A Substation-Based Optimal Photovoltaic Generation System Placement Considering Multiple Evaluation Indices

Abstract

The placement of the photovoltaic generation system (PVGS) and operation of the on-load tap changer (OLTC) should have great impacts on the system loss and voltage quality, which are the main concerns of the distribution operator. Considering these multiple evaluation indices and other constraints, this paper proposed a substation-based optimal PVGS placement and OLTC operation model. The objective function that was used to evaluate the optimal PVGS placement and OLTC operation consists of a minimization of system loss and voltage quality. The model’s constraints contain the voltage and line flow limits, voltage deviations, voltage unbalance, etc. Uncertainties of the load and irradiance are also included in the model. A nondominated sorting genetic algorithm II (NSGA II) is used to solve this multi-objective optimization problem. Comparisons of the substation-based and feeder-based planning are also studied in this paper. The test results demonstrate the substation-based planning could obtain a better solution.

1. Introduction

Along with the growing distributed generation (DG) technology, the installation of DGs have become a feasible alternative in the system planning of distribution systems. The DGs, such as the PV and wind energies, are friendly to the environment because they produce less pollution than the conventional generators. If DGs are programmed properly, they can reduce system loss and improve voltage profile. However, DGs can also cause undesirable effects on the system, such as fluctuations in the voltage profile, increased fault current, and reverse flow direction [1]. In order to facilitate a higher level of DG penetration, the distribution system operators face several challenges that are related to system reliability, stability, and quality.

There are about 400 GW grid-connected PV capacities installed worldwide. More than half of the capacity is interconnected at the distribution network within our power systems [2]. The benefits of DGs can be enhanced when the DGs are planned properly. In contrast, crude decisions on DG planning may not only eliminate the distribution utilities of such benefits but also lead to negative economic and operational impacts [3]. Therefore, this issue has drawn a lot of attention. These studies can be categorized by solution methods such as analytical approaches and metaheuristic algorithms. They can also be classified by the objective functions such as loss minimization, voltage quality, and multi-objective.

The plant propagation algorithm (PPA) is proposed in [4] to reduce the active power loss and boost the magnitude of the minimal bus voltage simultaneously. A total of four rounds of DGs are investigated in this paper, from considering one DG to four DGs. The installation of DGs to improve the voltage profile and system loss has been discussed in [5]. The GA algorithm is used to determine the DG placements. Analytical approaches for optimal DG placement are proposed in [6,7,8], but they are only suitable for one DG placement. The goal in [6] is to minimize the total average power loss and maintain the voltages along the feeder in the acceptable range 1 ± 0.05 p.u. The same exact loss formula is used in [7,8] to determine the optimal location and size of DG. The authors in [9] extend the one DG placement to multiple DG placements, aiming at a loss reduction. Based on the same exact loss formula, the first optimal location and size of DG are determined. Update the system data after the placing first DG, and then determine the next DG placement and then repeat this process until the maximal number of DGs is reached or any system constraint violation.

A PV placement method for reducing power loss and improving the voltage profile is proposed in [10]. The candidate PV installed places are located at the buses with minimal loss or violation of the system constraints. A total PV capacity is set, and the simulation results for different combinations of these PV candidates are studied. An integrated Geographical Information Systems (GIS), optimization and simulation framework for optimal PV size and location is proposed in [11]. The GIS module is used to evaluate rooftops and their capacities for PV installations. If the suitable rooftops have been obtained from the GIS module, the optimal module determines the optimal placement of the PV systems to maximize the net profit of PV installation. An analytical approach for the optimal location and capacity of DG in distribution networks to minimize real and reactive power losses is proposed in [12]. Changes of the active and reactive branch currents that are caused by the DG placement are formulated in this paper. A multi-objective of PV placement model is proposed in [13]. The DGs’ sitting positions are determined by the loss sensitivity factors, and the multi-objective particle swarm optimization (PSO) algorithm is used to solve this problem.

The optimal PV placement in a village distribution network is proposed in [14] with the objective of loss minimization and unbalance. The Grey Wolf Optimization (GWO) technique is used for obtaining the optimal PV placement. An optimal sizing and placement of rooftop solar PV is proposed in [15]. A genetic algorithm (GA) that is based on Newton-Raphson power flow with the objective of power loss minimization is proposed to find the solution. DG placement coordinated with the reconfiguration method to achieve energy loss minimization is proposed in [16,17]. In [16], a Harmony Search algorithm (HSA) is adopted to simultaneously reconfigure and identify the optimal placement of DG units in a distribution network. In [17], the GA and improved switch-exchange methods have been blended to find the optimal status of switches and locations and sizes of DG units. A model coordinating the optimal integration of distributed energy resources with existing voltage-regulating schemes is proposed in [18]. The improved genetic algorithm is adopted to solve this problem. Optimal DGs and capacitor banks (CB) allocation is proposed in [19]. A Tabu Search algorithm is used to find the positions of DGs and CBs. As the devices’ locations in the network are determined, an optimal power flow is used to find the capacities of DGs and CBs. A Chu-Beasley genetic algorithm is used to solve this optimal power flow problem.

