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10.1371/journal.pone.0308450
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Engineering and Technology
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Coordinated active-reactive power optimization considering photovoltaic abandon based on second order cone programming in active distribution networks
"PV abandon optimization in active distribution networks with 2nd order cone programming"
Peng Bo Data curation Software Writing – original draft Writing – review & editing 1
https://orcid.org/0009-0002-4499-6631
Wang Yongjie Conceptualization Funding acquisition Methodology Validation 1 2 *
1 Faculty of Energy and Electrical Engineering, Qinghai University, Xining, China
2 Qinghai Key Lab of Efficient Utilization of Clean Energy, Qinghai University, Xining, China
Abou Houran Mohamad Editor
Xi’an Jiaotong University, CHINA
Competing Interests: The authors have declared that no competing interests exist.

* E-mail: wangyongjie@qhu.edu.cn
19 9 2024
2024
19 9 e030845015 12 2023
24 7 2024
© 2024 Peng, Wang
2024
Peng, Wang
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

On the basis of predecessors’ coordination optimization of active and reactive power in distribution network, For the necessity of the optimal operation in the distribution network, part of power generated from photovoltaic (PV) cannot be sold to users, and cannot enjoy subsidies. Similarly, the network loss in the power transmission will also bring a certain economic loss. This paper comprehensively considers the economic loss caused by the network loss and PV abandon of the distribution system, and establishes a model to minimize the economic loss. To solve this problem efficiently, the method of DistFlow equation and mixed integer second order cone programming (MISOCP) is used to solve the problem, in this method, the original mixed integer nonlinear programming non-convex problem is transformed into a convex problem, which makes the optimization problem easy to solve. The modified IEEE 33 and IEEE 69 distribution networks are tested by the above method. The optimized results are able to meet the target and have very small relaxation gaps, and the voltage level is also optimized. This coordinated optimization approach helps to optimize the economic operation for active distribution networks with PVs.

State Key Laboratory of Power System Operation and Control Project SKLD22KM10 https://orcid.org/0009-0002-4499-6631
Wang Yongjie This work was supported by the Open Fund of the State Key Laboratory f Power System Operation and Control(SKLD22KM10). Data AvailabilityAll relevant data are within the paper and its Supporting information files.
Data Availability

All relevant data are within the paper and its Supporting information files.
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pmc1 Introduction

With the depletion of fossil energy, the global demand for reducing carbon emission is on a constant increase. As a result, the distributed generation (DG) enjoys more and more popularity. Having the characteristics of local power absorption and being able to improve the stability of the power system [1], DG has become a sustainable mode for the development of renewable energy. However, the access of the DG to the distribution network has a significant impact on the voltage distribution of the power distribution network [2]. Considering that more and more DGs are connected to the distribution network, there are many challenges to the operation and management. There are various types of DGs accessing the distribution network. Photovoltaic (PV) is one of the most often used renewable sources [3]. Photovoltaic power generation has the role of promoting green energy transformation, protecting the ecological environment, and mitigating climate change, and it is an important way for China to realize the goal of carbon peaking and carbon neutrality [4]. In recent years, the installed capacity of PV generation in China is on a constant increase [5], which takes a great proportion and occupies an important position in the northwest power grid of China. Moreover, the installation rate of energy storage system, static reactive power compensation devices and capacitor banks in distribution network rise day by day [6, 7].

Active power optimization (APO) and reactive power optimization (RPO) of the distribution network are two important aspects of the optimization of the distribution network.

Zafar MH [8] use deep learning algorithms to optimize hybrid power generation scenarios with photovoltaics. Reference [9], used the optimized MPPT algorithm to optimize the power distribution network. Reference [10] use Runge Kutta Method algorithm achieves power optimization in the hot spot power generation (TEG) scenario.

Considering the R/X of the distribution system is large and P-Q is not decoupled [11], active power optimization of the distribution network will not only ameliorate the cost and other economic conditions, but also affect the voltage of the distribution network. In [12], PV inverters are used in the distribution system for reactive power optimization to reduce the loss of the system.

Based on the capability of the DG inverters to adjust the power factor, the power loss and network voltage level can be optimized through certain amount of reactive power provided by the inverter of DG [13–15]. Besides, many countries have adopted new grid regulations, in which the power factor for the new PV inverters should be 0.85 or higher [16–18]. Considering the characteristics of DG power factor, the accessed static reactive compensation (SVC) and capacitor banks (CBs), together with DG can be treated as a whole for coordinated active-reactive optimization.

In order to efficiently solve the power flow in the distribution network, the DistFlow equation is used for the radial distribution system. which is robust and efficient [19]. The power flow equation to solve the optimization of distribution network contains DGs, CBs, and SVCs. This problem is a mixed integer nonlinear problem, which is essentially non-convex, and difficult to solve. In order to effectively solve the aforementioned optimization problems, predecessors proposed a variety of algorithms for solving different objectives. Among the heuristic algorithms to solve this problem, there are genetic algorithm, particle swarm optimization (PSO) [20], sparrow search optimization algorithm(SSOA) [21],grey wolf optimizer(GWO) [22],meta-heuristic optimization algorithm(MOA) [23], etc. [24] adopted PSO—PRIM to optimize the low voltage distribution network design. PSO algorithm tries to improve the candidate solution by iterative method, but it is easy to fall into the local optima, and it also takes a long time to reach a desirable result. In [25], Krill Herd Algorithm (KHA) is adopted for the optimal configuration of DG to achieve the minimized network loss. In [26, 27], the meta-heuristic algorithm is adopted to minimize the investment cost of distribution network. In [28], the multi-objective method is adopted to minimize the actual network loss and net re-active power flow through the optimal configuration and classification of DGs and capacitors. These kinds of heuristic algorithm are relatively easy to implement, and therefore, it has attracted wide attention. When solving nonlinear non-convex problems of mixed integers, the analytic algorithm is faster than the heuristic algorithm, and has a rigorous demonstration process. However, as for the non-convex power flow equation of distribution network, traditional methods such as interior point method and gradient descent can only guarantee to get the local optimal solution [11]. In article [29], semi-defined programming is used to relax the original model into a convex model. In [30], the whale optimization algorithm is used to optimize the power loss and voltage characteristics of the non-convex scenario. Compared with the above methods, the SOCP method, which relaxes the flow equation into a convex problem, has a simpler calculation process, and reduces the number of dummy variables and simplifies the complexity. Based on the above discussion, the SOCP method [31, 32] for the branch flow model is more convenient and is of higher quality. [33, 34] analyzed the feasibility of relaxation properties after second order cone (SOC) relaxation being carried out. When the gap between the obtained results and the original feasible region is small, the results have a high accuracy. [35] used the MISOCP to solve the optimal configuration of DG. In [36], the SOCP is used to provide a day-ahead optimal scheduling scheme considering two-way power flow for gas and electric power systems. In [37], the SOCP is used to optimize the distribution of energy storage capacity in the distribution network. In [38], two objectives of minimum network loss and generation cost of power system are established by using SOCP. These papers all use the SOCP method to relax the non-convex problems into convex problems, and have considerable accuracy.

The traditional optimization of parameter configurations usually takes the minimum network loss as the optimization goal. Nowadays, as more new devices are added and more versatile energy sectors like heating networks are integrated in the electric power network [39–42], the optimization goal of the distribution network becomes diverse. In order to improve the voltage distribution, [43] establishes an active and reactive power optimization model to minimize the operation cost. Considering that PV power generation is the main form of medium and low voltage DG, PV is selected as a typical DG in this paper. In [44], the total operating cost is taken as the objective function, and the optimization model of the distribution network with wind and solar clusters is constructed by considering various constraints such as network loss cost, unit operating cost, and energy storage equipment constraints, demand response constraints, etc. In [45], the optimization of PV to distribution network is discussed, and the optimization characteristics such as objective function, variables and constraints are summarized.

Through the analysis of various studies, it is found that for active photovoltaic distribution networks, few studies have comprehensively analyzed the economic loss of distribution networks considering the abandonment of photovoltaic power. The main economic loss of distribution networks containing photovoltaic power is the loss of photovoltaic abandonment and the economic loss of line loss in this distribution network. Therefore, we plan to focus the optimization objective on the economic loss caused by photovoltaic loss and photovoltaic abandonment. However, this optimization scenario is non-convex and difficult to solve using conventional planning algorithms or heuristic algorithms. Therefore, in this active-passive coordinated optimization economic loss scenario, we introduce MISOCP to solve this problem, which can convert non-convex problems into convex problems, which are easier to solve and easier to obtain the global optimal solution. In order to verify its accuracy, we finally analyzed the degree of relaxation, and the results were accurate enough.

