Model-Free Cooperative Control for Volt-Var Optimization in Power Distribution Systems

Abstract

Power distribution systems are witnessing a growing deployment of distributed, inverter-based renewable resources such as solar generation. This poses certain challenges such as rapid voltage fluctuations due to the intermittent nature of renewables. Volt-Var control (VVC) methods have been proposed to utilize the ability of inverters to supply or consume reactive power to mitigate fast voltage fluctuations. These methods usually require a detailed power network model including topology and impedance data. However, network models may be difficult to obtain. Thus, it is desirable to develop a model-free method that obviates the need for the network model. This paper proposes a novel model-free cooperative control method to perform voltage regulation and reduce inverter aging in power distribution systems. This method assumes the existence of time-series voltage and load data, from which the relationship between voltage and nodal power injection is derived using a feedforward artificial neural network (ANN). The node voltage sensitivity versus reactive power injection can then be calculated, based on which a cooperative control approach is proposed for mitigating voltage fluctuation. The results obtained for a modified IEEE 13-bus system using the proposed method have shown its effectiveness in mitigating fast voltage variation due to PV intermittency. Moreover, a comparative analysis between model-free and model-based methods is provided to demonstrate the feasibility of the proposed method.

1. Introduction

Traditional distribution networks have been evolving into active distribution networks (ADNs) with increasing deployment of DGs. Renewable DGs, when introduced into the system, can lead to certain challenges such as fast voltage fluctuations due to DG output intermittency. In traditional networks, feeder devices such as on-load tap changers (OLTCs) and capacitor banks (CBs) are used to mitigate these fluctuations. However, such devices, when used in renewable DG-integrated systems, might need frequent switching which would lead to increased operation and maintenance (O&M) costs.

To address this issue, various VVC techniques have been designed. These techniques can perform voltage regulation, peak load shaving, and various functions with the help of smart inverters capable of operation in four quadrants. The authors of [1,2] developed a voltage regulation method focused on the coordinated operation of OLTCs and the reactive power of DGs. A deep reinforcement learning (DRL)-based method was devised for VVC in [3]. The authors of [4,5] developed a kernel-based model and distributed model predictive control, respectively, to address voltage fluctuations through optimal reactive power dispatch. A dual-layer DRL-based method was proposed in [6,7] to perform voltage regulation. Cooperative control-based voltage regulation methods that harness inverter reactive power capability have been proposed in [8,9,10].

These previous methods include model-based methods. In other words, they require distribution network models including topology and impedance data, which may be difficult to obtain [11]. This calls for developing a VVC method agnostic to the network model, i.e., model-free VVC. The model-free method relies on the data obtained from metering devices such as advanced metering infrastructure (AMI), from which the voltage and power relationship can be derived for the feeder under control. A comprehensive review was provided by [12] focusing on the need to combine artificial intelligence (AI) with distributed energy resources (DERs) to create an efficient, economic, and adaptive energy system, along with adopting cybersecurity measures to strengthen the resilience and reliability of the power system.