DGs can affect the voltage quality of the distribution system. In [20,21], DGs are located and sized to improve the voltage stability. A multi-objective for sizing and placement of DGs from the DG owner’s and distribution company’s viewpoints are proposed in [22]. The Particle Swarm Optimization is used to solve this problem. Different voltage stability indices are used to determine the optimal PV placement in [23]. The PV is placed at the bus with the worst stability index, and the sizing increases step by step until the loss begins to increase. A voltage stability-based placement of distributed generation against extreme events is proposed in [24]. The proposed approach starts out by selecting a set of outage scenarios that are credible and significant. A subset of the load buses is chosen as candidate locations for placing new DGs. A voltage stability constrained optimal power flow and an integer programming are used to determine the desired numbers and locations of DGs to be placed in order to maintain feasibility under all the considered outage scenarios.

Few studies have clearly addressed the substation-based and feeder-based modeling issues. A joint chance constrained programming (JCCP) method is proposed in [25] to optimize expansion of substations, feeders, and DGs simultaneously. The objective function considers the substation cost, feeder cost, and other related costs to obtain the coordinated solution. A planning model to determine the capacities of DG units considering multi-DG configurations has been proposed in [26]. The impact of multiple DG configurations under the active network management schemes is investigated in this paper. The locations of DG units can affect the DG penetration. A multi-configuration multi-period OPF technique has been proposed to determine DG capacities for different configurations.

Table 1. Summary of the literature reviews and the proposed model.

Reference	Decision Variables	Objective Functions	Solution Methods
[4]	DGs’ locations and capacities	Min. loss and upgrade the minimum bus voltage	PPA
[5]	DGs’ locations and capacities	Min. loss	GA
[6]	DGs’ Locations	Min. loss	Analytic approach
[7]	DGs’ locations and capacities	Min. loss	Analytic approach
[8]	DGs’ capacities and power factors	Min. loss	Analytic approach
[9]	DGs’ capacities, locations, and power factors	Min. loss	Analytic approach
[10]	DGs’ locations and capacities	Min. loss	Analytic approach
[11]	Roof PVs’ locations and capacities	Max. Profit	Analytic approach
[12]	DGs’ locations and capacities	Min. real and reactive loss	Analytic approach
[13]	DGs’ locations and capacities	Min. system cost and loss, and improve voltage stability	PSO
[14]	DGs’ locations and capacities	Min. loss	GWO
[15]	Roof PVs’ locations and capacities	Min. loss	GA
[16]	Network reconfiguration and DGs’ capacities	Min. loss	HSA
[17]	Network reconfiguration and DGs’ capacities	Min. loss	Hybrid method
[18]	DGs’ locations, capacities, and OLTC	Min. loss	GA
[19]	DGs’ locations, capacities, and capacitors	Min. cost	Hybrid method
[20,21]	DGs’ locations and capacities	Improve voltage stability	Analytic approach
[22]	DGs’ locations and capacities	DG owner’s cost and profit, and distribution company’s cost	PSO
[23]	DGs’ locations and capacities	Improve voltage stability	Analytic approach
[24]	DGs’ locations and capacities	Maintain voltage stability under extreme events	Analytic approach
[26]	DGs’ locations and capacities	Max. DGs’ penetration	Analytic approach
The proposed model	DGs’ locations, capacities, and OLTC	Min. loss, and improve voltage stability	NSGA II

summarizes the literature reviews and the proposed model by decision variables, objective function, and solution methods.

In Taiwan, the government has set the target of installing 20 GW (about half of the peak load) PV capacity by 2025. They can cause undesired impacts on the power system if they aren’t planned well. Therefore, it is essential to build a planning model to arrange these distribution sources adequately. This paper is proposed to address this issue. The novelty and contribution of this paper summarized as follows:

• This paper highlights the importance of a substation-based PVGS placement model. It can avoid the dilemma of the OLTC operation. The test results show the substation-based model outperforms the feeder-based model.
• The proposed model takes the uncertainties of irradiance and load demands into account, and the objective function contains multiple performance indices to obtain comprehensive results.
• The decision variables contain the tap value of the OLTC, and the positions and capacity of the PVGSs. Optimal solutions can be obtained from the coordination of these decision variables.