For the optimal operation of distribution network, a part of the output of DG is abandoned. This part of the abandoned output, together with the transmission loss, will bring certain economic loss for the whole system. The price of PV grid usually falls along with the cost of PV products, and some countries have policies to provide a certain amount of subsidy [46]. Considering the current development of PV industry in Qinghai Province of China is relatively mature, this paper developed a general economic model for network loss and PV abandon. Therefore, a modified 33-node distribution system with PV DG is designed for the optimization containing the network loss and PV abandon according to certain weight. The DistFlow equation-based MISOCP is used for the comprehensive optimization of the static configuration of the distribution network with DGs, by minimizing the sum of PV abandon and network loss with respect to certain weights in a unit time. Moreover, based on the above method, another group of 69-node distribution system with PV generations was modified to carry out the same analysis and verification. And the simulation of the calculation example satisfies the demand, which has reference significance for the economic operation of distribution network.

2 Methods and constraints of model establishment

2.1. DistFlow branch flow model

For illustration purposes, a sample distribution network with a radial shape is introduced as shown in the Fig 1:

10.1371/journal.pone.0308450.g001 Fig 1 A 7-node radial system.

The Distflow form of the power flow equation is:

For any node j: ∑i∈u(j)(Pij−(Pij)2+(Qij)2(Vi)2rij)+Pj=∑k∈v(j)Pjk∑i∈u(j)(Qij−(Pij)2+(Qij)2(Vi)2xij)+Qj=∑k∈v(j)Qjk (1)

For the branch ij: (Vj)2=(Vi)2−2(rijPij+xijQij)+((rij)2+(xij)2)(Pij)2+(Qij)2(Vi)2 (2)

In the above formula, set u(j) represents the set of the head nodes of the branch with j as the end node in the power grid. Similarly, set v(j) represents the set of end nodes of the branch with j as the head node in the power grid. Pij and Qij respectively represent the active and reactive power at the first end of branch ij. Vi represents the voltage amplitude of node i. rij and xij represent the resistance and reactance of branch ij.

For the node j: Pj=Pj,DG−Pj,dQj=Qj,DG+Qj,com−Qj,d (3)

In the above formula, Pj and Qj are the net injection of active and reactive power of node j.Pj,DG and Qj,DG are respectively the active and reactive power of DG connected to node j, Pj,d and Qj,d are the active and reactive power of the load connected to node j and Qj,com is the reactive power of the reactive compensation device connected to node j, such as CBs, SVCs, etc.

2.2. Objective function based on Distflow

The objective function is defined as: minC1∑i=1Nbus∑j∈v(i)rij|Iij|2+C2∑k=1NDG(PDGkmax−PDGk) (4)

In this formula, Nbus and NDG are the number of nodes and the number of DGs respectively, PDGkmax is the maximum active power that can be generated by each DG, and PDGk is the actual active power of grid-connection for the current period of time. The definition of set v(i) is similar to Eq (1), representing the set of end nodes of the branch starting with i in the power grid. |Iij|2 is the square of the amplitude of the branch current, which can be obtained from Eq (2): |Iij|2=(Pij)2+(Qij)2(Vi)2 (5)

The objective function consists of two parts: transmission line loss and photovoltaic module abandonment loss. These two parts are selected as representatives of economic losses based on their significant impact on the economy and efficiency of the power system. In the objective function, the first part with the coefficient C1 represents the economic loss caused by the power transmission loss of all branches of the whole network. Transmission line losses are mainly due to energy losses caused by resistance and other factors during the transmission process. Specifically, when the current flows in the transmission line, it encounters resistance, causing part of the electrical energy to be converted into heat energy, resulting in losses. This loss is proportional to the square of the current and is related to the resistance of the line. Reducing transmission line losses can not only improve the transmission efficiency of the power grid, but also reduce the operating costs of power generation and transmission, thereby achieving improved economic benefits. The second part with the coefficient C2 represents the economic loss from the subsidies and the direct loss due to unsold electricity from PVs. The photovoltaic module abandonment loss refers to the amount of power that has to be abandoned because the photovoltaic power generation capacity exceeds the actual load demand or the grid’s absorption capacity is insufficient. The reason behind this loss is the volatility and intermittent characteristics of photovoltaic power generation, which makes the photovoltaic power generation exceed the demand or carrying capacity of the grid in certain periods. The abandonment phenomenon leads to the waste of potential power generation, which in turn affects the economy and sustainability of photovoltaic power generation. By optimizing the operation of the power grid and reducing the abandonment loss of photovoltaic modules, renewable energy can be used more effectively and the overall economic benefits can be improved.

The selection of transmission line losses and photovoltaic module abandonment losses as representatives of economic losses aims to fully reflect the main economic factors in the operation of the distribution network. Transmission line losses involve the problem of energy transmission efficiency in traditional power systems, while photovoltaic module abandonment losses reflect the problem of renewable energy utilization in modern power systems. By optimizing these two parts, the overall economic benefits of the distribution network can be maximized, while promoting the sustainable development of the power system.

In conclusion, C1 is the coefficient of network loss to indicate a particular period of time the average electricity price in one hour, C2 is the coefficient of PV abandon to indicate a particular period of time the average electricity price in one hour plus the subsidies per kilowatt. That is

C1 = electricity price per kilowatt in a given hour.

C2 = electricity price per kilowatt in a given hour + subsidy per kilowatt.

The sum of two parts with coefficients represents the loss to the overall economic operation of the system due to PV abandon and network loss in a given hour. In China, the price of PV electricity is jointly subsidized by the national government and local government. The price of electricity generated by the distributed PV power station can be different in different areas, but the national subsidy for each generation unit is the same. In 2023, the national average electricity price in China will be about 0.594¥/kWh. In line with China’s West-to-East Power Transmission Policy, Qinghai Province integrates various types of power generation locally, resulting in high power generation and relatively low electricity charges. On this basis, together with the "guided price for electricity" policy carried out by Qinghai province in 2022, the actual price range for bidding in 2023 is 0.2427–0.4492¥/kWh. By 2023, the average electricity price in Qinghai Province, China is 0.390¥/kWh, so we set it as the network loss coefficient for this study. Qinghai Province, China is one of the main locations of China’s photovoltaic power generation industry, so its subsidies are relatively high. In 2023, the official price of photovoltaic grid-connected power in Qinghai Province is 0.301¥/kWh, and the combined grid-connected power price subsidy of the local government and the national government is 0.050¥/kWh. The electricity price in an hour according to the local electricity price in Qinghai Province, the PV feed-in tariff and the feed-in tariff mitigation are shown in Table 1.

10.1371/journal.pone.0308450.t001 Table 1 The price information for the PVs and the grid.

Electricity Price (¥/kWh)	PV feed-in tariff(¥/kWh)	Feed-in tariff mitigation(¥/kWh)	
0.390	0.301	0.050	

The price of electricity from the power grid includes industrial and commercial price, large industrial price and residential price, and varies greatly for different users on the user side. The network loss means the economic loss of electricity that the grid fails to sell to users, and the PV abandon means the economic loss from the abandoned power in PV. The feed-in tariff mitigation is for the feed-in PV. The coefficient of the two factors are set as 0.390 and 0.351 respectively. That is: C1=0.390C2=0.301+0.050=0.351 (6)

Then we normalized C1 and C2, and were treated 0.526 and 0.474. Therefore, the sum of the first and the second part multiplied by the coefficient of 0.526 and 0.474 represents the general model of economic operation.

2.3. Constraints

1. Power balance constraints:(1), (2), (3). In distribution network optimization models, power balance constraints are imposed to ensure that the power supply at each node in the grid matches the demand. These constraints ensure that at any given moment, the total amount of power generation in the grid equals the total amount of load demand, plus losses in the system.

2. Safety constraints of node voltage and branch current:

In the distribution network optimization model, node voltage and branch current constraints are imposed to ensure the safety, stability and reliability of the power grid. These constraints ensure that the voltage is within the specified range, prevent equipment damage and power quality degradation, and ensure that the current does not exceed the rated capacity of the equipment and lines, prevent overload and overheating, thereby ensuring the safe operation of the power grid and the long life of the equipment. Through these constraints, the optimization model ensures that all parameters of the power grid are always within the safe range while pursuing economic benefits. The constraints are as follows: Vimin≤Vi≤VimaxIij≤Iijmax (7)

In the formula above, Vi is the voltage amplitude of node i, Vimin and Vimax respectively are the maximum and minimum limits of voltage amplitude, Iij is the current amplitude of the branch, and Iijmax is the overload critical current amplitude of the branch. In this simulation study, the upper and lower limits of node voltage are set to 1.03 and 0.97 times. The upper and lower limits of branch current are 1.2kA and -1.2kA.