Some model-free studies have been proposed in the field of VVC. The authors of [13] developed a strategy to perform hourly voltage regulation, maximize generated active power, minimize active power loss by using VVC and volt/watt control (VWC) by selecting optimal parameters using parallel genetic algorithm (GA) or parallel particle swarm optimization (PSO) while considering various PV uncertainties. Then a neural network based on Local Outlier Factor (LOF), Long Short-Term Memory (LSTM), and Gate Recurrent Unit with Normalization (GN) was developed to reduce the search time to find the optimal control strategy. A model-free VVC was performed by [14,15] for wind farms using the Koopman operator-based method and zeroth-order feedback optimization, respectively. An extremum seeking approach was developed by [16] for VVC which required only local optimization to minimize power losses. A model-free VVC framework was developed by [17] based on statistical analysis of measurement data using K-nearest neighbor (KNN) with Principal Component Analysis (PCA) method. This method was effective even in the presence of measurement noise due to the noise filtering feature of the proposed method. A DRL-based model-free VVC was proposed by the authors of [18,19,20,21,22,23] where [18,19] used the multi-agent DRL framework, and Ref. [20] used a two-agent framework, and Refs. [21,22] relied on a single-agent framework. A multi-agent framework was used for the faster time scale, whereas a single agent was selected for the slower time scale optimization in [23]. Ref. [18] devised a cooperative bi-level framework for VVC between DSOs and customers. A reward-based function was developed by [19,21] to perform voltage regulation and minimize power loss where [19] considered the capacitor, voltage regulators, and smart inverters, whereas [21] only considered PVs. Ref. [22] proposed a Markov decision-based VVC scheme which used a safe deep reinforcement learning (SDRL) algorithm to ensure safety constraints are followed during the learning process. Dual-layer optimization was proposed by [20,23] to regulate voltage where the first layer obtained schedules for OLTCs and CBs, whereas the second layer focused on optimal dispatch for PV inverters. The authors of [24] presented a robust regression-based feedback optimization algorithm and a revised alternating direction multiplier method (ADMM) to perform VVC for ADNs containing multiple virtual power plants (VPPs). Refs. [13,14,15,16,17,18,19,20,21,22,23,24] focused on using PVs as DERs, refs. [14,15] focused on wind energy, and ref. [19] chose a hybrid model combining PV and wind. However, they did not consider EVs in their study for volt-var control. Refs. [25,26] devised VVC strategies while considering electric vehicles, aimed at minimizing power losses and voltage limit violations. A three-level VVC framework was developed by [27] to mitigate voltage violations and minimize real power loss and peak demand. A recurrent neural network was proposed by [28] to solve a VVC problem modeled as the Markov decision process for voltage regulation. Some studies have focused on power converter design and control using reinforcement learning [29,30] which can be used for various power sector applications such as volt-var control.

However, the above-mentioned model-free volt-var control studies do not consider fast-paced voltage regulation, i.e., on the second level, and the aging of smart inverters unlike the proposed study in this paper. Hence, the paper makes the following contributions. A novel model-free VVC framework based on a cooperative control algorithm has been proposed for voltage regulation on the second level while considering PV inverter aging due to reactive operation. This framework does not rely on feeder parameters and is based on the voltage and power relationship model derived from an ANN. Cooperative control is a distributed optimization method that focuses on reactive power optimization of DGs by using their local information and information from neighboring nodes. Moreover, an adaptive gain-based method [31] is used to calculate the gradient gains to help convergence.

The remainder of the paper is organized as follows. Section 2 presents the ANN that models the relationship between nodal power injections and node voltages. Section 3 describes the cooperative control algorithm and the VVC problem. Section 4 illustrates and discusses the results obtained using the proposed method. It also provides a comparative analysis between model-based and model-free control methods. Section 5 provides the conclusion.

2. ANN That Models Voltage and Power Relationship

This section describes the process of creating an ANN that models the relationship between node voltages and nodal power injections based on measurement data without using the power grid topology and impedance data.

2.1. Structure of the Neural Network

The power distribution network has been modeled on the phase level, where a neural network has been defined individually for each phase to derive the relationship between voltage and nodal power injection. The adopted neural network model is shown in Figure 1. The network contains one input layer, one hidden layer, and one output layer. The number of neurons in the input and output layers depends on the number of inputs and outputs, i.e., m and n, respectively, whereas the number of neurons in the hidden layer is defined by the user. The activation function chosen for the hidden layer is the sigmoid function (σ) shown in (1).

$$\sigma \left(x_k\right)={\displaystyle \frac{1}{1+e^{-x_k}}}$$

(1)

Here, $x_k$ is the output of $k^{th}$ neuron in the hidden layer.