The organization of the paper is as follows. Performance indices are described in Section 2. The proposed model and solution method are proposed in Section 3. The test results and discussions are shown in Section 4. Finally, a concluding remark is given.

2. Performance Indices

The performance indices that are used to evaluate the optimal PVGS placement profile are described in this section. The objective function contains the system loss and voltage quality, while the voltage deviation and voltage imbalance are deemed as constraints.

2.1. System Loss

The system loss formula is shown in Equation (1). System loss is affected by the capacity and position of the PVGS:

$$Line Loss={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\ell =1}^{N_{\ell }}{I}_{t,\ell }^2\ast R_{\ell }}}$$

(1)

where $I_{t,\ell }$ is the current of branch $\ell$ at time t, $R_{\ell }$ is the resistance of branch $\ell$, and $N_{\ell }$ is the number of branches.

2.2. Voltage Quality

The voltage quality is evaluated by Equation (2). This objective function is to regulate the overall voltage in the system closing to 1.0 p.u.

$$VolQ={\displaystyle \sum _{t=1}^T\frac{{\displaystyle \sum _{i=1}^{N_n}\left|V_i^t-1\right|}}{N_n}}$$

(2)

where $V_i^t$ is the voltage magnitude of bus I at time t, and N_n is the number of buses.

2.3. Voltage Deviation

The voltage deviation is used to estimate the voltage change before and after the PVGS connection to the distribution system. The maximal voltage deviation at time t is shown in Equations (3) and (4). The voltage deviation is limited to 3% in Taiwan.

$$MaxVolDe{v}_t=Max\left(\left|\frac{V_i^t(PV)-V_i^t(NO PV)}{V_i^t(NO PV)}\right|\ast 100\%\right) for i=1~N_n$$

(3)

$$MaxVolDev=Max(MaxVolDe{v}_t) for t=1~T$$

(4)

where $V_i^t(NO PV)$ and $V_i^t(PV)$ are the voltages of bus i at time t without PVGSs and with PVGSs, respectively; N_n is the number of buses; and T is the number of time intervals.

2.4. Voltage Imbalance

The voltage imbalance can be estimated as the maximal deviation from the average of the three-phase voltages, divided by the average of the three-phase voltages, expressed as a percentage, as shown in Equation (5). The maximal voltage imbalance is expressed as Equations (6) and (7):

$${\mathrm{VI}}_i^t=\frac{\max \mathrm{deviation} \mathrm{from} \mathrm{average} \mathrm{voltage} \mathrm{of} \mathrm{bus} \mathrm{i} \mathrm{at} \mathrm{time} \mathrm{t}}{\mathrm{average} \mathrm{voltage} \mathrm{of} \mathrm{bus} \mathrm{i} \mathrm{at} \mathrm{time} \mathrm{t}}\times 100\%$$

(5)

$$MaxV{I}_t=Max(V{I}_i^t) for i=1~N_n$$

(6)

$$MaxVI=Max(MaxV{I}_t) for t=1~T$$

(7)

where $MaxV{I}_t$ is the maximal voltage imbalance at time t.

3. The Proposed Optimal PV Placement Model

3.1. The Substation-Based vs. Feeder-Based Model

Many studies discussed the optimal PV placement problem based on the feeder-based solution; this will cause some problems. Figure 1 shows a substation with two simplified feeders. If the PV is installed on the Feeder 1, Equation (8) shows the voltage rise of bus 1 ($\Delta V_1$), which is related to the PV output, load power, and substation voltage. In order to accommodate more PV on the Feeder 1 and maintain the voltage of bus 1 not exceeding the upper limit, the OLTC should be operated to lower the substation voltage V_s as shown in Equation (9):

$$\Delta V_1=V_1-V_s=\frac{R_1\left(P_{PV1}-P_{D1}\right)+X_1\left(\pm Q_{PV1}-Q_{D1}\right)}{V_1}$$

(8)

$$V_1=V_s+\frac{R_1\left(P_{PV1}-P_{D1}\right)+X_1\left(\pm Q_{PV1}-Q_{D1}\right)}{V_1}\le V_{\lim }^{+}$$

(9)

where $V_{\lim }^{+}$ is the upper voltage limit.

Equations (10) and (11) show the voltage drop of the Feeder 2. As there is no PV installed on this feeder, customers on this feeder prefer to raise the substation voltage V_s to improve the voltage profile and maintain the voltage not below the lower limit.

$$\Delta V_2=V_s-V_2=\frac{R_1{P}_{D2}+X_1{Q}_{D2}}{V_s}$$

(10)

$$V_2=V_s-\frac{R_1{P}_{D2}+X_1{Q}_{D2}}{V_s}>V_{\lim }^{-}$$

(11)

where $V_{\lim }^{-}$ is the lower voltage limit.