3. Power constraint

In the distribution network optimization model, active and reactive power balance constraints are key to ensure that the power supply and demand of each node in the power grid are matched. These constraints ensure the stable operation and power quality of the power grid. In order to suppress the influence of the power fluctuation of the active distribution network on the transmission network, the power constraints of the root node of the distribution network need to be taken into account, that is: P0min≤P0≤P0maxQ0min≤Q0≤Q0max (8)

In this formula, P0 is the power into the distribution network from the root node. P0min and P0max are the upper and lower bounds of active power set by the control center respectively. The constraints of reactive power can be calculated similarly.

4. Constraint of discrete reactive power compensation device

In the distribution network optimization model, the constraints of discrete reactive compensation devices are to ensure that the reactive compensation devices are within a reasonable operating range and to correctly enable or disable these devices during the optimization process to achieve reactive power balance and voltage stability. The switching state of CB is a discrete decision variable, and the following linearized model is adopted in this paper: tiQi,comstep=Qi,comdis0≤ti≤nti∈N* (9)

In the formula, ti is an integer variable, Qi,comstep is the step size of each CB, Qi,comdis is the output of the ith CB, and n is the maximal step number of each CB. This simulation has four gears: 0, 1, 2, 3.

5. Constraint of continuous reactive power compensation device

In the distribution network optimization model, the constraints of continuous reactive power compensation devices (such as static VAR compensators SVC or static VAR generators STATCOM) are to ensure that these devices provide appropriate reactive power within their adjustable range to achieve reactive power balance and voltage stability. The constraints are as follows: Qi,commin≤Qi,comcon≤Qi,commax (10)

In this formula, Qi,comcon represents the output of a certain continuous reactive compensation device. Qi,commin and Qi,commax are the upper and lower limits of outputs for SVCs.

The above equation reflects the reactive power compensation capacity constraints of compensating devices with continuously and independently adjustable power (SVCs, etc.).

6. DG operation constraint

In the distribution network optimization model, the operating constraints of DG are to ensure that these power sources operate within their technical and physical limitations and coordinate their operation with the grid to ensure the stability and reliability of the power system. The constraints are as follows: 0≤Pi,DG≤Pi,DGpre0≤Qi,DG≤Qi,DGpreQi,DGpre=Pi,DGpretanφ (11)

The remaining variables in formula (11) are already defined above. Pi,DG and Qi,DG represent the active power output and reactive power output of a certain DG respectively. The steady state operation of DG in this paper adopts PQ type. DG accesses the power grid through power electronic equipment or conventional rotary motor interface, and the power of DG has achieved the independent adjustment of both active and reactive power. To make full use of distributed clean energy, this paper uses the MPPT as the DG operation mode. The active and reactive power output by DG are set as an adjustable variable within the interval [0,Pi,DGpre] and [0,Qi,DGpre] separately where Qi,DGpre=Pi,DGpretanφ, and φ is the power factor angle, considering the flexibility of the power electronic equipment of PV inverter interface.

To sum up, the control variables in the reactive voltage optimization problem of the active distribution networks are the distributed generation Qi,DG and the operational power of continuous and discrete reactive power compensation device Qi,comcon,Qi,comdis. The above problem is a typical mixed integer nonlinear non-convex programming problem, which belongs to NP-hard problem.

3 Solution method

SOCP is a powerful optimization technique for minimizing linear objective functions under complex constraints. SOCP constraints include second-order cone constraints and linear equality constraints, which can describe complex situations such as norm restrictions. Compared with linear programming and quadratic programming, SOCP can handle a wider range of problem types.

A typical application scenario involves solving non-convex problems. Non-convex problems are usually difficult to solve directly to the global optimal solution due to their complexity. In this case, mathematical derivation and relaxation procedures are particularly important. By converting the non-convex constraints in the original problem into second-order cone constraints, we relax the problem into a convex optimization problem. This relaxation process makes the originally difficult non-convex problem solvable. Specifically, through this conversion, the non-convexity of the original problem is simplified, allowing us to efficiently find the global optimal solution of the problem using the existing SOCP solution algorithm.

This non-convex-to-convex relaxation process not only improves the solution efficiency, but also ensures the accuracy of the solution, making SOCP a powerful tool for solving complex optimization problems. In practical applications, this method is widely used in engineering, finance, and machine learning, and has achieved significant application value by accurately modeling and effectively solving optimization problems under complex constraints. Next, we analyze SOCP step by step and apply it to the relaxation work in the current scenario.

3.1. The standard form of SOCP

The standard form of SOCP is defined as: minxi{cTx|Ax=b,xi∈K,i=1,2…,N} (12)

In the above formula, x ∈ RN is a variable, b ∈ RM, c ∈ RN, AM×N ∈ RM×N are coefficient constants, and K is a second order cone or a rotating second order cone of the following form:

a) second order K={xi∈RN|y≥∑i=1Nxi2,y≥0} (13)

b) rotating second order K={xi∈RN|yz≥∑i=1Nxi2,yz≥0} (14)

SOCP can be regarded as an extension of linear programming, which is essentially a convex optimization. SOCP has positive properties such as the optimality of solutions and computational efficiency. The existing CPLEX, GROBI, MOSEK and other algorithm packages can be used to achieve favorable results, and many SOCP-based optimization problems can be completed within a polynomial time.

3.2. MISOCP for optimization of distribution network

Considering the strong non-convex form of formulas (1) and (2) the above problem belongs to NP-hard problem, where finding the optimal solution is difficult and the solving efficiency is low. However, the problem can be relaxed using the SOCP. Two new variables can be introduced: squared voltage amplitude v2,i and squared branch current amplitude i2,ij: v2,i=Vi2i2,ij=|Iij|2=(Pij)2+(Qij)2Vi2 (15)

Replace the related terms in both the objective function and constraints with the above variables, and the objective function becomes minC1∑i=1Nbus∑j∈v(i)riji2,ij+C2∑k=1NDG(PDGkmax−PDGk). Then we can add i2,ij=(Pij)2+(Qij)2v2,i to the constraint. Under a set of sufficient conditions, such as the strict increment function of the objective function being i2,ij, the above equation can be transformed as follows: i2,ij≥(Pij)2+(Qij)2v2,i (16)

Then, after equivalent transformation, Eq (15) is written in the form of a standard SOC, that is: 2Pij2Qiji2,ij−v2,i2≤i2,ij+v2,i (17)

Thus, the power flow equation can be transformed into the following form: ∑i∈u(j)(Pij−i2,ijrij)+Pj=∑k∈v(j)Pjk∑i∈u(j)(Qij−i2,ijxij)+Qj=∑k∈v(j)Qjkv2,j=v2,i−2(rijPij+xijQij)+(rij2+xij2)i2,ij2Pij2Qiji2,ij−v2,i2≤i2,ij+v2,i (18)

Then the original reactive power optimization problem is finally transformed into the following model: minw1∑i=1Nbus∑j∈v(i)riji2,ij+w2∑k=1NDG(PDGkmax−PDGk)s.t.(3),(6),(7),(8),(9),(10),(11),(18) (19)

3.3. Solvability analysis

For the following mathematical problems: minf(x)s.t.gi(x)≤0,i=1,2,…,mhj(x)=0,j=1,2,…,l (20)

The criteria for this model to be convex programming are: the objective function f(x) being a convex function (the Hessen matrix is semi-positive definite), all equations in the constraint hj(x) being linear functions, and all Hessen matrix of inequality constraint gi(x) being semi-positive definite.