2.2. Training of the Neural Network

The ANN is trained using the Levenberg–Marquardt backpropagation algorithm. Under this algorithm, the gradient ($g$) is computed as follows:

$$g=J^{T}e$$

(2)

Here, $J$ is the Jacobian matrix containing the first derivatives of the network errors with respect to the weights and biases. $e$ is a vector of network errors.

After calculating the Jacobian matrix and the gradient, the weights of the inputs $\left(W\right)$ are updated as shown in (3).

$$W\left(k+1\right)=W\left(k\right)-{\left[J^{T}J+\mu I\right]}^{-1}g$$

(3)

Here, $\mu$ is a scalar value that dynamically changes to speed up convergence.

The training data for ANN was divided into three parts: 70% of the data was used for training the network, 15% was for network validation, and the remaining 15% was for testing the trained network. The training data was obtained by running power flow studies using the simulation software OpenDSS (version 10.1.0.1).

Different numbers of neurons in the hidden layer were tested to identify the most suitable neural network.

$$MSE={\displaystyle \frac{1}{C}}\sum _{i=1}^C{\left(Y_i-\widehat{Y_i}\right)}^2$$

(4)

Here, $C$ is the total number of outputs. $Y_i$ and $\widehat{Y_i}$ are the actual and predicted outputs, respectively.

The cross-validation technique has been used here to avoid overfitting. Under this method, 15% of the training data is considered for the validation dataset which is used to ensure that the trained model performs well without causing overfitting. Moreover, one of the stopping criteria for the Levenberg–Marquardt algorithm (algorithm used for network training) is the validation check in which the network training stops if the number of validation checks exceeds 6.

2.3. Inputs and Outputs for Training the Neural Network

The inputs for training the neural network $X_{train}$ include the substation voltage and active and reactive power of the load nodes in the power distribution network, which are assumed to be obtained through the metering system. The first row of the matrix is the substation voltage. The remaining rows include the net active and reactive power of load nodes as shown in (5).

$$X_{train}={\left[\begin{array}{cccc}{V}_{S1}& V_{S2}& \dots & V_{SK}\\ P_{11}^{net}& P_{12}^{net}& \dots & P_{1K}^{net}\\ Q_{11}^{net}& Q_{12}^{net}& \dots & Q_{1K}^{net}\\ ⋮& ⋮& \dots & ⋮\\ P_{N1}^{net}& P_{N2}^{net}& \dots & P_{NK}^{net}\\ Q_{N1}^{net}& Q_{N2}^{net}& \dots & Q_{NK}^{net}\end{array}\right]}_{\left(2N+1\right)⨯K}$$

(5)

Here, $V_{Sk}$ is the substation voltage at time moment $k$ (from 1 to the total number of moments $K$). $P_{nk}^{net}$ and $Q_{nk}^{net}$ are the net active and reactive power at node $n$ (from 1 to the total number of nodes $N$) and moment $k$, which are obtained as

$$P_{nk}^{net}=P_{nk}-P_{nk}^{gen}$$

(6)

$$Q_{nk}^{net}=Q_{nk}-Q_{nk}^{gen}$$

(7)

$P_{nk}$ and $Q_{nk}$ are the load active and reactive power, whereas $P_{nk}^{gen}$ and $Q_{nk}^{gen}$ are the active and reactive power generated by the DG connected at node $n$.

The outputs, the ground truth, for training the neural network are the node voltages, which are assumed to be obtained through the metering system and are

$$Y_{train}={\left[\begin{array}{cccc}{V}_{11}& V_{12}& \dots & V_{1K}\\ V_{21}& V_{22}& \dots & V_{2K}\\ ⋮& ⋮& \dots & ⋮\\ V_{N1}& V_{N2}& \dots & V_{NK}\end{array}\right]}_{\left(N⨯K\right)}$$

(8)

All the inputs and outputs are in per unit, a commonly used normalization technique in power system analysis, and no additional normalization process is performed.