Previous discussion points out the dilemma that is faced by the feeder-based modeling. The OLTC should be operated to lower the substation voltage for Feeder 1, but on the contrary, Feeder 2 prefers to raise the substation voltage. Therefore, a substation-based PV placement model is proposed in this paper to address this problem, and a comparison of the substation-based and feeder-based results is also presented in this paper. The following section described the proposed model and solution method.

3.2. The Proposed Model and Solution Method

Capacities and locations of PVGSs can affect the system loss and voltage deviation. The system performance can be improved significantly if properly planned. The paper proposes a mathematic model to determine the optimal capacities and locations of PVGSs, considering uncertainties of load and irradiance profiles. The objective function is to minimize system loss and voltage quality index, and the constraints include the voltage limits, line flow limits, system imbalance, etc. The multi-objective evolutionary algorithms that use nondominated sorting and sharing have been criticized for three drawbacks, which are computational complexity, nonelitism, and the need for specifying a sharing parameter. The nondominated sorting in genetic algorithm II (NSGA-II) can alleviate the above three problems. A nondominated sorting in genetic algorithm II [27,28] is used to solve this multi-objective optimization problem to find a much better spread of solutions and better convergence near the true Pareto-optimal front. This section describes the uncertainties of load and irradiance profiles first, and then addresses how to apply the NSGA II to solve the proposed model.

3.2.1. Uncertainties of Load and Irradiance Profiles

Load and irradiance have seasonal characteristic. In order to address this point, four seasons’ possible load and irradiance profiles are considered in the proposed model. Real data of irradiance and load demand have been collected lasting one year. In order to obtain the representative irradiance and load profiles, a scenario reduction algorithm [29,30] is used to deduce a set of load and irradiance profile classes and occurrence probabilities from historical demand data and irradiance data, respectively, for the four seasons. Take the irradiance profile as an example. Figure 2a–d show the representative daily irradiance profiles and occurrence probabilities for four seasons, respectively.

3.2.2. The Proposed Model

The PVGS’s position and capacity can affect the system loss and cause voltage-related problems. The proposed model regards the system loss and voltage quality as the objective function, and the constraints include the voltage limits, line flow limits, maximal voltage deviation, system imbalance, etc. This is a multi-objective problem, and an NSGA II algorithm is used to solve this problem. The proposed model and solution method are described below.

The optimal planning of PVGS in a distribution network determines the location and capacity of these devices. The objectives of the system loss and voltage quality considering uncertainties are shown in Equations (12) and (13), respectively. The proposed model is shown in Equation (14) with the objective function of minimizing the system loss as well as the voltage quality index. The constraints include load flow equation, node voltage and line current limits, and maximal voltage deviation and voltage imbalance restrictions. The tap values of the OLTC are treated by season and time. According to the climate characteristics in Taiwan, two groups are divided in one year, i.e., the winter and spring seasons regarded as one group, and the summer and autumn seasons form the other group. For each season group, three OLTC setting periods are considered in one day, i.e., (24:00~8:00), (9:00~17:00), and (18:00~23:00):

$$f_1={\displaystyle \sum _{s=1}^4{\displaystyle \sum _{lo=1}^{lono}{\displaystyle \sum _{irr=1}^{irrno}{p}_1(s,lo)\ast p_2(s,irr) \ast }}}System Loss(s,lo,irr,PVCap,Tap) \ast 365$$

(12)

$$f_2={\displaystyle \sum _{s=1}^4{\displaystyle \sum _{lo=1}^{lono}{\displaystyle \sum _{irr=1}^{irrno}{p}_1(s,lo) \ast p_2(s,irr) \ast }}}VolQ(s,lo,irr,PVCap,Tap)\ast 365$$

(13)

Where s is the season index, lono and irrno are the number of load scenarios and irradiance scenarios, respectively, PVCap is the installed PV capacities, Tap is the tap value of the OLTC, and p₁(s,lo) and p₂(s,irr) are the probabilities of the loth load scenario and irrth irradiance scenario in season s.

$$Min \left(f_1,f_2\right)$$

S.T.

