Integration of Distributed Generation and Plug-in Electric Vehicles on Power Distribution System by Using Queuing Theory

Abstract

Plug-in electric vehicles (PEVs) and distributed generation (DG) can positively and negatively impact the distribution system. Therefore, this paper presents the modeling and analysis of DG and PEVs’ penetration levels of the three-phase unbalanced radial distribution system. The study aims to optimize the distribution system’s DG sizing and PEV charging to minimize total power loss. The test system is the 4th circuit of the Nonsung service station along Thaharn Road, Muang District, Udon Thani, Thailand. According to objective function and constraints, such control variables as installation buses and rated outputs of DG and the penetration levels of PEVs were obtained to evaluate the total power loss. Significantly, the charging demand of PEVs is an uncertain load estimated by queuing theory integration with the minimization tool called the differential evaluation (DE) method. According to the result comparison of a four case simulation, the total power losses of queuing theory and DE application are minimum. Finally, total power losses conform to the regulation of the Provincial Electricity Authority (PEA), Thailand.

1. Introduction

The DG units are installed in the distribution systems to compensate the network operation. The power output of DG can solve the problem of reliability and the power quality of the system. The general definition of DG is the power unit connected to the distribution system [1]. The relation of the power distribution system of the Provincial Electricity Authority (PEA), Thailand, and DG is shown in Figure 1.

Referring to the characteristics of a power source, two types of DG can be classified as follows [2].

(1) Non-renewable energy sources such as gas turbine generators and fuel cells.

(2) Renewable energy sources such as biomass, wind, and solar cells.

The benefits that the distribution system will receive from DG are non-destructive energy generation systems, economic profits, power loss reduction, improving the voltage profile, voltage stabilization, increased reliability, and extending the time of system updates. Moreover, using DG as a source of renewable energy will benefit the PEA and the consumers such as greenhouse gas reduction, increasing the system capacity, flexibility, and convenience for changing system components [3]. DG will bring many benefits in terms of economic issues for consumers and the PEA, the technical parameters of the distribution system, and it is environmentally friendly; however, the increase of DG at a very high rate may have a detrimental effect on the PEA’s distribution system [4]. The installation of DG is the increasing of the power source; this will cause the high level of fault currents. On the other hand, the fault may be less than the level where relays can be detected. Moreover, more power sources in the distribution system will result in an overvoltage problem.

Most of the literature presented on determining the size and installation location of DG mainly considers static load, time-varying load, different load model, and DG as a renewable energy source [5]. However, impact studies of the penetration level of PEVs integrated with DG is still in the minority of research, while the demand for PEVs continues to grow. Thus, the potential impact of PEVs and DG on the PEA’s distribution system is an issue that cannot be ignored. The PEV’s demand may affect the distribution system such as increasing the overall loss of the system [6] and overload conditions in transformer distribution [7]. According to the IEEE Std 519-1992, the charging of PEVs will cause a harmonic distortion that exceeds this standard [8]. For the very high load intensity feeder, the uncertain PEVs demand and the base load may generate the new peak load curve that is close to or higher than the capacity that the distribution system can supply [9].

Smart grids aim to develop a modern distribution system, optimize power transmission, and preserve the reliability and stability of the system. Therefore, the importance of the opportunities and issues related to DG is critical [10]. According to the abovementioned, this paper presents the study and develops the knowledge of DG’s location and output power and the penetration level of PEV demand. The control variables comply with the standards set by the PEA and do not cause any damage to the consumer. The distribution system used for this work is the feeder 4th circuit, Nonsung station, PEA, which supplies electricity in the Udon Thani Municipality, Thailand.

According to the report of [11], on 28 February 2023, Udon Thani, Thailand, had 496 PEVs and 25 charging stations. As an annual one-off, this PEV number is a 158% increment. To illustrate, the graph in Figure 2 shows the monthly number of PEVs from 2018 to 2023. Thus, PEVs are growing in popularity as more efficient low-emission alternatives to conventional fuel-based automobiles. With the increasing popularity of electric vehicles, there is a growing need for a PEV charging infrastructure to support the expanding PEV market. However, the introduction of PEV charging has implications for the power losses of the distribution system [12]. Power losses in a distribution system occur due to various factors, including resistance in transmission lines, transformer losses, and losses in distribution lines. Adding PEV charging stations to the distribution system can lead to increased power losses due to the additional load on the system.