2.4. Trained Neural Network

When successfully trained, a power distribution network is represented by a neural network function called $net$. The function when given the input values in the same order as given during network training will yield outputs in the same order as the outputs obtained during network training. The $net$ function is shown in (9).

$$Y_{out}=net\left(X_{in}\right)$$

(9)

Here, $X_{in}$ is the input and $Y_{out}$ is the output. The sensitivity of the output versus input change can be calculated based on the perturbation of the input. For this study, the ANN inputs were changed by 1% to calculate the sensitivity of each output with respect to the inputs.

3. Model-Free Cooperative Control

This section presents the VVC problem and describes the cooperative control algorithm using the voltage sensitivity versus power injection derived in Section 2 and without utilizing the power grid model.

3.1. Reactive Power Utilization Ratio (${\alpha }_{qi}$)

The proportion of reactive power supplied by a DG is denoted by the reactive power utilization ratio (${\alpha }_{qi}$), i.e., the decision variable, shown in (10).

$${\alpha }_{qi}={\displaystyle \frac{Q_{G_i}}{\overline{{Q_G}_i}}}$$

(10)

$Q_{G_i}$ is the reactive power generated by the $i^{th}$ DG. The available reactive power capacity of a DG is denoted by $\overline{Q_{G_i}}$ that is calculated using its rated apparent power and active power determined by solar irradiance. A positive and negative value of ${\alpha }_{qi}$ denotes generation and consumption of reactive power, respectively. The range of ${{\alpha }_q}_i$ is from −1 to 1.

3.2. Objective Function

The cooperative control method aims to minimize the objective function shown in (11), which includes the voltage deviation from the desired value (1 per unit is used here)at selected nodes and the reactive power generation/consumption by the DGs to reduce inverter aging. This is achieved by fairly utilizing the reactive power of the DGs. The node without a DG is referred to as a non-DG node.

$$F=\sum _{i=1}^{N_{DG}}{f}_i+\sum _{k=1}^{N_{NDG}}{f}_k=\sum _{i=1}^{N_{DG}}\left[w_{v_{DG}}{\left(V_i-1\right)}^2+w_{\alpha }{\alpha }_{q_i}^2{\overline{Q_{G_i}}}^2\right]+\sum _{k=1}^{N_{NDG}}{w}_{v_{NDG}}{\left(V_k-1\right)}^2$$

(11)

Here,

• $V_i$ is the per unit voltage at DG node $i$;
• $V_k$ is the per unit voltage at non-DG node $k$;
• $w_{v_{DG}}$ is the weight associated with voltage deviation minimization at DG nodes;
• $w_{v_{NDG}}$ is the weight associated with voltage deviation minimization at non-DG nodes;
• $w_{\alpha }$ is the weight associated with minimizing the generation/consumption level of the reactive power;
• $N_{DG}$ is the total number of DGs;
• $N_{NDG}$ is the total number of non-DG nodes.

In (11), the first part of cost function $f_i$ relates to voltage deviation from the desired value, and the second part relates to the reactive power generation/consumption of DGs. In the second part of $f_i$, ${\alpha }_{q_i}^2{\overline{Q_{G_i}}}^2$ has been used instead of ${\alpha }_{q_i}\overline{Q_{G_i}}$ to consider the case of consumption of reactive power by a DG as ${\alpha }_{qi}$ will be negative. For non-DG nodes, only the voltage deviation is considered.