$$\begin{array}{l}{P}_{PVi,t}(s,irr)-P_{di,t}(s,lo)=V_{i,t}(s,lo,irr){\displaystyle \sum _{j=1}^{N_n}{V}_{j,t}(s,lo,irr)\left(G_{ij}\mathit{cos}({\theta }_{i,t}-{\theta }_{j,t})+B_{ij}\mathit{sin}({\theta }_{i,t}-{\theta }_{j,t})\right)} \forall i,j,t,s,lo,irr\\ Q_{PVi,t}(s,irr)-Q_{di,t}(s,lo)=V_{i,t}(s,lo,irr){\displaystyle \sum _{j=1}^{N_n}{V}_{j,t}(s,lo,irr)\left(G_{ij}\mathit{sin}({\theta }_{i,t}-{\theta }_{j,t})-B_{ij}\mathit{cos}({\theta }_{i,t}-{\theta }_{j,t})\right)} \forall i,j,t,s,lo,irr\\ V_{\min }\le V_{i,t}(s,lo,irr)\le V_{\max } \forall i,t,s,lo,irr\\ \left|I_{\ell ,t}(s,lo,irr)\right|\le I_{\ell ,\max } \forall \ell , t,s,lo,irr\\ MaxVolDev(s,lo,irr)\le VolDe{v}_{\max } \forall s,lo,irr\\ MaxVI(s,lo,irr)\le V{I}_{\max } \forall s,lo,irr\\ {\displaystyle \sum _{i=1}^{N_n}P{V}_i}=PVCap\end{array}$$

(14)

where N_n and $N_{\ell }$ are the number of buses and branches, respectively, T is the time intervals, P_PVi,t and P_di,t, and Q_PVi,t and Q_di,t are the real and reactive power output of PVGS and load demand at bus i at time t, ${\theta }_{i,t}$ is the phase angle at bus i at time t, G_ij + jB_ij is the ij term in the Y matrix, VolDev_max, and VI_max are the maximal allowable voltage deviation and voltage imbalance, respectively, and PV_i is the PV capacity at bus i.

3.2.3. Apply NSGA II Algorithm to Solve This Problem

Many decision-making problems intend to optimize multiple objectives simultaneously. The best solution in single objective optimization problem is usually the global minimum or the global maximum. But in the case of multiple objectives, it usually cannot find a solution that is best for all the objectives. In a typical multi-objective optimization problem, there exists a set of solutions which are superior to the rest of the solutions when all the objectives are considered but are inferior to other solutions in one or more objectives. These solutions are known as Pareto-optimal solutions or nondominated solutions. The rest of the solutions are known as dominated solutions.

A general multi-objective optimization problem contains a number of objectives and subjects to a number of inequality and equality constraints. The problem can be described as follows:

$$\begin{array}{l}Minimize/Maximize f(x)=\langle f_1(x) f_2(x)\cdots f_n(x)\rangle \\ Subject to\\ g_i(x)\le 0 j=1,2,\cdots ,J\\ h_k(x)=0 k=1,2,\cdots ,K\end{array}$$

(15)

A decision vector x₁ is called Pareto-optimal if there is no other decision vector x₂ that dominates it. Take the minimization problem for example, the solution x₁ dominates x₂ if:

• $\forall i\in \left\{1,2,\cdots ,N_o\right\}:f_i(x_1)\le f_i(x_2)$
• $\exists j\in \left\{1,2,\cdots ,N_o\right\}:f_j(x_1)<f_j(x_2)$

where N_o is the number of objective functions.

Figure 3 shows the flow chart of the proposed algorithm. Initially, a random parent population is created, and the fast nondominated sorting approach is applied to the parent population. The population is sorted based on the nondomination. Each solution is assigned a fitness value that is equal to its nondomination level. An offspring population is reproduced based on the fitness values by Roulette Wheel Selection. The next generation is obtained by applying the NSGA II procedure to the combined parent and offspring populations. The gene’s representation, fast nondominated sorting approach, crowding distance assignment, and NSGA II procedure are depicted below.

The Gene’s Representation

Decision variables of the proposed model contain the tap values of the OLTC and the positions and capacity of the PVGSs. The optimal solutions can be obtained from the coordination of these decision variables. The gene format containing two parts is shown in Figure 4. The first part is the six tap values to show the operation of the OLTC for different season and time period groups. Each tap value is represented by an integer number ranging from −16 to 16. The second part is responsible for the encoding of the PV position and capacity. For each bus i, the PVGS capacity is represented by the multiple of the unit capacity as shown in Equation (16).

$$Capi=c_i\ast unit capacity$$

(16)

where $0\le c_i\le c_{\max }$.

Nondominated Sorting Genetic Algorithm

A. Fast Nondominated Sorting Approach

The fast nondominated sorting approach is shown below. There are two subjects for each solution that should be calculated: (1) dominated count n_p, the number of solutions dominating the solution p, and (2) S_p, a set of solutions that the solution p dominates. All the solutions in the first nondominated front will have their domination count as zero. For each solution p with n_p = 0, each member (q) of its set S_p is visited and its domination count is reduced by one. Repeating this process, if for any member q the domination count becomes zero, it is put in a separate list Q. These members belong to the second nondominated front. The above procedure is continued with each member of Q and the third front is identified. This process continues until all the fronts are identified (Algorithm 1).

