should be between 20% and 40% of the system load.

4.2. Multi-Objective Reactive Power Optimization Solution Process

The main steps of the ISAMOPSO for distribution grid reactive power optimization with DG are as follows:

• Step 1: Read the system data, set the population size $N$ and the maximum iteration number $M_t$ and the related parameters of the improvement strategy, and initialize the population $P$, personal best $P_t$ and global best $P_g$.
• Step 2: Update the personal best and leader for each particle. Carry out power flow computation and obtain the objective function values of each particle. The non-dominated solutions of the population $P$ are confirmed according to the non-dominated ranking strategy, and the positions of the next generation of particles and the individual optimal positions are updated according to Equation (13).
• Step 3: Generate $P’s$ off-spring population $Q$ according to Equations (11) and (13) based on the current velocity and position of each particle in $P$, and then combine $P$ and $Q$ together and obtain population $R$ whose size is $2N$.
• Step 4: Identify the non-dominated solutions from $R$, and store them in $ND$.
• Step 5: Generate the particle population $P_{new}$ for the next iteration and the evaluation criterion Formula (17) for the leader particles.
• Step 6: Based on the state of velocity in the population, determine whether it constitutes a perturbation condition, and adjust the population again.
• Step 7: Determine whether the maximum number of iterations $M$ is reached; if so, proceed to the next step; otherwise, go to Step 3 for the next iteration.
• Step 8: Output the non-dominated solutions as the final Pareto-optimal solutions.

4.3. Analysis and Discussion Results

According to the conclusions on the influence of DG capacity and location in the IEEE-33 node distribution system, the DG is, respectively, incorporated into the 15th node and 31st node. It is assumed that each DG can output active power or compensate for the reactive power of the distribution network.

In the example of the IEEE-33 node reactive power compensation distribution system [28], the active load is 3715 kW, the reactive load is 2300 kvar, the system reference voltage is 102.66 kV, and the reference capacity is 10 MVA. Based on the discussion of the capacity of DG in Section 4.1, the active output of the DG is all 1 MW, and the reactive power output range is −0.1~0.5 Mvar. Based on the load of the 33-node reactive power compensation distribution system, the shunt capacitor bank with a compensation capacity of 0.15 Mvar $\times$ 4 is incorporated into the 6th node, and the shunt capacitor bank with a compensation capacity of 0.15 Mvar $\times$ 7 is incorporated into the 24th node [29]. The on-load voltage regulating transformer with nine transformer taps is connected between 0 and 1 node which ratio range is 0.9–1.1. and the transformer tap adjustment step is 0.025. The range of node voltage variation is 0.85–1.15 per unit. The active network loss of the system before optimization is 0.1885 MW, and the node voltage deviation is 1.0435. The IEEE -33 node reactive power compensation distribution system is shown in Figure 6.

4.3.1. Static Reactive Power Optimization

During the process of static reactive power optimization process, the objective functions are the minimum distribution network loss, node voltage deviation, and reactive power compensation cost. The reactive power output of the DG, the capacity of the shunt capacitor, and the transformer tap of the on-load regulator transformer were selected as the control variables.

The optimal solution set in the objective space is shown in Figure 7. There is a certain non-linear relationship among the three objectives, which are in conflict with each other, and no optimization result can make each objective function reach the optimal at the same time.

The optimal solution set in the objective space is shown in Figure 7. The five minimum Euclidean distance compromise solutions were selected from the optimal solution set as 5 optimization schemes (W1~W5), and the control variables and objective function values were calculated, which are shown in

Table 5. Five static reactive power optimization schemes.