For the optimization model proposed in this paper, if discrete control variables such as energy storage and reactive power compensation device are not taken into account, the objective function, all equality constraints, and all inequality constraints except Eq (16) can be all considered as linear. The mathematical form of Eq (16) can be abstractly written as x12+x22+x32≤x4. This form of inequality satisfies the definition of the SOC, whose Hessen matrix (the second derivative matrix) is: 1/(x12+x22+x32)(1/2)−x12/(x12+x22+x32)(3/2)−(x1*x2)/(x12+x22+x32)(3/2)−(x1*x3)/(x12+x22+x32)(3/2)0−(x1*x2)/(x12+x22+x32)(3/2)1/(x12+x22+x32)(1/2)−x22/(x12+x22+x32)(3/2)−(x2*x3)/(x12+x22+x32)(3/2)0−(x1*x3)/(x12+x22+x32)(3/2)−(x2*x3)/(x12+x22+x32)(3/2)1/(x12+x22+x32)(1/2)−x32/(x12+x22+x32)(3/2)00000 (21)

The eigenvalues of the above Hessen matrix are: 00x14+2*x12*x22+2*x12*x32+x24+2*x22*x32+x34/x12+x22+x325/2 (22)

It is not difficult to find that the eigenvalues of the matrix are all non-negative, that is, the Hessen matrix is semi-positive definite, which indicates that the feasible region of the SOC is convex. In other words, the above problem is a SOC convex programming problem.

After adding discrete variables, this model is no longer a strictly convex programming problem, but the convexification of the power flow equation, a strong non-convex source, changes the mathematical properties of the problem, and the solvability and optimality are improved accordingly.

If integer variables are excluded, as shown in the Fig 2, the feasible domain of the original problem Coriginal will be relaxed into a feasible domain Csoc that is a convex SOC. At this point, the original problem essentially becomes a convex programming. Due to the introduction of SOC relaxation, the optimal solution S found in Csoc is a lower bound solution of the original problem. If the optimal solution is a point in the feasible region Coriginal, it is the optimal solution of the original problem. The equal sign in Eq (16) can be guaranteed to be accurate enough to satisfy all constraints of the original problem when the original problem finds the optimal solution.

10.1371/journal.pone.0308450.g002 Fig 2 Simplified schematic diagram of feasibility domain.

Primitive non-convex feasible region. The feasible region of a loose convex cone.

If the integer variables are taken into account, the original problem is extended to a SOC optimization problem with mixed integer variables. Existing algorithm packages such as CPLEX, Gurobi and MOSEK can find the optimal solution of the original problem by cutting-plane method or branch and bound method.

4. Case analysis and discussion

As mentioned above, for the distribution network, we relax the Eq (15) into the Eq (16). The original feasible region is relaxed into a wider feasible region, and then the problem to be solved in the relaxed feasible region has a strong convexity. Thus, we turned the original non-convex optimization problem into a convex problem.

In order to verify the performance, the above procedure is programmed using MATLAB and algorithm packages such as CPLEX. The hardware environment of the system is AMD Ryzen 7 5800H,CPU@3.2GHz and 8GB memory, with Windows 11 64bit installed as the operating system, and the development environment is MATLAB R2022A.

The feasibility and relaxation accuracy of the proposed method are firstly tested on a smaller scale, that is a modified IEEE33 distribution network, and the optimal scheduling strategy for each adjustable reactive power compensation device is obtained. Then, the model is extended to a modified IEEE 69 distribution network, which contains more branches and nodes, in order to test the accuracy, optimality and efficiency of the proposed method under different test systems.

4.1. Modified IEEE 33 distribution network test example

The modified IEEE 33 distribution network is shown in Fig 3.

10.1371/journal.pone.0308450.g003 Fig 3 Modified IEEE 33 distribution network.

The system consists of 33 branches and operates radially. The voltage class is 12.66kV, the total active power of the load is 3715kW, and the total reactive power is 2300kVar. Nodes 6, 18, 19 and 31 are respectively connected to PVs. The active power of each PV is set as 1.4MW, and the power factor cos φ is 0.95, thus the reactive power is 0.4602 MVar. Given the small geographical distance of each PV, it is considered that the predicted power of the four PVs is equal. Nodes 22 and 25 are respectively connected with CBs, where the maximum step number is 3, and the step size is 100kVar. Node 9 is connected with a continuous adjustable SVC, and the compensation range of SVC is -300kVar to 300kVar.

The load of each bus and the impedance of each branch is shown in Table 2:

10.1371/journal.pone.0308450.t002 Table 2 The load of each bus and the impedance of each branch in modified IEEE 33.

Bus number	Pd (kW)	Qd(kVar)	branch from-to	Impedance(Ω)	
1	0	0	1–2	0.0922+j0.047	
2	100	60	2–3	0.493+j0.2511	
3	90	40	3–4	0.366+j0.1864	
4	120	80	4–5	0.3811+j0.1941	
5	60	30	5–6	0.819+j0.707	
6	60	20	6–7	0.1872+j0.6188	
7	200	100	7–8	0.7114+j0.2351	
8	200	100	8–9	1.03+j0.74	
9	60	20	9–10	1.044+j0.74	
10	60	20	10–11	0.1966+j0.065	
11	45	30	11–12	0.3744+j0.1238	
12	60	35	12–13	1.468+j1.155	
13	60	35	13–14	0.5416+j0.7129	
14	120	80	14–15	0.591+j0.526	
15	60	10	15–16	0.7463+j0.545	
16	60	20	16–17	1.289+j1.721	
17	60	20	17–18	0.732+j0.574	
18	90	40	2–19	0.164+j0.1565	
19	90	40	19–20	1.5042+j1.3554	
20	90	40	20–21	0.4095+j0.4784	
21	90	40	21–22	0.7089+j0.9373	
22	90	40	3–23	0.4512+j0.3083	
23	90	50	23–24	0.898+j0.7091	
24	420	200	24–25	0.896+j0.7011	
25	420	200	6–26	0.203+j0.1034	
26	60	25	26–27	0.2842+j0.1447	
27	60	25	27–28	1.059+j0.9337	
28	60	20	28–29	0.8042+j0.7006	
29	120	70	29–30	0.5075+j0.2585	
30	200	600	30–31	0.9744+j0.963	
31	150	70	31–32	0.3105+j0.3619	
32	210	100	32–33	0.341+j0.5302	
33	60	40			

Based on the above 33 nodes corresponding to the data and parameters, the solutions and optimal scheduling strategy obtained by calling CPLEX in MATLAB are shown in the following table and figures:

The information about optimal solution is shown in Table 3.

10.1371/journal.pone.0308450.t003 Table 3 Optimal solution in modified IEEE 33.

Solution time (s)	Objective Function: C1*Network loss+ C2*PV abandon (¥)	
0.526*Network loss	0.474 *PV abandon	Sum	
1.07	15.16	879.83	894.99	

After solving the problem, the minimum value of objective function is 894.99, which represents the minimum economic loss per hour caused by the network loss and PV abandon. It has certain reference value for the comprehensive economic operation of the active distribution network with PVs.

The outputs of PVs, SVC and CBs under the optimal solution are respectively shown in the Figs 4–6:

10.1371/journal.pone.0308450.g004 Fig 4 PV outputs in modified IEEE 33.

10.1371/journal.pone.0308450.g005 Fig 5 SVC output in modified IEEE33.

10.1371/journal.pone.0308450.g006 Fig 6 CBs output in modified IEEE33.

The proportions of discarded active power and on-gird power of PV are shown in Fig 7.

10.1371/journal.pone.0308450.g007 Fig 7 The proportions of abandoned and on-grid power from all PVs.

Voltage distribution is an important index of power quality, the curves of bus voltage amplitude before and after the optimization are shown in Fig 8. There has been a very significant improvement in the bus voltage amplitude after the coordinated active-reactive power optimization.

10.1371/journal.pone.0308450.g008 Fig 8 Bus voltage amplitude before and after optimization in modified IEEE 33.

The example provides a set of optimal solution information for the modified IEEE 33 to operate with the minimum economic loss, and the bus voltage amplitude is also improved.

Finally, we explore the impact of different C1 and C2 in different regions on the optimal solution. By appropriately simulating the results of different regions, we can perform sensitivity analysis on the optimization model. Under the premise of ensuring that the sum of C1 and C2 is 1, C1 starts from 0 and takes a step of 0.1 to explore the changes in the optimal solution under different weighting factors. In addition, we use C1 and C2 as indicators to explore the accuracy of SOCP relaxation, and count the maximum gap value of the model under different weighting factors to verify the accuracy of the model. The results are shown in the Fig 9 below:

10.1371/journal.pone.0308450.g009 Fig 9 Optimal solution and relaxation gap changes under different weighting factors.

By changing the weighting factors of C1 and C2, it can be found that when the weight of C1 increases, the economic loss of the optimal solution usually decreases. This is because the model will consider reducing the transmission line loss more during the optimization process, and the reduction of this part of the loss can bring significant economic benefits. However, this may also lead to an increase in the abandonment loss of photovoltaic modules, because the model may tend to reduce the load of the transmission line and ignore the full utilization of photovoltaic power generation. Therefore, the weight setting of C1 and C2 needs to find a balance between reducing transmission line losses and photovoltaic module abandonment losses. The weighting factor values of C1 and C2 determined in this study are in line with the local situation in Qinghai Province.