Algorithm 1. Fast-Nondominated-Sort(P) [28]

for each p$\in$P

$S_p=\varnothing$

np = 0

for each q$\in$P

if (p < q) then

$S_p=S_p\cup \left\{q\right\}$

else if (q < p) then

np = np + 1

if np = 0 then

prank = 1

$F_1=F_1\cup \left\{p\right\}$

i = 1

while $F_i\ne \varnothing$

Q = $\varnothing$

for each p$\in$Fi

for each q$\in$Sp

nq = nq − 1

if nq = 0 then

qrank = i + 1

$Q=Q\cup \left\{q\right\}$

i = i + 1

Fi = Q

B. Crowding Distance Assignment

The crowding distance assignment is shown below (Algorithm 2). The crowding distance computation requires sorting the population according to each objective function value in ascending order. For each objective function, the boundary solutions are assigned an infinite distance value. All other intermediate solutions are assigned a distance value that is equal to the absolute normalized difference in the function values of two adjacent solutions. The calculation process is finished for each objective function. The overall crowding distance value is calculated as the sum of each individual distance corresponding to each objective function.

Algorithm 2. Crowding-Distance-Assignment(I) [28]

$l=\left|I\right|$

for each I, set I[i]distance = 0

for each objective m

I = sort(I,m)

I [1]distance = I[l]distance = $\infty$

for i = 2 to (l − 1)

I[i]distance = I[i]distance + (I[i + 1] . m − I[i − 1] . m)/$\left(f_m^{\max }-f_m^{\min }\right)$

C. NSGA II Procedure

The NSGA II procedure is shown in Figure 5. Initially, a random parent population is created. This parent population is sorted by the nondominated sorting algorithm. The fitness value of each solution is equal to its nondomination level. Thus, a minimization of fitness is used in this procedure. The Roulette Wheel Selection, recombination, and mutation operators are used to create an offspring population Q₀ with size N. For the tth generation, a combined population $R_t=P_t\cup Q_t$ is formed first. The population R_t is of size 2N. Then, the population R_t is sorted according to the non-dominated soring algorithm. Elitism is guaranteed because all the previous and current population members are included in R_t. F₁ is the best non-dominated set. If the size of F₁ is smaller than N, all the members of F₁ are chosen for the new population P_t+1. The remaining members of the population are chosen from subsequent nondominated fronts in the order of their ranking. Therefore, solutions from the set F₂ are chosen next, followed by solutions from the set F₃, and so on. In order to choose exact population members N, the solutions of the last front $F_{\ell }$ are sorted using the crowded distance sorting algorithm in descending order and choosing the best solutions that are needed to fill all the population slots. The new population P_t+1 with size N is now used for selection, crossover, and mutation to create a new offspring population Q_t+1 of size N. This procedure continues until convergence.

4. Test Results and Discussion

The proposed multiobjective function of optimal PVGS planning is solved and discussed in this section. The objective function takes system loss and the voltage quality into account. The uncertainties of load and irradiance are also considered in this model. One unit of PVGS’s size is 100 kW. A real system of the Taiwan Power Company is used as the test system. The test substation system including two feeders that are shown in Figure 6. Many studies verify their proposed methods on a single feeder instead of a substation, which could lead to a performance downgrade. The test results of the substation-based and feeder-based analyses are proposed and compared in this section.

4.1. The Substation-Based Test Results

The optimal PVGSs’ placements for the target installed capacity of 1000 kW, 3000 kW, and 5000 kW has been studied. Figure 7a–c show the Pareto frontiers that were obtained from the optimization of the target installed capacity. These figures show that the two objectives are contradicting. If it is required to decrease the loss, the voltage quality index will increase, and vice versa. The system loss decreases as the PV capacity increases because the PVGSs can support part of the load. The voltage quality index is also improved as the PVGSs support the voltage profile.

The concept of the Utopian point is used in this study to find the most preferable solution. The Utopia point is an imaginary point including the best results that can be obtained for the two objectives as shown in Figure 7. The Utopian point can be used as an “ideal” standard for the criteria values to find the best solution from the Pareto optimal set. The Utopian point method minimizes the distance from the points on the Pareto front to the Utopian point as shown in Equation (17).

Table 2. Summary of the optimal compromising solution, optimal loss solution, and optimal voltage quality solution.