Result	DG1 (Mvar)	DG2 (Mvar)	C1 (Mvar)	C2 (Mvar)	K (pu)	${\mathit{f}}_{\mathit{P}\mathit{l}\mathit{o}\mathit{s}\mathit{s}}$ (MW)	${\mathit{f}}_{\mathbf{\Delta }\mathit{V}}$	${\mathit{f}}_{\mathbf{c}\mathbf{o}\mathbf{s}\mathit{t}}$ (million)
W1	0.0335	0.2965	0.15	0.15	0.975	0.0645	0.1581	0.0594
W2	0.1325	0.3214	0.15	0.15	0.975	0.0663	0.1595	0.0569
W3	0.1618	0.2933	0.3	0.45	0.975	0.0512	0.1287	0.0701
W4	0.3187	0.3396	0.45	0.75	1	0.0547	0.2741	0.0421
W5	0.3571	0.4163	0.45	0.3	1	0.0582	0.2803	0.0486

. It was observed that, while the active power loss ($f_{Ploss}$) and voltage deviation ($f_{\mathsf{\Delta }V}$) are all smallest in optimization scheme W3, but the smallest reactive power compensation ($f_{\cos t}$) is in W4 and the minimum Euclidean distance is W1. Because there is a certain non-linear relationship among the three objectives, which are in conflict with each other, and no optimization result can make each objective function reach the optimal at the same time. Therefore, it is necessary to provide multiple solutions to meet different requirements. The results of the minimum Euclidean distance of the ISAMOPSO algorithm were compared with the minimum Euclidean distance of the NSGA-II algorithm, as shown in

Table 6. Comparison of the optimization results of the algorithms.

Comparison Object	Before Optimization	NSGA-II	ISAMOPSO
$f_{Ploss}$ (MW)	0.1885	0.0728	0.0645
$f_{\mathsf{\Delta }V}$	1.0435	0.1945	0.1581
$f_{\cos t}$ (million)	-	0.0637	0.0594

, the network loss decreased by 11.4%, the voltage deviation decreased by 18.7%, and the compensation cost decreased by 6.75%, which proves that the ISAMOPSO algorithm is more applicable to the reactive power optimization problem than the NSGA-II algorithm.

Using the five static reactive power optimization schemes, the voltage amplitudes of the 33 nodes are different and compared with the value before static reactive power optimization, which is shown in Figure 8.

The results show that the voltage amplitude of each node optimized by the ISAMOPSO algorithm is within its upper and lower limits. Moreover, the voltage amplitude difference between nodes is reduced, and the voltage distribution after optimization is smoother. This shows that the distribution network is more stable. As known from the calculation, the active power network loss maximum decreased by 77.82%, and the maximum voltage deviation was reduced by 86% compared with the value before optimization. The effectiveness of the ISAMOPSO algorithm for multi-objective reactive power optimization is proved.

4.3.2. Dynamic Reactive Power Optimization

Affected by natural environmental factors such as wind speed, light intensity, and temperature, the output of DG will change over time, and the load of the actual distribution network is also changing. In this section, according to the natural environmental factors of a typical day in a certain region, the photovoltaic and wind power output at 24 different times of the day is predicted. This is incorporated into the IEEE-33 node distribution system as the access capacity of DG. The natural environmental factors for a typical day in a region are shown in

Table 7. Typical daily environmental factors and power output of DG.

Time	Light Intensity (W/m2)	Temperature (℃)	Wind Speed (m/s)	PV Output (kW)	WT Output (kW)	Time	Light Intensity (W/m2)	Temperature (℃)	Wind Speed (m/s)	PV Output (kW)	WT Output (kW)
0	0	17.33	7.15	0	219.4	12	494.44	27.46	5.90	287.9	134.3
1	0	17.29	8.32	0	313.6	13	619.44	30.10	6.72	315.1	188.1
2	0	17.10	7.53	0	248.4	14	605.56	29.26	5.82	403.4	129.5
3	0	16.73	8.16	0	299.9	15	565.56	29.18	5.18	392.7	93.01
4	0	16.32	10.37	0	513	16	522.22	28.35	6.74	365.2	189.8
5	91.67	16.06	9.25	0	399	17	394.44	26.30	5.65	334.8	119.2
6	161.11	16.11	8.86	53.78	361.6	18	81.67	23.93	6.78	247.9	192.6
7	245.56	16.66	9.55	95.25	428.3	19	0	21.55	5.48	49.25	109.8
8	333.33	17.90	8.77	146.8	353.4	20	0	19.50	6.22	0	154.9
9	341.67	19.91	7.43	202.1	241	21	0	18.03	5.96	0	138.3
10	391.11	22.42	5.90	208.8	134.5	22	0	17.09	6.11	0	147.7
11	457.22	25.05	6.34	242.4	162.4	23	0	16.52	6.81	0	194.9