We found that with the increase of C1, that is, the increase in the proportion of network loss, the relaxation gap tends to decrease as a whole, and a sharp drop occurs at 0.5. In this study, C1 is set to be 0.526, and the corresponding maximum relaxation gap is 1.23 × 10−6. This indicates that network loss is the main influencing factor of the relaxation gap. Under the premise of fixed resistance, small network loss means relatively small current, so the difference in the relaxation gap will also be smaller. In the weighting factors selected in this study, the relaxation gap is very small, which can indirectly prove that our simulation optimization model has high accuracy.

4.2. Modified IEEE 69 distribution network test example

The modified IEEE 69 distribution network is shown in Fig 10.

10.1371/journal.pone.0308450.g010 Fig 10 Modified IEEE 69 distribution network.

The system consists of 69 branches and operates radially. The voltage class is 12.66kV, the total active power of the load is 3715kW, and the total reactive power is 2300kVar. Nodes 7, 12, 35, 50, 53 and 69 are respectively connected to PVs. The active power of each PV is set as 0.75MW, and the power factor cos φ is 0.95, thus the reactive power is 0.2465MVar. Given the small geographical distance of each PV, it is considered that the predicted PV power of the six PVs is equal. Nodes 21, 41, 57 and 56 are respectively connected with four CBs, where the maximum compensation of each CB is 300kVar, and the step size is 100kVar. Nodes 27 and 45 are connected with two SVCs, and the compensation range of each SVC is -300kVar to 300kVar. Based on the above data and configuration parameters, the solutions and optimal scheduling strategy are shown in the following table and figure. The method of solving this case is consistent with that in modified IEEE33.

The load of each bus and the impedance of each branch is shown in Table 4:

10.1371/journal.pone.0308450.t004 Table 4 The load of each bus and the impedance of each branch in modified IEEE 69.

Bus number	Pd (kW)	Qd(kVar)	branch from-to	Impedance(Ω)	
1	0	0	1–2	0.0050+j0.0012	
2	0	0	2–3	0.0050+j0.0012	
3	0	0	3–4	0.0015+j0.0036	
4	0	0	4–5	0.0251+j0.0294	
5	0	0	5–6	0.3660+j0.1864	
6	2.6	2.2	6–7	0.3811+j0.1941	
7	40.4	30	7–8	0.0922+j0.0470	
8	75	54	8–9	0.0493+j0.0251	
9	30	22	9–10	0.8190+j0.2707	
10	28	19	10–11	0.1872+j0.0691	
11	145	104	11–12	0.7114+j0.2351	
12	145	104	12–13	1.0300+j0.3400	
13	8	5.5	13–14	1.0440+j0.3450	
14	8	5.5	14–15	1.0580+j0.3496	
15	0	0	15–16	0.1966+j0.0650	
16	45.5	30	16–17	0.3744+j0.1238	
17	60	35	17–18	0.0047+j0.0016	
18	60	35	18–19	0.3276+j0.1083	
19	0	0	19–20	0.2106+j0.0696	
20	1	0.6	20–21	0.3416+j0.1129	
21	114	81	21–22	0.0140+j0.0046	
22	5.3	3.5	22–23	0.1591+j0.0526	
23	0	0	23–24	0.3463+j0.1145	
24	28	20	24–25	0.7488+j0.2457	
25	0	0	25–26	0.3089+j0.1021	
26	14	10	26–27	0.1732+j0.0572	
27	14	10	3–28	0.0044+j0.0108	
28	26	18.6	28–29	0.0640+j0.1565	
29	26	18.6	29–30	0.3978+j0.1315	
30	0	0	30–31	0.0702+j0.0232	
31	0	0	31–32	0.3510+j0.1160	
32	0	0	32–33	0.8390+j0.2816	
33	14	10	33–34	1.7080+j0.5646	
34	19.5	14	34–35	1.4740+j0.4873	
35	6	4	4–36	0.0034+j0.0084	
36	0	0	36–37	0.0851+j0.2083	
37	79	56.4	37–38	0.2898+j0.7091	
38	384.70	274.5	38–39	0.0822+j0.2011	
39	384.70	274.5	8–40	0.0928+j0.0473	
40	40.5	28.3	40–41	0.3319+j0.1114	
41	3.6	2.7	9–42	0.1740+j0.0886	
42	4.35	3.5	42–43	0.2030+j0.1034	
43	26.4	19	43–44	0.2842+j0.1447	
44	24	17.2	44–45	0.2813+j0.1433	
45	0	0	45–46	1.5900+j0.5337	
46	0	0	46–47	0.7837+j0.2630	
47	0	0	47–48	0.3042+j0.1006	
48	100	72	48–49	0.3861+j0.1172	
49	0	0	49–50	0.5075+j0.2585	
50	1244	888	50–51	0.0974+j0.0496	
51	32	23	51–52	0.1450+j0.0738	
52	0	0	52–53	0.7105+j0.3619	
53	227	162	53–54	1.041+j0.5302	
54	59	42	11–55	0.2012+j0.0611	
55	18	13	55–56	0.0047+j0.0014	
56	18	13	12–57	0.7394+j0.2444	
57	28	20	57–58	0.0047+j0.0016	
58	28	20	3–59	0.0044+j0.0108	
59	26	18.55	59–60	0.0640+j0.1565	
60	26	18.55	60–61	0.1053+j0.1230	
61	0	0	61–62	0.0304+j0.0355	
62	24	17	62–63	0.0018+j0.0021	
63	24	17	63–64	0.7283+j0.8509	
64	1.2	1	64–65	0.3100+j0.3623	
65	0	0	65–66	0.0410+j0.0478	
66	6	4.3	66–67	0.0092+j0.0116	
67	0	0	67–68	0.1089+j0.1373	
68	39.22	26.3	68–69	0.0009+j0.0012	
69	39.22	26.3			

The information about optimal solution is shown in Table 5.

10.1371/journal.pone.0308450.t005 Table 5 Optimal solution in modified IEEE 69.

Solution time (s)	Objective Function: C1*Network loss+ C2*PV abandon (¥)	
0.526*Network loss	0.474 *PV abandon	Sum	
1.87	12.22	308.37	320.59	

After solving the problem, the minimum value of objective function is 320.59, which represents the minimum economic loss per hour caused by the network loss and PV abandon, and the outputs of PVs, SVCs and CBs in the optimal solution are respectively shown in the Figs 11–13:

10.1371/journal.pone.0308450.g011 Fig 11 PVs outputs in modified IEEE 69.

10.1371/journal.pone.0308450.g012 Fig 12 SVCs outputs in modified IEEE69.

10.1371/journal.pone.0308450.g013 Fig 13 CBs output in modified IEEE69.

The proportions of discarded active power and on-gird power of PVs are shown in Fig 14.

10.1371/journal.pone.0308450.g014 Fig 14 The proportions of abandoned and on-grid power from all PVs.

The curves of bus voltage amplitude before and after the optimization are shown in Fig 15. It can be seen that the bus voltage amplitude also has a significant improvement after the coordinated active-reactive power optimization.

10.1371/journal.pone.0308450.g015 Fig 15 Bus voltage amplitude before and after optimization in modified IEEE 69.

The example provides a set of optimal solution information for the modified IEEE 69 to operate with the minimum economic loss, and the the bus voltage amplitude is also improved.

4.3. Accuracy analysis of relaxation

In order to verify the accuracy of Eq (16), the gap distribution caused by SOC relaxation is shown in the Figs 16 and 17:

10.1371/journal.pone.0308450.g016 Fig 16 Scatter diagram of relaxation gap of modified IEEE 33.

10.1371/journal.pone.0308450.g017 Fig 17 Scatter diagram of relaxation gap of modified IEEE 69.

At the optimal solution after relaxation, the infinite norm of the SOC relaxation gap vector of the branch is defined as follows: devi=i2−(P)2+(Q)2v2∞ (23)

In the modified IEEE 33 optimal solution, devi=1.236×10−6 (24)

In the modified IEEE 69 optimal solution, devi=1.410×10−5 (25)

It is not difficult to find from the above equation that the SOC relaxation adopted in this paper is accurate enough.

Finally, we randomly generated 10000 sets of data for Monte Carlo algorithm quantization to verify the results. The optimal solution can be obtained only under the configuration conditions of the above solution. It can be seen that the calculation example is valid and reasonable.