	PV Capacity	1000 kW		3000 kW		5000 kW
Objective		Loss (kWh)	VolQ (p.u.)	Loss (kWh)	VolQ (p.u.)	Loss (kWh)	VolQ (p.u.)
optimal compromising		465,193	717.4	398,120	603.9	356,780	497.2
optimal loss		458,265	1415.7	391,616	1308.8	350,182	1318.5
optimal voltage quality		477,329	230.2	411,092	145.2	372,621	111.0

summarizes the optimal compromising solution, optimal loss solution, and the optimal voltage quality solution.

$$Dis\tan ce(i)=\sqrt{\frac{{\left(f_1^i-f_1^{\min }\right)}^2}{{\left(f_1^{\max }-f_1^{\min }\right)}^2}+\frac{{\left(f_2^i-f_2^{\min }\right)}^2}{{\left(f_2^{\max }-f_2^{\min }\right)}^2} }$$

(17)

where $f_1^{\min }$ and $f_1^{\max }$ are the minimal and maximal line loss of the Pareto-optimal solutions, respectively; $f_2^{\min }$ and $f_2^{\max }$ are the minimal and maximal voltage quality indices of the Pareto-optimal solutions, respectively; and $f_1^i$ and $f_2^i$ are the line loss and voltage quality index of the ith Pareto-optimal solution, respectively.

Figure 8 shows the optimal PV placements of different capacity for different objectives including the optimal compromising objective, minimal loss, and minimal voltage quality index. Most of the PVGSs are placed on the end buses. In order to maintain the voltage quality, the PV placement for minimizing the voltage quality index is more even than the placement for minimal loss. These results are also expressed in a tabular form as shown in

Table 3. Summary of the optimal PV placements of the substation-based model for different objectives.

	PV Capacity	1000 kW	3000 kW	5000 kW
Objective		(Position (Bus No.), Capacity (kW))	(Position (Bus No.), Capacity (kW))	(Position (Bus No.), Capacity (kW))
optimal compromising		(120, 500), (220, 100), (222, 200), (223, 100), (224, 100)	(113, 100), (115, 400), (119, 800), (120, 200), (206, 900), (220, 300), (223, 200), (225, 100)	(112, 500), (113, 200), (114, 200), (115, 400), (119, 1100), (120, 100), (205, 1900), (219, 300), (223, 100), (225, 200)
optimal loss		(119, 500), (120, 100), (206, 100), (222, 100), (223, 100), (225, 100)	(112, 100), (115, 400), (119, 1000), (120, 100), (205, 100), (206, 700), (220, 300), (223, 200), (225, 100)	(112, 600), (113, 100), (114, 300), (115, 300), (119, 1100), (120, 100), (204, 100), (205, 1900), (219, 300), (223, 100), (225, 100)
optimal voltage quality		(119, 600), (204, 200), (205, 100), (222, 100)	(115, 200), (119, 900), (205, 300), (206, 800), (220, 300), (223, 200), (225, 300)	(102, 200), (104, 200), (107, 700), (111, 100), (112, 300), (113, 400), (119, 1200), (205, 1800), (225, 100)

.

Figure 9a–c shows the tap value settings of different objectives for the target installed capacity of 1000 kW, 3000 kW, and 5000 kW, respectively. According to the weather pattern in Taiwan, the proposed model forms the winter and spring seasons as one group, while the summer and autumn seasons as the other group. The OLTC adopts the time-base control scheme. There are three time periods, (24:00~8:00), (9:00~17:00), and (18:00~23:00), that are divided for each group. These three time periods are in accordance with the time periods of the off-peak, peak, and half-peak load demand. They can reflect the load variation and avoid too many operating times of OLTC. The test results show the higher tap setting values for the objective of minimizing loss, while the lower tap setting values for the objective of improving voltage quality. As high tap values can raise the voltage level and decrease the current leading to loss reduction, the tap values reach the maximal value of 1.05 p.u. in most of the scenarios. The voltage quality index is defined as the deviation quantity from 1.0 p.u.; therefore, the optimal tap value of OTC for optimal voltage index is about 1.01 p.u. to maintain the bus voltages closing to 1.0 p.u. As the optimal compromising solution minimizes the energy loss and voltage index simultaneously, the tap values are located between them.

4.2. The Feeder-Based Test Results

The feeder-based solutions refer the PVGSs’ placement to a single feeder and the scope of the objective function just contains the considered feeder. Figure 10a–c and Figure 11a–c show different objectives of optimal PV placements of the Feeder1-based and Feeder2-based cases for the target installed capacity of 1000 kW, 3000 kW, and 5000 kW, respectively. The optimal locations are located at the end buses of the feeder when the installed capacity is low and include some front end buses gradually as the installed capacity increases. These results are also expressed in tabular forms as shown in

Table 4. Summary of the optimal PV placements of the Feeder1-Based model for different objectives.