The real-time coordination strategy for charging PEVs in a smart grid environment was proposed by [13]. The goal is to minimize power losses and improve the voltage profile while ensuring sufficient PEVs demand and availability of renewable energy sources. The effectiveness of the proposed coordination strategy is demonstrated through simulations on a test distribution network. The results show that the proposed approach can effectively reduce power losses and improve the voltage profile while satisfying the charging demand of PEVs. Furthermore, the other coordinated strategy of [12] is also to minimize peak power demand and power. The results show that it effectively reduces peak power demand, power losses, and energy costs. According to the charging starting time (OCST) and time-of-use (ToU) system, ref. [12] proposed the coordinated charging scheduling of electric vehicles considering the optimal charging time for network power loss minimization. As a result, this work verifies that incorporating OCST can significantly reduce network power losses.

Otherwise, the network reconfiguration and optimal charging scheduling of PEVs are crucial to ensure the efficient use of the power grid [14]. This work addressed two vital issues for power loss minimization. Optimal charging scheduling of PEVs is crucial to ensure efficient use of the power grid, especially during peak hours when there is a high electricity demand. By scheduling PEV charging, the load can spread over time, reducing the need for additional infrastructure investment and avoiding overloading the grid. Subsequently, network reconfiguration minimized the total power loss leading to a more efficient and reliable power system [15].

The novel approach for the optimal allocation of PEV charging spots and capacitors in the distribution network presented by [16] is another approach to power loss minimization. This work proposed a novel method for the optimal allocation of PEV charging spots and capacitors in the distribution network to improve voltage and power loss. The proposed approach is based on the two meta-heuristic optimization algorithms, Quantum Behaved and Gaussian Mutational Dragonfly Algorithm (QGDA). The objectives are voltage improvement and power loss minimization. The voltage and power loss can be improved by optimizing the allocation of PEV charging spots and capacitors, leading to a more efficient and reliable power system [17].

A two-stage approach for PEV charging planning and network reconfiguration to minimize power loss in low and medium-voltage distribution networks was proposed by [15]. The first stage involves optimizing the PEV charging schedule to reduce the peak load on the grid, while the second stage involves network reconfiguration to minimize power loss. The crucial point is considering PEV charging planning and network reconfiguration as separate but interconnected optimization problems. By minimizing the power loss, the overall efficiency of the power system can be improved, leading to a more sustainable and reliable power grid.

According to the literature reviewed above, this paper will focus on the total power loss minimization of DGs and PEVs integration into the power distribution system using the queuing theory. All associated parameters will be modeled by the queuing theory and optimized using the differential evaluation (DE) algorithm. With proper DGs planning and an efficient and smart charging infrastructure, the total power losses can be minimized, and the transition to PEVs can be made more sustainable.

2. Queuing Theory

The queuing theory is the mathematical model of the service systems consisting of the arriving rate of customers, the service rate of service units, the waiting time for customers, etc. The charging PEVs are the uncertain situations in three-phase unbalanced radial distribution systems. Thus, the number of PEVs, time intervals of charging, and charging demand can be determined using the probability called queuing theory. For the integration of PEVs demand, the load flow analysis tools will be applied for a demonstration of the system parameters or optimization objective function values.

2.1. Components and Symbols of Queuing Model

Generally, a queuing system consists of servers and customers. If customers arrive while the server services other customers, they must wait in the waiting line. They will receive assistance after the front customers receive their service requirements from the server. The whole system causes customers to leave the system without waiting, resulting in customer loss. The estimation of PEVs demand depends on the components of the queuing system, characterized as follows: [18]

(1) The arrival process of customers. Generally, the inter-arrival times of customers are independent and have a standard distribution.

(2) The behavior of customers. Customers will attend the queues if it is available; otherwise, they may switch the queue.

(3) The service times. Usually, the service times will be assumed to be independent and identically distributed and independent of the inter-arrival times.

(4) The service discipline. Customers can be served one-by-one or in batches. There are many possibilities for the order in which the customers enter the service, such as first come first served, random order, last come, first served, priorities, or process sharing.

(5) The capacity of the queuing system may be a single or a group of the service unit.