5. Conclusions

This paper established a DistFlow model suitable for radial distribution networks and introduced SOC relaxation to convert non-convex problems into convex ones, further extending it to a MISOCP model. The numerical examples verified the method’s high relaxation accuracy, short solution time, and strong optimality. Future work plans to study the network topology of radial distribution networks, and then reconstruct and optimize them based on the original ones; then further combine the time scale to perform dynamic planning and reconstruction based on the optimal value of the real-time distribution network. The current study does not account for variables introduced by multiple daily sessions or constraints on the number of switching times of discrete devices. Future research should focus on these aspects to develop a more comprehensive optimized scheduling strategy. Additionally, solving more complex mathematical models will be necessary to address these challenges.

Supporting information

S1 File (M)

S2 File (M)

10.1371/journal.pone.0308450.r001
Decision Letter 0
Abou Houran Mohamad Academic Editor
© 2024 Mohamad Abou Houran
2024
Mohamad Abou Houran
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Submission Version0
23 May 2024

PONE-D-23-42185Coordinated Active-Reactive Power Optimization Considering Photovoltaic Abandon based on Second Order Cone Programming in Active Distribution NetworksPLOS ONE

Dear Dr. wang,

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Reviewer #1: The paper entitled Coordinated Active-Reactive Power Optimization Considering Photovoltaic Abandon based on Second Order Cone Programming in Active Distribution Networks

• The paper is well organised

• Did authors have considered the uncertainties in load. Justify it.

• Why authors considered 1 and 2, as 0.526 and 0.474. Justify it.

• Why authors have selected specific buses for PV and CBs installation? Justify it

• More recent literature could be included especially from relevant publications

Reviewer #2: I have now completed the review and these are the suggestions.

Major Comments:

The introduction section needs to be more focused and provide a clear motivation for the work presented. The background information on distributed generation and reactive power optimization is extensive, but the specific gap or problem addressed by this study should be clearly highlighted.

Include these works in literature review:

Khan NM, Khan UA, Asif M, Zafar MH. Analysis of deep learning models for estimation of MPP and extraction of maximum power from hybrid PV-TEG: A step towards cleaner energy production. Energy Reports. 2024 Jun 1;11:4759-75.

Al-Tawalbeh N, Zafar MH, Radzi MA, Zainuri MA, Al-Wesabi I. Novel initialization strategy: Optimizing conventional algorithms for global maximum power point tracking. Results in Engineering. 2024 Jun 1;22:102067.

Mansoor M, Abou Houran M, Al-Tawalbeh N, Zafar MH, Akhtar N. Thermoelectric power generation system intelligent Runge Kutta control: A performance analysis using processor in loop testing. Energy Conversion and Management: X. 2024 May 1:100612.

The objective function formulation needs further clarification and justification. The rationale behind the coefficients C1 and C2, and the inclusion of PV abandon and network loss components, should be explained in more detail.

The constraint formulation section could be improved by providing a clearer explanation of the various constraints considered, such as voltage limits, branch current limits, and reactive power compensation device constraints.

The solution methodology section could benefit from a more detailed explanation of the Second-Order Cone Programming (SOCP) technique used. The mathematical derivations and relaxations involved should be explained more clearly, especially for readers unfamiliar with SOCP.

The case study section should provide more information on the modified IEEE 33 and IEEE 69 distribution networks used for testing. Details such as network topology, load profiles, and DG/PV locations should be clearly stated.

The results and discussion section could be improved by providing a more comprehensive analysis of the obtained solutions. The impact of different weighting factors (C1 and C2) on the optimal solution and the trade-off between network loss and PV abandon should be discussed.

The accuracy analysis of the relaxation technique used should be more thorough. Additional metrics or statistical measures could be used to quantify the accuracy and potential errors introduced by the relaxation.

The conclusions section should be more concise and focused on the key findings and contributions of the study. Potential limitations and future work directions should also be discussed.

Minor Comments:

Some of the equations and variable definitions could be improved for better readability and clarity. For example, the use of subscripts and superscripts should be consistent throughout the manuscript.

The abbreviations and acronyms used in the manuscript should be defined at their first occurrence or in a separate list of abbreviations.

The literature review section could be more comprehensive and include more recent references related to the problem addressed in the study.

The quality of some figures, particularly those showing network topologies, could be improved for better clarity and visual representation.

The manuscript could benefit from a thorough language and grammar check to improve the overall readability and clarity of the text.

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Reviewer #1: Yes: Dr. Pamshetti Vijay Babu

Reviewer #2: No

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10.1371/journal.pone.0308450.r002
Author response to Decision Letter 0
Submission Version1
10 Jul 2024

First, we would like to sincerely thank the referee for the insightful comments and invaluable suggestions. Here are the responses to the reviewer’s comments. We have numbered the reviewer’s comments to facilitate referring to them in the manuscript.(The word version of the reviewer's comments is also uploaded as an attachment in the system.)

Responses to Referee 1:

Comment1：Did authors have considered the uncertainties in load. Justify it.

This study sets the time scale to a certain time section or a small time window, and simulates and optimizes the distribution network at this moment. Therefore, this paper does not discuss the uncertainty of the load, but we plan to conduct research related to dynamic programming and network reconstruction based on this paper, and then we will consider multiple time scales and focus on the uncertainty of the load.

Comment2：Why authors considered 1 and 2, as 0.526 and 0.474. Justify it.

In order to make the C1 and C2 coefficients easier to understand, we added an explanation of the origin of the network loss and abandoned light coefficients in lines 199-209 of the manuscript. We chose this coefficient, which is fundamentally based on the local electricity price situation and the photovoltaic industry and its subsidies in Qinghai Province, China.

Comment3：Why authors have selected specific buses for PV and CBs installation? Justify it

In fact, the allocation of specific DG and other devices based on the IEEE 33 and IEEE 69 examples is optional. As long as it is reasonable, different authors can adjust the number, location and even parameters of the devices according to the actual conditions in different regions. We just used such a matching method and conducted our simulation optimization research based on this custom method to reflect the superiority of the mathematical model processing and optimization method application used in our paper.

Comment4：More recent literature could be included especially from relevant publications

Following your suggestion, I have cited two recent articles on electrical engineering published by PLOS ONE in my manuscript. The text is located at lines 33-37, and the citation is located at lines 541-544..

Responses to Referee 2:

Comment 1：The introduction section needs to be more focused and provide a clear motivation for the work presented. The background information on distributed generation and reactive power optimization is extensive, but the specific gap or problem addressed by this study should be clearly highlighted.

We have revised the review to clarify the research motivation and current research questions, which are located on pages 104-115.

Comment 2：Include these works in literature review:

（1）Khan NM, Khan UA, Asif M, Zafar MH. Analysis of deep learning models for estimation of MPP and extraction of maximum power from hybrid PV-TEG: A step towards cleaner energy production. Energy Reports. 2024 Jun1;11:4759-75.

（2）Al-Tawalbeh N, Zafar MH, Radzi MA, Zainuri MA, Al-Wesabi I. Novel initialization strategy: Optimizing conventional algorithms for global maximum power point tracking. Results in Engineering. 2024 Jun 1;22:102067.

（3）Mansoor M, Abou Houran M, Al-Tawalbeh N, Zafar MH, Akhtar N. Thermoelectric power generation system intelligent Runge Kutta control: A performance analysis using processor in loop testing. Energy Conversion and Management: X. 2024 May 1:100612.

Your suggestion is very useful, which fills in the weak points in our review, which is on pages 43-46 of the main text. Reference at：590-602.

In order to better enrich the review, we have added three additional articles, which are located in lines 64-66 of the manuscript. The specific added works are:

（1）M. H. Zafar, U. A. Khan and N. M. Khan, "A sparrow search optimization algorithm based MPPT control of PV system to harvest energy under uniform and non-uniform irradiance," 2021 International Conference on Emerging Power Technologies (ICEPT), Topi, Pakistan, 2021, pp. 1-6

（2）Muhammad Hamza Zafar, Noman Mujeeb Khan, Adeel Feroz Mirza, Majad Mansoor,Bio-inspired optimization algorithms based maximum power point tracking technique for photovoltaic systems under partial shading and complex partial shading conditions,Journal of Cleaner Production,Volume 309,2021, 127279

（3）Muhammad Hamza Zafar, Noman Mujeeb Khan, Adeel Feroz Mirza, Majad Mansoor, Naureen Akhtar, Muhammad Usman Qadir, Nauman Ali Khan, Syed Kumayl Raza Moosavi,A novel meta-heuristic optimization algorithm based MPPT control technique for PV systems under complex partial shading condition,Sustainable Energy Technologies and Assessments,Volume 47,2021,101367.