3.3. Inputs and Outputs for the ANN

The inputs for the ANN at time moment $k$, i.e., $X_k$ comprise the substation voltage and net active and reactive power at load nodes, as shown in (12).

$$X_k={\left[\begin{array}{c}{V}_{Sk}\\ P_{1k}^{net}\\ Q_{1k}^{net}\\ ⋮\\ P_{Nk}^{net}\\ Q_{Nk}^{net}\end{array}\right]}_{\left(2N+1\right)⨯1};k=1,2,3,\dots ,N_{sec}$$

(12)

Here, $N_{sec}$ is the total number of seconds considered for the study. $P_{nk}^{net}$ and $Q_{nk}^{net}$ are the net active and reactive power as obtained in (13) and (14), respectively.

$$P_{nk}^{net}=\left\{\begin{array}{c}{P}_{nk},fornon-DGnodes\\ P_{nk}-{\lambda }_k{P}_{{DGR}_i},forDGnodes\end{array}\right.$$

(13)

All the DGs in this study are assumed to be solar PVs. ${\lambda }_k$ is the solar irradiance at moment $k$, scaled between 0 and 1. $P_{{DGR}_i}$ is the rated active power of the $i^{th}$ DG.

$$Q_{nk}^{net}=\left\{\begin{array}{c}{Q}_{nk},fornon-DGnodes\\ Q_{nk}-{\alpha }_{qi}\left(\sqrt{S_{D{G}_i}^2-{\left({\lambda }_k{P}_{{DGR}_i}\right)}^2}\right),forDGnodes\end{array}\right.$$

(14)

$P_{nk}\mathrm{a}\mathrm{n}\mathrm{d}{Q}_{nk}$ are the active and reactive power of the load at node $n$ at moment $k$. $S_{D{G}_i}$ is the apparent power of the $i^{th}$ DG.

The outputs for the ANN are the node voltages as shown in (15).

$$Y_k={\left[\begin{array}{c}{V}_{1k}\\ V_{2k}\\ ⋮\\ V_{Nk}\end{array}\right]}_{\left(N⨯1\right)};k=1,2,3,\dots ,N_{sec}$$

(15)

3.4. Communication Topology

In cooperative control, the communication network between the participating nodes is defined using the communication topology matrices shown in (16) and (17).

The communication from DG node $j$ to $i$ is represented by a square matrix $d_{ij}$ of order ($N_{DG}$ × $N_{DG}$), shown in (16).

$$d_{ij}=\left\{\begin{array}{c}0;nocommunicationbetweennodeiandj\\ {\displaystyle \frac{W_{ij}{S}_{ij}}{\sum _{l=1}^{N_{DG}}\left(W_{il}{S}_{il}\right)}};informationflowsfromnodejtoi\end{array}\right.$$

(16)

Here, $W_{ij}>0$ is the weight of communication channel between node $i$ and $j$. For a symmetric system, where all channels have equal weight, $W_{ij}$ is one. $S_{ij}$ represents communication status between nodes $i$ and $j$. $S_{ij}$ is one if there is a communication between node $i$ and $j$ and is zero otherwise.

The information flowing from a non-DG node $k$ to a DG node $i$ is denoted by the matrix $d_{NDGik}$, shown in (17), of order ($N_{DG}$ × $N_{NDG}$).

$$d_{NDGik}=\left\{\begin{array}{c}0;nocommunicationbetweennodeiandk\\ {\displaystyle \frac{W_{ik}{S}_{ik}}{\sum _{l=1}^{N_{NDG}}\left(W_{il}{S}_{il}\right)}};informationflowfromnodektoi\end{array}\right.$$

(17)

Here, $W_{ik}>0$ represents the weight of communication channel between node $i$ and $k$. For a symmetric system, $W_{ik}$ is one. $S_{ik}$ represents communication status between nodes $i$ and $k$. $S_{ik}$ is one if there is a communication between node $i$ and $k$ and is zero otherwise.

Although there are various possible communication topologies when considering the information flow direction and nature of interaction between DGs, a unidirectional intra-phase DG communication has been assumed for this study, as shown in Figure 2. This communication topology is called Unidirectional Individual Cooperative Control (UDICC). In the figure, DG1–DG3 belong to phase A, whereas DG4–DG6 and DG7–DG9 belong to phase B and C, respectively. In this study, selected non-DG nodes are communicating with each of the DG nodes.