The symbol of the queuing system based on Kendall’s notation is as follows:

$$A/B/C/X/Y/Z$$

(1)

where A is the arrival probability distribution, B is the service probability distribution, C is the number of servers, X is the maximum queue size, Y is the customer population, and Z is the type of queuing discipline. The essential parameters of the queuing system for PEVs demand estimation can be described as follows:

(1) The arrival rate (λ) is the average number of customers entering the charging station in a time interval.

(2) The service rate (μ) is the average number of customers the charging station can service in a time interval.

(3) The service time (τ) is the average time service for charging a customer.

(4) The traffic intensity (ρ) is the ratio between the average number of customers and the average number of customers that servers can provide during the time interval.

(5) The average waiting time of customers in the queue (W).

(6) The total waiting time of customers in the queuing system (T).

(7) The average number of customers in the queue (N_q).

(8) The average number of customers in the queuing system (N).

(9) The probability of k customers exists in the queuing system (p_k) where k = 0, 1, 2, …, X, and X is the number of customers at that time.

All parameters mentioned above will be modeled on the uncertain PEVs charging demand as described in the next section.

2.2. M/M/c/K Queue

The M/M/c/K queue is the queuing system that has c servers. The limited capacity is equal to K customers. When arrival customers are less than c, they will receive the service immediately. Otherwise, customers may be waiting in the queue or exit the system. The structure of the M/M/c/K queue is shown in Figure 3. The probability of k customers can be calculated by using Equation (2). In the case that all servers service other customers or are not ready to serve, some customers may not enter the queue and exit the system. Thus, the probability that the system is out of service can be calculated using Equation (4).

$$p_k=\left\{\begin{array}{cc}{\left(k!\right)}^{-1}{\left(\frac{\lambda }{\mu }\right)}^k{p}_0;& k<c\\ {\left(c! c^{k-c}\right)}^{-1}{\left(\frac{\lambda }{\mu }\right)}^k{p}_0;& k\ge c\\ 0;& k>K\end{array}\right.$$

(2)

where

$$p_0=\frac{1}{{\sum }_{k=0}^{c-1}\frac{1}{k!}{\left(\frac{\lambda }{\mu }\right)}^k+\frac{1}{c!}{\left(\frac{\lambda }{\mu }\right)}^c\cdot \left(\frac{c\mu }{c\mu -\lambda }\right)}$$

(3)

$$p_B=p_{k=K}$$

(4)

The number of customers in the queue and average waiting time can be calculated by Equations (5) and (6), respectively.

$$N_Q=\frac{p_0}{c!}{\left(\frac{\lambda }{\mu }\right)}^c\pi \left(\frac{\rho }{{\left(1-\rho \right)}^2}\right)\left[1-{\rho }^{K-c+1}-\left(1-\rho \right)\left(K-c+1\right){\rho }^{K-c}\right]$$

(5)

$$W=\frac{N_Q}{\lambda \left(1-p_B\right)}$$

(6)

Considering only the customers in the queue and who are currently undergoing service, the equation for calculating the total number of customers in the queuing system is:

$$T=\frac{N}{\lambda \left(1-p_B\right)}$$

(7)

This work applied the M/M/c/K queue to estimate PEV loads in radial distribution systems. Based on the field data of charging stations, the details of the modeling and the results are shown in the next section.

3. Differential Evaluation Optimization

The DE algorithm proposed by [19] has been confirmed to be more efficient and accurate than all annealing algorithms and genetic algorithms [20]. This work applied DE algorithms that optimized the DG’s rated output and installation bus, and PEV penetration levels. The process of this algorithm is shown in Figure 4.

Suppose the solutions of the optimization problem are real number D values; the details of the DE algorithm is described as follows:

3.1. Initialization

The initialization process is detailed as follows:

(1) Initiate the population vector of a solution where N > 4 as Equation (8).

$$x_{i,G}=\left[\begin{array}{cccc}{x}_{1,i,G}& x_{2,i,G}& \cdots & x_{D,i,G}\end{array}\right],i=\mathrm{1,2},3,...,N$$

(8)

where G is the generation number.

(2) Define the lower and upper bound of solutions as Equation (9).

$$x_j^L\le x_{j,i,1}\le x_j^U$$

(9)

where $x_j^L$ is the lower bound of the jth solution, $x_{j,i,1}$ is the jth solution of the ith population and 1st generation, and $x_j^U$ is the upper bound of the jth solution.