Comment 3：The objective function formulation needs further clarification and justification. The rationale behind the coefficients C1 and C2, and the inclusion of PV abandon and network loss components, should be explained in more detail.

In response to your point of view, we have modified the expression of the objective function and explained in more detail the meaning of the two parts of the objective function: abandoned light and network loss, as well as the reason for choosing these two as representatives. The specific modified text is located in lines 162-189 of the manuscript.

Comment 4：The constraint formulation section could be improved by providing a clearer explanation of the various constraints considered, such as voltage limits, branch current limits, and reactive power compensation device constraints.

A new text explanation section has been added to the manuscript for each constraint, which can make it easier for readers to understand the composition of the constraint, and specific constraints such as voltage and current have been added. The specific modified text is located in lines 224-279 of the manuscript.

Comment 5：The solution methodology section couldbenefit from a more detailed explanation of the Second-Order Cone Programming (SOCP) technique used. The mathematical derivations and relaxations involved should be explained more clearly, especially for readers unfamiliar with SOCP.

On pages 295-313, before the mathematical explanation of the second-order cone model, a new paragraph is added to introduce the SOCP technology in a more popular and straightforward manner. By explaining its principles and relaxation process in a more superficial way, readers can more easily understand the subsequent SOCP content.

Comment 6：The case study section should provide more information on the modified IEEE 33 and IEEE 69 distribution networks used for testing. Details such as network topology, load profiles, and DG/PV locations should be clearly stated.

The manuscript originally contained detailed information, but we later deleted it after considering the length of the article. After listening to your suggestions, we restored it to the manuscript, which can be found in: lines 411-413 and 474-476, showing the revised network data of IEEE 33 nodes and 69 nodes respectively.

Comment 7：The results and discussion section could be improved by providing a more comprehensive analysis of the obtained solutions. The impact of different weighting factors (C1 and C2) on the optimal solution and the trade-off between network loss and PV abandon should be discussed.

After listening to your suggestions, we chose to discuss the impact of different weighting factors in the IEEE33 example. We used different C1C2 weighting factors to perform simulation optimization, statistics and drawing analysis. Finally, we also summarized some rules, and the network loss coefficient factor is inversely proportional to the optimal solution. Located on pages 435-459.

Comment 8：The accuracy analysis of the relaxation technique used should be more thorough. Additional metrics or statistical measures could be used to quantify the accuracy and potential errors introduced by the relaxation.

In solving your Comment 7, we decided to use the change of weighting factors to statistically analyze the maximum node relaxation gap value of the optimized model with different weighting factors. Through statistical calculation, we found that the results are of reference significance. With the increase of C1 (C2=1-C2), the relaxation gap tends to decrease, and it drops three orders of magnitude from 0.007 at about 0.51 and then stabilizes. The maximum gap of the node at 0.526 set in this paper is 1.23e-6 (p.u.). In comparison, the relaxation gap is small, which can prove that the research optimization simulation has high accuracy. The specific location is the same as comment 7, located on pages 435-459.

Comment 9：The conclusionssection should be more concise and focused on the key findings and contributions of the study. Potential limitations and future work directions should also be discussed.

According to your suggestion, we have revised the original conclusion section and re-edited it to ensure that the content is not redundant and to focus more on its findings, limitations and future research prospects. Located on pages 511-521.

Comment 10：Minor Comments:

（1）Some of the equations and variable definitions could be improved for better readability and clarity. For example, the use of subscripts and superscripts should be consistent throughout the manuscript.

（2）The abbreviations and acronyms used in the manuscript should be defined at their first occurrence or in a separate list of abbreviations.

（3）The literature review section could be more comprehensive and include more recent references related to the problem addressed in the study.

（4）The quality of some figures, particularly those showing network topologies, could be improved for better clarity and visual representation.

（5）The manuscript could benefit from a thorough language and grammar check to improve the overall readability and clarity of the text.

Thank you for your additional suggestions, which are also very important to us. We checked and regulated the equations in the manuscript to ensure that they are clear and readable. We ensured the abbreviations of special terms were set, and after confirmation, all abbreviations were fully defined and explained when they first appeared. In addition to adding the corresponding six articles, we also added two recent articles in the electrical field of PLOSONE and updated the references. We improved the clarity of the exported images to ensure that you can observe the result data more clearly. Finally, we checked the grammar of the entire article to eliminate potential errors and prevent the possibility of misunderstanding by future readers.

Attachment Submitted filename: Response to Reviewers 2.docx

10.1371/journal.pone.0308450.r003
Decision Letter 1
Abou Houran Mohamad Academic Editor
© 2024 Mohamad Abou Houran
2024
Mohamad Abou Houran
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Submission Version1
24 Jul 2024

Coordinated Active-Reactive Power Optimization Considering Photovoltaic Abandon based on Second Order Cone Programming in Active Distribution Networks

PONE-D-23-42185R1

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10.1371/journal.pone.0308450.r004
Acceptance letter
Abou Houran Mohamad Academic Editor
© 2024 Mohamad Abou Houran
2024
Mohamad Abou Houran
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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==== Refs
References