3.5. Gradient Components

The change in objective function $f_i$ with respect to change in ${{\alpha }_q}_i$, i.e., $g_{ii}$ is calculated in (18). The change in objective function $f_k$ with respect to the change in the ${{\alpha }_q}_i$, i.e., $g_{ki}$ is calculated as shown in (19).

$$g_{ii}={\displaystyle \frac{\partial f_i}{\partial {\alpha }_{q_i}}}=2w_{v_{DG}}\left(V_i-1\right)\overline{Q_{G_i}}{S}_{V_{ii}}+{2w}_{\alpha }{{\alpha }_q}_i{\overline{Q_{G_i}}}^2$$

(18)

Here, $S_{V_{ii}}$ is the voltage sensitivity at node $i$ with respect to the reactive power at node $i$.

$$g_{ki}={\displaystyle \frac{\partial f_k}{\partial {\alpha }_{q_i}}}=2w_{v_{NDG}}\left(V_k-1\right)\overline{Q_{G_i}}{S}_{V_{ki}}$$

(19)

Here, $S_{V_{ki}}$ is the voltage sensitivity at node $k$ with respect to the reactive power at node $i$.

The voltage sensitivity shown in (18) and (19) is calculated based on the ANN model as shown in (9).

Thus, the derivative of the objective function with respect to ${\alpha }_{q_i}$ is

$$g_i=g_{ii}+\sum _{k=1}^{N_{NDG}}\left\{d_{NDGik}{g}_{ki}\right\}$$

(20)

3.6. Gradient Gain

The gain for $g_i$ is calculated using the adaptive gradient method [31] as follows:

$${\beta }_i=L_i{\left[g_i^2+\xi \right]}^{-{\displaystyle \frac{1}{2}}}$$

(21)

Here, $L_i$ is the learning rate for the $i^{th}$ DG, and $\xi$ is a small value employed to avoid division by zero.

3.7. Updating Reactive Power Utilization Ratio Based on Consensus Algorithm

An optimal ${\alpha }_{qi}$ for $i^{th}$ DG is achieved through a consensus algorithm-based iterative process, as shown in (22).

$${\alpha }_{qi}\left(n+1\right)=\left(\sum _{l=1}^{N_{DG}}{\{d}_{il}{\alpha }_{ql}\left(n\right)\}\right)-{\beta }_i{g}_i$$

(22)

As per the consensus algorithm, cooperative control aims to regulate voltage by equitably utilizing each DG’s reactive power. The convergence criterion for the iterative process is that the utilization ratio between consecutive iterations no longer changes significantly, i.e., the difference between the utilization ratio at consecutive iterations is less than a predefined tolerance (0.001 in this study) [32].

4. Results and Discussions

4.1. Power Distribution Network

This study was performed for a modified 13-bus system shown in Figure 3. The system contains nine DGs that include three single-phase DGs on buses 670 and 671 and one single-phase DG on buses 645, 646, and 692. The DGs considered in this study are assumed to be single-phase elements to address the phase voltage imbalance.

4.2. Inputs and Outputs for the Neural Networks for the Modified IEEE 13-Bus System

As mentioned in Section 2.1, a separate neural network has been created for each phase of the modified 13-bus system shown in Figure 3. This has been performed with the aim of implementing cooperative control on the phase level [8,9]. The nodes that have a DG and/or a load connected to them are the nodes providing inputs and outputs to the neural networks. The inputs and outputs are obtained as shown in (12)–(15). The substation bus for the modified IEEE 13-bus system is the bus RG60.

Table 1. Buses considered in the neural network for each phase.

Phase	Buses
	DG Buses	Non-DG Buses
A	670, 671, 692	634, 652, 675
B	645, 670, 671	634, 646, 675
C	646, 670, 671	611, 634, 675, 692

shows the list of buses to be included in the neural network.