3.2. Mutation

The mutation process is detailed as follow:

(1) Random 3 vectors $x_{r_1,G},x_{r_2,G},\mathrm{a}\mathrm{n}\mathrm{d}{x}_{r_3,G}$ from $x_{i,G}$ where r₁, r₂, and r₃ must be unequal to i.

(2) Calculate the donor vector by using Equation (10) with the mutation factor F∈ [0, 2].

$$v_{i,G+1}=x_{r_1,G}+F\left(x_{r_2,G}-x_{r_3,G}\right)$$

(10)

3.3. Recombination

The trial vector (u_i,G+1) can be calculated from the member of the target vector (v_i,G+1) and donor vector by using Equation (11).

$$u_{j,i,G+1}=\left\{\begin{array}{cccc}{v}_{j,i,G+1},& \mathrm{if}& ran{d}_{j,i}\le CR& \mathrm{or} j=I_{rand}\\ x_{j,i,G},& \mathrm{if}& ran{d}_{j,i}>CR& \mathrm{or} j\ne I_{rand}\end{array}\right.$$

(11)

where j = 1, 2, 3,…, D is the generation number, rand_j,i is the random number from the interval [0, 1], CR is the recombination constant random from the interval [0, 1], and I_rand is the index number random from [1, D] to satisfy v_i,G+1 ≠ x_i,G.

3.4. Selection

The selection process is the comparison of objective function values due to the target vector (x_i,G) and the trial vector (u_i,G+1). The better will be selected for the next iteration as a constraint in Equation (12).

$$x_{i,G+1}=\left\{\begin{array}{cc}{u}_{i,G+1},& \mathrm{if} f\left(u_{i,G+1}\right)\le f\left(x_{i,G}\right)\\ x_{i,G},& \mathrm{otherwise}\end{array}\right.,i=\mathrm{1,2},3,...,N$$

(12)

4. PEVs Demand Estimation

Load flow analysis requires a queuing theory model for the charging load of PEVs [20,21,22]. The uncertain parameters of PEVs charging are the arrival time, charging time period, and charging power. The algorithm of the PEVs demand estimation can be described as follows.

(1) Assign the arrival rate (λ) and the service rate (μ). These data can be obtained from the charging station or optimization tool.

(2) Assign the type and number (N_max) of PEVs.

(3) Assign the maximum charging power (P_PEV,max) for each PEV.

(4) A random value from [0, 1] by using the uniform distribution.

(5) Calculate the probability of PEVs charging, p(n), from Equation (2) where n = 1, 2, 3,…, N_max.

(6) Calculate the minimum number (N_min) of PEVs from vector [0, 1, 2, …, N_max] to satisfy the following condition,

$$r\left[0,1\right]<p\left(n\left(k\right)\right)$$

(13)

(7) Calculate the charging time (t_c,i) for each PEV by using Equation (14) where t_max is the maximum charging time and I = 1, 2, 3, …, N_min

$$t_{c,i}=\left\{\begin{array}{cc}{t}_{max},& r(0,1]<{\mu n}_{max}{e}^{-{\mu n}_{max}{t}_{max}}\\ -\frac{1}{{\mu n}_{max}}ln\left(\frac{r\left(\mathrm{0,\; 1}\right]}{{\mu n}_{max}}\right),& r(0,1]\ge {\mu n}_{max}{e}^{-{\mu n}_{max}{t}_{max}}\end{array}\right.$$

(14)

(8) Calculate the charging power for the k-th type of PEVs by using the following equation,

$$P_D\left(k\right)=P_{PEV,max}\left(N_{min}-\sum _{i=1}^{N_{min}}{e}^{-\frac{\alpha t_{c,i}}{t_{max}}}\right)$$

(15)

(9) Repeat step (4) until all type of PEVs are considered.

(10) Calculate total PEVs charging demand by using the following equation,

$$P_{PEV}=\sum _{k=1}^{N_{max}}{P}_D\left(k\right)$$

(16)

5. Integration of DG and PEVs on Power Distribution System

The system data of the study were obtained from the unbalanced radial distribution system of the Provincial Electricity Authority (PEA), Udon Thani, Thailand [23]. The selected feeder is the 4th circuit of the Nonsung station along Thaharn Road, in the Muang District, Udon Thani, Thailand. The topology of the feeder is shown in Figure 5. The Udon Thani Municipality buys electricity from the DG through the feed line of the 4th circuit. The information of the Energy Regulatory Commission states that the renewable energy potential of the Udon Thani Province in 2018 has the capacity of generating electricity as shown in the rated reactive power of the biomass DG (Q_pv1, Q_pv2, Q_pv3, Q_pv4) with the maximum power factor equal to 0.95. The maximum rated apparent power of the 4 DGs are 12.375 MVA 18.750 MVA 24.500 MVA, and 37.500 MVA, respectively [24].