. Based on the data given in Table 7, the output of PV and wind power was predicted, and the predicted results are shown in Figure 9.

During the process of the dynamic reactive power optimization, the objective functions are the minimum distribution network loss, node voltage deviation, and maximum total consumption ratio of DG.

Considering load time-varying over 24 h a day, the access capacity of DG, the capacity of the shunt capacitor, and the transformer tap of the on-load regulator transformer were selected as the control variables. The ISAMOPSO was used for dynamic reactive power optimization. The optimal solution set in the three objective spaces of dynamic reactive power optimization is shown in Figure 10.

The three objective functions have conflicting distributions in the objective space with non-linear relationships and present a spatial curve in the three-dimensional space. The uniform distribution and high diversity of the points on the curve further illustrate the effectiveness of the ISAMOPSO algorithm in solving the multi-objective reactive power optimization problem.

In this paper, a compromise solution with the minimum Euclidean distance from the ideal point (0, 0, 0) was selected from the optimal solution set as dynamic reactive power optimization schemes.

The power consumed by photovoltaic power generation and wind power generation at each time after grid connection is very close to their respective predicted power outputs. The specific situation is shown in Figure 11 and Figure 12.

The total daily consumption ratio of photovoltaic power generation is 85.61%, and that of wind power generation is 82.75%.

The total consumption and power output of DG can be seen in Figure 13. The total consumption ratio of the system after optimization is very high at 0.8383, which proves that the algorithm improves the ability of the distribution network to absorb DG.

The whole-day average network loss decreased by 21.08% compared with before optimization, which can be seen in Figure 14.

The whole-day average voltage deviation decreased by 4.18% compared with before optimization, which can be seen in Figure 15.

The comparison of the voltage distribution of different nodes at 24 different times in Figure 16 shows that the voltage at each node after optimization is smoother than before optimization and the system operation is more stable. This further proves the effectiveness of the ISAMOPSO algorithm to solve the multi-objective reactive power optimization problem and the rationality of the mathematical optimization model established.

5. Conclusions

To solve the reactive power compensation problem of distribution networks with DG for economical operation, reduction in network loss, and reduction in voltage deviation, this study investigates the optimal size and siting of DG, establishes a multi-objective optimization model for optimal regulation in the distribution system, and uses the ISAMOPSO algorithm as an optimization tool. The algorithm is compared with other algorithms and successfully applied to the IEEE-33 node system. In the static reactive power optimization experiment, the network loss decreases by 11.4%, the voltage deviation decreases by 18.7%, and the compensation cost decreases by 6.75% in the solution set obtained by the ISAMOPSO algorithm compared to the solution set obtained by the NSGA-II algorithm with the minimum Euclidean distance. In the dynamic reactive power optimization experiment, the multi-objective reactive power optimization strategy proposed in this paper takes into account the security and stability of the distribution network operation and the absorption capacity of DG. After optimization by the ISAMOPSO algorithm, the DG consumption rate reached 0.8383, the network loss for the whole day decreased by 21.08% compared with that before optimization and the voltage deviation for the whole day reduced by 4.18% compared with that before optimization. Thus, the ISAMOPSO algorithm proposed in this paper has good computational capability for solving non-convex, multi-objective, multi-constrained high-dimensional non-linear reactive power optimization problems in power systems, and can provide a set of Pareto-optimal solution sets as optimization schemes, each of which can optimize the operational performance of the system and provide a flexible reactive power optimization scheme for decision-makers.