1 P. Kaur, S. Kaur and R. Khanna, Optimal Placement and Sizing of DG Comparison of Different Techniques of DG Placement. 2015 Annual IEEE India Conference (INDICON), New Delhi, India.2015.
2 A. Shukla and K. Verma, Optimal Voltage Regulation of a Distribution Network by Output Power Management of DGs. 2016 IEEE 1st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES), Delhi, India. July 2016.
3 Hussein MM , Mohamed TH , Mahmoud MM , Aljohania M , Mosaad MI , et al . (2023) Regulation of multi-area power system load frequency in presence of V2G scheme. PLOS ONE 18 (9 ): e0291463. doi: 10.1371/journal.pone.0291463 37695790
4 Zhang Z , Hu L (2024) Is there a stronger willingness to pay for photovoltaic power generation with high education in China. PLOS ONE 19 (4 ): e0296714 doi: 10.1371/journal.pone.0296714 38568920
5 Yao M. and Cai X. , An Overview of the PV Industry Status and Perspective in China IEEE Access, Vol.7 , pp.181051–181060, 2019.
6 Arouma O. , Adolphe I.O. , Robert A.O. , Kenneth A.Z. , Antoine V. , Ramanou B. , et al . Technico-economic optimization of Distributed Generation (DG) and Static Var Compensator (SVC) positioning in a real radial distribution network using the NSGA-II genetic algorithm. 2019 IEEE PES/IAS Power Abuja, Nigeria, 20–23 Aug.2019.
7 N. Krstic, Reduction of Energy and Power Losses in Distribution Network Using Energy Storage Systems. 2020 55th International Scientific Conference on Information, Communication and Energy Systems and Technologies (ICEST), Serbia, 10–12 Sept. 2020.
8 Khan NM , Khan UA , Asif M , Zafar MH . Analysis of deep learning models for estimation of MPP and extraction of maximum power from hybrid PV-TEG: A step towards cleaner energy production. Energy Reports. 2024 Jun 1;11 :4759–75.
9 Al-Tawalbeh N , Zafar MH , Radzi MA , Zainuri MA , Al-Wesabi I . Novel initialization strategy: Optimizing conventional algorithms for global maximum power point tracking. Results in Engineering. 2024 Jun 1;22 :102067.
10 Mansoor M , Abou Houran M , Al-Tawalbeh N , Zafar MH , Akhtar N . Thermoelectric power generation system intelligent Runge Kutta control: A performance analysis using processor in loop testing. Energy Conversion and Management: X. 2024 May 1 :100612.
11 Wang L. , Wang X. , Jiang C. , Yin S. and Yang M. , Dynamic Coordinated Active–Reactive Power Optimization for Active Distribution Network with Energy Storage Systems. Applied Sciences, vol. 9 , p. 1129. 2019.
12 Vlahinic S. , Frankovic D. , Komen V. and Antonic A. , Reactive Power Compensation with PV Inverters for System Loss Reduction. Energies, vol.12 , p. 4062,2019
13 M.Q. Duong and G.N. Sava, Coordinated Reactive Power Control of DFIG to Improve LVRT Characteristics of FSIG in Wind Turbine Generation. 2017 International Conference on Electromechanical and Power Systems (SIELMEN), lasi,Romania, 11–13 Oct. 2017.
14 Cagnano A. and De Tuglie E. , Bronzizi M . Multiarea Voltage Controller for Active Distribution Networks. Energies, vol.11 , p.583,2018
15 Cagnano A. and De Tuglie E. , A decentralized voltage controller involving PV generators based on Lyapunov theory. Renew. Energy, vol. 86 , pp. 664–674, 2016
16 Z. Lv, M. Shi and X. Yuan, Empirical Study of System-level PV power plant power factor adjustment relying on PV inverter. 2014 17th International Conference on Electrical Machines and Systems (ICEMS), Hangzhou, China, Oct. 22–25, 2014.
17 Braun M. et al . Is the distribution grid ready to accept large-scale photovoltaic deployment? State of the art, progress, and future prospects. Prog. Photovoltaics Res. Appl., vol. 20 , pp.681–697, 2012
18 European Network of Transmission System Operators for Electricity (ENTSO-E). Draft Requirements for Grid Connection Applicable to all Generators, Brussels, March 2011. Brussels, Belgium, http://www.entsoe.eu [accessed on 12 May 2011].
19 Baran M.E. and Wu F. , OPTIMAL SIZING OF CAPACITORS PLACED ON A RADIAL DISTRIBUTION SYSTEM IEEE Trans. Power Delivery, vol. 4 , pp. 735–743, 1989
20 X. Xu, K. Li, Y. Liu and H. Jia, Integrated Optimal Power Flow for Distribution Networks in Local and Urban Scales. 2016 UKACC 11th International Conference on Control (CONTROL), Belfast, UK, 31st August–2nd September, 2016.
21 M. H. Zafar, U. A. Khan and N. M. Khan, "A sparrow search optimization algorithm based MPPT control of PV system to harvest energy under uniform and non-uniform irradiance," 2021 International Conference on Emerging Power Technologies (ICEPT), Topi, Pakistan, 2021, pp. 1–6
22 Muhammad Hamza Zafar , Noman Mujeeb Khan , Adeel Feroz Mirza , Majad Mansoor ,Bio-inspired optimization algorithms based maximum power point tracking technique for photovoltaic systems under partial shading and complex partial shading conditions,Journal of Cleaner Production,Volume 309 ,2021, 127279
23 Muhammad Hamza Zafar , Noman Mujeeb Khan , Adeel Feroz Mirza , Majad Mansoor , Naureen Akhtar , Muhammad Usman Qadir , Nauman Ali Khan , Syed Kumayl Raza Moosavi ,A novel meta-heuristic optimization algorithm based MPPT control technique for PV systems under complex partial shading condition,Sustainable Energy Technologies and Assessments,Volume 47 ,2021,101367.
24 J.P. Aviles, O. Erives and O. Micheloud, Optimal Design of Low Voltage Distribution Networks Using a PSO-PRIM Algorithm. 2018 IEEE Third Ecuador Technical Chapters Meeting (ETCM), Cuenca, Ecuador, 15–19 Oct. 2018.
25 Sultana S. and Roy P.K. Oppositional krill herd algorithm for optimal location of distributed generator in radial distribution system. Electrical Power and Energy Systems, vol.73 , pp. 182–191, 2015.
26 M.O.W. Groud, H.N. Luong, J. Morren and P.A.N. Bosman, Practice-Oriented Optimization of Distribution Network Planning using Metaheuristic Algorithms. 2014 Power System Computation Conference, Wroclaw, Poland, 18–22 Aug. 2014.
27 F. Zhang, K. Liu and Q. Zhang, The Research on the Planning Model of Rural Distribution Network Containing Diversification Loads. 2015 5 th International Conference on Electric Utility Deregulation and Restructuring and Power Technologies (DRPT), Changsha, China, 26–29 Nov. 2015.
28 Biswas P.P. , Mallipeddi R. , Suganthan P.N. and Amaratunga G.A.J. , A multiobjective approach for optimal placement and sizing of distributed generators and capacitors in distribution network. Applied Soft Computing, vol. 60 , pp.268–280, 2017.
29 S. Li, T. Yu, T. Pu, J. Ming and S. Fan, Coordinated optimization control method of transmission and distribution network. 2016 IEEE PES Asia-Pacific Power and Energy Conference, Xi’an, China, 25–28 Oct. 2016.
30 Mahdavi E. ; Asadpour S. ; Macedo L.H. ; Romero R. Reconfiguration of Distribution Networks with Simultaneous Allocation of Distributed Generation Using the Whale Optimization Algorithm. Energies 2023, 16 , 4560. doi: 10.3390/en16124560
31 Zheng W. , Wu W. and Sun H. , A Fully Distributed Reactive Power Optimization and Control Method for Active Distribution Networks. IEEE Transactions on Smart Grid. Vol.7 , pp. 1021–1033,2014.
32 Mohamadreza B. , Reza H.M. and Mehrdad G. , Second–order cone programming for optimal power flow in VSC–type–AC–DC grids. IEEE Transactions on Power Systems. Vol.28 , pp. 4282–4291, 2013.
33 Baradar M. and Hesamzadeh M.R. , AC Power Flow Representation in Conic Format. IEEE Transactions on Power Systems. vol.30 , pp.546–547, 2015.
34 Yang Z. , Zhong H. , Xia Q. and Kang C. , Solving OPF using linear approximations: fundamental analysis and numerical demonstration. IET Generation, Transmission & Distribution, vol. 11 , pp.4115–4125, 2017.
35 Gil-Gonzalez W. , Garces A. , Montoya O.D. and Hernandez J.C. , A Mixed-Integer Convex Model for the Optimal Placement and Sizing of Distributed Generators in Power Distribution Networks. Applied. Sciences, vol.11 , p. 627,2021.
36 Sun Y. , Zhang B. , Ge L. , Sidorov D. , Wang J. and Xu Z. , Day-ahead Optimization Schedule for Gas-electric Integrated Energy System Based on SOCP. CSEE JOURNAL OF POWER AND ENERGY SYSTEMS. vol.6 , pp.142–151, 2020.
37 Xu T. , Meng H. , Zhu J. , Wei W. , Zhao H. , Yang H. , et al . Optimal Capacity Allocation of Energy Storage in Distribution Networks Considering Active/Reactive Coordination. Energies, vol. 14 , p. 1611, 2021.
38 Ding T. , Cheng L. , Li F. , Chen T. and Liu R. , A bi-objective DC-optimal power flow model using linear relaxation-based second order cone programming and its Pareto Frontier. Electrical Power and Energy Systems. Vol. 88 , pp. 13–20, 2017.
39 Zheng W. , Hou Y. , and Li Z. , "A Dynamic Equivalent Model for District Heating Networks: Formulation, Existence and App plication in Distributed Electricity-Heat Operation," IEEE Transactions on Smart Grid, vol.12 , pp. 2685–2695, 2021.
40 Zheng W. , Wu W. , Li Z. , Sun H. , and Hou Y. , "A Non-Iterative Decoupled Solution for Robust Integrated Electricity-Heat Scheduling Based on Network Reduction," IEEE Transactions on Sustainable Energy, vol.12 , pp. 1473–1488, 2021.
41 Zheng W. and Hill D. J. , "Incentive-Based Coordination Mechanism for Distributed Operation of Integrated Electricity and Heat Systems," Applied. Energy, 285 , 116373,2021.
42 Zheng W. and Hill D. , "Distributed Real-Time Dispatch of Integrated Electricity and Heat Systems with Guaranteed Feasibility," IEEE Transactions on Industrial Informatics. in press.
43 Yong C.C. ; Li Y. ; Zeng Z.L. ; Zhang Z.W. ; Zhang Z.Y. ; Liu Y.L. Coordinated Active and Reactive Power Optimization Considering Load Characteristics for Active Distribution Network. Chinese Journal of Electrical Engineering, vol. 6 , pp. 97–105, 2020.
44 Zheng F. ; Meng X. ; Wang L. ; Zhang N. Operation Optimization Method of Distribution Network with Wind Turbine and Photovoltaic Considering Clustering and Energy Storage. Sustainability 2023, 15 , 2184. doi: 10.3390/su1503218
45 Dubravac M , Fekete K , Topić D , Barukčić M . Voltage Optimization in PV-Rich Distribution Networks—A Review. Applied Sciences. 2022; 12 (23 ):12426. doi: 10.3390/app122312426
46 Dong H. ; Zeng B. ; Wang Y.Q. ; Liu Y.X. ; Zeng M. China’s Solar Subsidy Policy. IEEE Power and Energy Magazine. Vol.18 , pp. 49–60, 2021.