The rated active power of photovoltaic DG (P_pv1, P_pv2, P_pv3). The maximum rated active power of the three photovoltaic DGs are 1.50 MW, 1.00 MW, and 7.46 MW, respectively.

Table 1. Rated power of biomass power plants.

DG Number	Rated Power
	MW	MVA
1	9.90	12.38
2	15.00	18.75
3	19.60	24.50
4	30.00	37.50

and

Table 2. Rated power of photovoltaic power plants.

DG Number	Rated Power
	MW	MVA
1	1.50	1.50
2	1.00	1.00
3	7.46	7.46

are divided into four biomass power plants and three solar power plants [16]. According to the survey data of gas stations in the Udon Thani Municipality, the data used to model the queuing system of PEV charging stations are as follows.

(1) The number of PEVs charging simultaneously (c) equals 10 servers.

(2) The number of PEVs waiting in the queue (N_q) is 10 vehicles.

(3) The number of PEVs that perform other activities at the station to wait queues (λτ) is 30 vehicles.

(4) The total number of PEVs in the queuing system (N) is 50 vehicles.

(5) The average arrival PEVs (λ) is the control variable of the optimization problem.

(6) The average service time (μ) is the control variable of the optimization problem.

The queuing model for this work is the M/M/10/50 queue. The arrival of PEVs is the Markovian model and Poisson probability distribution [17]. The service time of the charging station is the exponential probability distribution. The objective function is the total power loss (P_loss,br) of branches. The details of the 18 control variables described as follows.

The installation bus numbers of biomass DG (N_bio1, N_bio2, N_bio3, N_bio4). The bus number of this test system is in the range of 1–205. Bus installation bus numbers of photovoltaic DG (N_pv1, N_pv2, N_pv3). The rated reactive power of the biomass DG (Q_pv1, Q_pv2, Q_pv3, Q_pv4) with the maximum power factor equal to 0.95. The maximum rated apparent power of the 4 DGs are 12.375 MVA, 18.750 MVA, 24.500 MVA, and 37.500 MVA, respectively. The rated active power of photovoltaic DG (P_pv1, P_pv2, and P_pv3). The maximum rated active power of the three photovoltaic DGs are 1.50 MW, 1.00 MW, and 7.46 MW, respectively.

The boundary of the control variables is based on the data of the test feeder and charging stations in Germany and Sweden. The base load curve of test feeder for April 2018 is shown in Figure 6. The simulation of the optimal power loss is calculated every 30 min from 0–24 h for 7 days. Therefore, the number of simulations is 336 times. The optimization problem calculating the optimal value of the total power loss when PEA purchases electricity from the distributed generation by considering the penetration level of the load PEVs are shown in Equation (17).

$$\mathrm{Minimize}{\sum }_{br=1}^{N_{br}}{P}_{loss,br}$$

(17)

$$\begin{array}{ccc}\mathrm{Subject} \mathrm{to}& 0\le P_{k,bio}\le P_{m,bio}^{max}& m=1,2,3,4\\ & 0\le Q_{k,bio}\le Q_{m,bio}^{max}& m=1,2,3,4\\ & 0\le P_{k,pv}\le P_{m,pv}^{max}& m=1,2,3\\ & 0.95\le \left|V_k\right|\le 1.05 \mathrm{p}.\mathrm{u}.& k=1,2,3,\dots ,N_{bus}\\ & {\sum }_{br=1}^{Nbr}{S}_{br}+{\sum }_{n=1}^{Nbus}{S}_{k,d}-{\sum }_{n=1}^{Nbus}{S}_{k,s}=0& br=\mathrm{1,2},3,\dots ,N_{br}\\ & 1\le N_i\le N_{bus}& i=\mathrm{1,2},3,\dots ,N_{maxiter}\\ & \lambda \in \left[\begin{array}{ccc}{\lambda }_{min}& {\lambda }_{avg}& {\lambda }_{max}\end{array}\right]\\ & \mu \in \left[\begin{array}{ccc}{\mu }_{min}& {\mu }_{avg}& {\mu }_{max}\end{array}\right]\end{array}$$