	PV Capacity	1000 kW	3000 kW	5000 kW
Objective		(Position (Bus no.), Capacity (kW))	(Position (Bus No.), Capacity (kW))	(Position (Bus No.), Capacity (kW))
optimal compromising		(115, 200), (119, 500), (120, 300)	(104, 100), (108, 100), (112, 800), (113, 100), (114, 600), (119, 1000), (120, 300)	(102, 800), (103, 400), (104, 400), (107, 200), (108, 200), (111, 300), (112, 700), (113, 200), (114, 600), (119, 900), (120, 300)
optimal loss		(115, 200), (119, 500), (120, 300)	(104, 100), (108, 100), (112, 700), (113, 200), (114, 700), (119, 1000), (120, 200)	(102, 700), (103, 300), (104, 400), (107, 400), (111, 400), (112, 700), (113, 300), (114, 600), (119, 900), (120, 300)
optimal voltage quality		(115, 200), (119, 700), (120, 100)	(103, 700), (111, 100), (112, 900), (113, 100), (114, 200), (119, 1000)	(102, 1000), (103, 900), (107, 400), (111, 400), (112, 1100), (114, 100), (119, 800), (120, 300)

and

Table 5. Summary of the optimal PV placements of the Feeder2-Based model for different objectives.

	PV Capacity	1000 kW	3000 kW	5000 kW
Objective		(Position (Bus No.), Capacity (kW))	(Position (Bus No.), Capacity (kW))	(Position (Bus No.), Capacity (kW))
optimal compromising		(206, 100), (207, 100), (219, 300), (222, 200), (225, 300)	(204, 600), (205, 600), (206, 400), (219, 500), (222, 300), (223, 100), (224, 200), (225, 300)	(202, 1500), (204, 900), (205, 600), (206, 700), (207, 100), (219, 400), (220, 100), (222, 200), (223, 200), (225, 300)
optimal loss		(206, 400), (207, 100), (219, 300), (222, 100), (223, 100)	(204, 500), (205, 500), (206, 800), (219, 500), (222, 300), (223, 100), (224, 100), (225, 200)	(202, 1300), (204, 1200), (205, 400), (206, 900), (219, 500), (222, 200), (223, 200), (225, 300)
optimal voltage quality		(202, 600), (206, 200), (207, 100), (220, 100)	(203, 100), (205, 100), (206, 900), (219, 1000), (222, 400), (223, 100), (224, 100), (225, 300)	(202, 2000), (204, 1100), (205, 300), (206, 800), (219, 200), (220, 100), (222, 300), (223, 100), (225, 100)

. Figure 12a–c and Figure 13a–c show the tap value settings for different objectives of the Feeder1-based and Feeder2-based cases for the target installed capacity of 1000 kW, 3000 kW, and 5000 kW, respectively. For the feeder-based model, the PVGSs are all placed on one single feeder; therefore, the tap values of the feeder-based model are lower than those of the substation-based model due to the voltage concerned that is described in Section 3.1.

Table 6. The comparison of optimal multi-objective values between the different models.

	PV Capacity	1000 kW		3000 kW		5000 kW
Model		Loss (kWh)	VolQ (p.u.)	Loss (kWh)	VolQ (p.u.)	Loss (kWh)	VolQ (p.u.)
Substation-Based		465,193	717.4	398,120	603.9	356,780	497.2
Feeder1-Based		470,368	553.6	416,377	739.8	404,100	104.1
Feeder2-Based		483,891	128.2	419,666	709.2	399,540	724.4

shows the comparison of optimal multi-objective values between the substation-based, Feeder1-based, and Feeder2-based cases. The substation-based model has a lower system loss comparing to the feeder-based models, and the voltage quality index decreases as the installed capacity increases. The voltage quality indices for the feeder-based models have irregular trends due to the fact that the scope of the feeder-based model is just the considered feeder.

5. Conclusions

Along with a large quantity of PVGSs that are being deployed in the distribution system, it is important to mitigate their impacts with optimal planning. This paper proposes a substation-based mathematic model to optimize the OLTC operation and deployment of PVGSs to minimize the system loss and improve the voltage profile. A nondominated sorting genetic algorithm II is used to solve this multi-objective optimization problem. The concept of a Utopian point is used to find the most preferable solution. The test results show that most of the PVGSs are placed on the end parts of the feeders, and the OLTC is operated at higher, and lower tap values for the objective of minimizing loss, and minimizing the maximal voltage index, respectively. Comparisons between the substation-based and feeder-based models are proposed in this paper, and the test results show that the substation-based model has a lower system loss and coincident trend for the voltage quality index.