where P_loss,bar is the total power loss of the feeder, N_br is the total branch number, N_bus denotes the total bus number, P_k,bio and Q_k,bio are the real and reactive power, respectively, of the biomass generation supplies at bus k, $P_{m,bio}^{\max }$ and $Q_{m,bio}^{\max }$ are the rated real and reactive power, respectively, of the mth biomass generation, P_k,pv denotes the real power of the photovoltaic generation supplies at bus k, $P_{m,pv}^{\max }$ is the rated real power of the m^th photovoltaic generation, V_k is the voltage of the kth bus, S_br, S_k,d, S_k,s are the complex power loss of the brth branch, complex power of load at the kth bus, complex power of generation of the kth bus, respectively, λ_min, λ_avg, λ_max are the minimum, average, maximum of the arrival rate of PEVs at charging station, respectively, and μ_min, μ_avg, μ_max are the minimum, average, maximum of the service rate of PEVs at the charging station, respectively.

The two cases of the simulation for optimal power loss are the random technique and queuing model application. The algorithm details of the random technique are described as follows.

(1) Input load data for each 30 min in 24 h through 7 days; thus, the number of load data is 336 values.

(2) Calculate PEVs load demand by using a random method for 336 time intervals which the total system load must not exceed 80% of the maximum base load.

(3) Assign the optimal power loss equal to infinity (P_loss,opt = ∞).

(4) Assign the locations and rated generation of DGs equal to the empty set.

(5) For k = 1–336:

(5.1) Calculate the locations and rated generation of DGs for the k-th time interval resulting optimal power loss (plus_{, k}) by using the DE method.

(5.2) If P_loss,k is less than the previous value (P_loss,k−1).

(5.2.1) Assign the optimal power loss (P_loss,opt) equal to P_loss,k.

(5.2.2) Replace the locations and rated generation of DGs by the new values obtained in k-th time interval.

The second manner of the power loss optimization is the application of the queuing theory. Each iteration of the calculation must consider the minimum value, average value, and maximum value of the two parameters of the charging station as shown in Equation (18).

$$\lambda =\left[\begin{array}{ccc}{\lambda }_{min}& {\lambda }_{avg}& {\lambda }_{max}\end{array}\right]$$

(18)

$$\mu =\left[\begin{array}{ccc}{\mu }_{min}& {\mu }_{avg}& {\mu }_{max}\end{array}\right]$$

Therefore, for the 1–336 time intervals, the total power loss must be calculated to find the optimal values according to the three queuing parameters, as detailed in the following algorithm.

(1) Input load data for each 30 min in 24 h through 7 days; thus, the number of load data is 336 values.

(2) Calculate the PEVs demand randomly for 336 periods, each time when combined with the system base load must not exceed 80% of the maximum base load.

(3) Assign the optimal power loss equal to infinity (P_loss,opt = ∞).

(4) Assign the locations and rated generation of DGs equal to empty set.

(5) For k = 1–336.

(5.1) For m = 1−3:

(5.1.1) Calculate the PEVs charging load for the time period k using the service rate of PEVs (μ_m) and the service rate of the charging station (λ_m). The combination of PEVs and base load of the system must not exceed 80% of the maximum base load.

(5.1.2) Calculate the locations and rated generation of DGs for the k-th time interval, m-th queuing data, and resulting optimal power loss (plus_{, k, m}) by using the DE method.

(5.1.3) If P_loss,k,m is less than the previous value (P_loss,k,m−1):

(5.1.3.1) Assign the optimal power loss (P_loss,opt) equal to P_loss,k,m−1.

(5.1.3.2) Replace the locations and rated generation of DGs by the new values obtained in the k-th time interval and m-th queuing data.

(5.2) If P_loss,k,m is lower than the previous total power loss, P_loss,k−1,m,

(5.2.1) Assign the optimal power loss (P_loss,opt) equal to P_loss,k,m.

(5.2.2) Replace the locations and rated generation of DGs by the new values obtained in the k-th time interval and m-th queuing data.

According to the algorithm mentioned above, the simulation results of the optimal total power loss and rated generation of DGs are shown in the next section.

6. Simulation Results

The simulation tools of this work are the load flow analyzer of a three-phase unbalanced radial distribution system, DE calculation package, and queuing modeling package. For the comprehension of circumstances, the simulation was performed every 30 min in 24 h through 7 days. The queuing data are generated by the random method, minimum, average, and maximum values. Thus, the number of simulation results is 336 × 4 = 1344 data points. Consequently, the total power loss of the simulation results are shown in Figure 7, Figure 8, Figure 9 and Figure 10. The first case in Figure 7 is the simulation result of the optimal power loss when the calculated PEV demand is random to PEVs number, arrival time, and start–stop charging time. Thus, the results of the optimal power loss with random PEV demand are the maximum power loss for all cases. To investigate the total power loss associated with queuing parameter values, Figure 8, Figure 9 and Figure 10 shows the optimal power loss for the case of minimum, average, and maximum queuing parameters, respectively. According to these four results, the maximum queuing parameters cause the maximum optimal power loss of the grid. In summary, controlling PEV demand is the critical point of the distribution system operation.

The optimal power loss for random PEVs demand in Figure 7 is the maximum for all four cases. A comparison of the results for all four issues of PEV demand is shown in

Table 3. The simulation results for four cases of the optimization manners.

The Control Variables and Objective Function Values	Random PEVs Demand	Calculating PEVs Demand by Queuing Theory
		(λmin, μmin)	(λavg, μavg)	(λmax, μmax)
Nbio1	101	108	153	193
Nbio2	124	91	140	123
Nbio3	165	147	144	115
Nbio4	128	192	85	137
Pbio1 (kW)	4.02	8.00	8.16	5.73
Qbio1 (kvar)	7.07	13.95	8.00	11.04
Pbio2 (kW)	1.43	16.90	2.62	4.86
Qbio2 (kvar)	13.10	22.10	23.51	9.22
Pbio3 (kW)	619.66	850.49	1371.81	1933.08
Qbio3 (kvar)	2747	2388	2624.77	669.82
Pbio4 (kW)	6246	2105	5188.04	2019.94
Qbio4 (kvar)	6406	104.87	8532.08	372.22
Npv1	9	51	51	98
Npv2	59	25	200	51
Npv3	14	163	62	132
Ppv1 (kW)	352	461	616.00	225.00
Ppv2 (kW)	140	113	955.00	608.00
Ppv3 (kW)	2680	2080	3900.00	3950.00
Ploss,opt (kW)	1510.00	435.00	701.00	952.00

. In more detail, the loss power values for the bus numbers purchasing electricity from distributed bioenergy and solar power plants using the power difference evolution method of PEA are compared. The final optimal values of the bus numbers that the PEA purchases electricity from distributed biomass and photovoltaic generation units with the DE method is shown in Equation (19).

$$N_{bio}=\left[\begin{array}{c}85 91 101 108 115 123 124 128 \dots \\ 137 140 144 147 153 165 192 193\end{array}\right]$$

(19)

$$N_{pv}=\left[9 14 25 51 51 51 59 62 98 132 163 200\right]$$

All bus numbers in the computation are close locations in the system, and three calculation rounds yield the same result for bus number 51. Therefore, it can be concluded that these results confirm the reliability of the simulation.

7. Conclusions

This paper proposed the application of the queuing theory and DE algorithm to perform optimization problems for the total power loss in an unbalanced three-phase radial power distribution system. The control variables of the optimization problems related to DGs and PEVs; for example, the location and sizing of DGs, average arrival rate, and service rate of PEVs. The M/M/c/K queue was applied to handle the uncertainty, and PEVs demand was modeled. Moreover, the situations were divided as four scenarios to determine the boundary of solutions. The analysis was divided into two cases: random control variables and DE algorithm optimization with three instances of queuing data. By comparing the results from both cases, the queuing theory modeled PEVs demand with the acceptance penetration rate. Furthermore, the DE algorithm minimized the total power loss of distribution for all DG and PEVs integration cases. In summary, the queuing theory is crucial for PEVs demand estimation and the DE algorithm is sufficient for the total power loss minimization of the unbalanced three-phase distribution system. Finally, these optimization results correspond to the PEA’s standards and can be applied to the smart grids of electric network systems.