4.3. Model-Free Cooperative Control

This study horizon is one hour, i.e., 3600 s. The solar PVs connected to the distribution system are rated at 300 kW each and follow the second-based irradiance profile, scaled between 0 and 1, as shown in Figure 4. The irradiance profile displaying significant fluctuations within an hour was chosen from the irradiance data released by the National Renewable Energy Laboratory (NREL) [33].

The number of neurons in the hidden layer is changed from 5 to 13 to find the optimal number of neurons in the hidden layer in the ANN. The MSE for 5 to 13 neurons for the phase A network was 1.59 × 10⁻⁷, 9.81 × 10⁻⁸, 8.41 × 10⁻⁸, 7.92 × 10⁻⁸, 7.79 × 10⁻⁸, 7.66 × 10⁻⁸, 7.54 × 10⁻⁸, 6.5 × 10⁻⁸, and 6.49 × 10⁻⁸, respectively. As can be seen, after 12 neurons, the MSE change is insignificant, and we choose 13 neurons. The MSEs for phases B and C reflect a similar trend.

Table 2. Weights considered for model-free cooperative control.

Case	Weights			Objective
	${\mathit{w}}_{{\mathit{v}}_{\mathit{D}\mathit{G}}}$	${\mathit{w}}_{{\mathit{v}}_{\mathit{N}\mathit{D}\mathit{G}}}$	${\mathit{w}}_{\mathit{\alpha }}$
1	1	1	0	Minimizes the voltage deviation
2	0	0	1	Minimizes the generation and consumption of reactive power
3	1	1	0.1	Reduce both voltage deviation and reactive power generation and consumption

shows the weights considered for the objective function shown in (11) for implementing the method. Case 1 tries to minimize voltage deviation by harnessing the reactive power capacity of inverters. Case 2 limits the utilization of the inverters’ reactive power capability. Case 3 tries to reduce both voltage deviation and inverters’ reactive power operation. Note that the voltage-related terms and reactive power-related terms in the objective function (11) are not directly comparable, although we add them together to demonstrate a simple way of considering both factors. How we form a more practical objective function and choose appropriate weights for the voltage terms and reactive power terms in the objective function will depend on practical application considerations and warrant further research. A learning rate of 0.04 is used in this study. The typical iteration number required for convergence for all cases was between 19 and 25 for a tolerance of 0.001.

Selected DG and non-DG nodes are chosen to display the results for the studied cases. For the DG node, phase A of bus 692 is selected, whereas phase A of bus 675 is considered for the non-DG node. The results for phases B and C are similar to those of phase A.

The base voltage profile considered for all cases is obtained for the scenario where all DGs have a unity power factor, i.e., the reactive power of all the DGs is zero.

Figure 5 depicts the ${{\alpha }_q}_i$ for all three cases described in Table 2. Case 1 focuses solely on minimizing the voltage deviation. As a result, the ${{\alpha }_q}_i$ for the DGs can take any value deemed necessary to ideally achieve a unity voltage profile. Case 2 focuses solely on minimizing the generation/consumption of the reactive power of DGs. Thus, the ${{\alpha }_q}_i$ will take a zero value. In case 3, both voltage deviation and reactive power utilization reduction are considered. As a result, the ${{\alpha }_q}_i$ for case 3 is smaller than that of case 1 but higher than case 2.

Figure 6 shows the voltage profile for all three cases at phase A of bus 692, i.e., node 692.1. It can be observed that the best optimal voltage profile is obtained for case 1, as utilization of inverter reactive power is not intentionally restricted apart from rating limits. The voltage profile for case 2 aligns with the base voltage profile due to intentionally limiting the inverter reactive power utilization to zero. The voltage profile for case 3 is better than case 2 but is inferior to case 1 due to the objective trade-off between voltage deviation and reactive power utilization.

Figure 7 shows the voltage profile at phase A of bus 675, namely node 675.1, being a non-DG node. It can be observed that the non-DG node also exhibits similar behavior to that shown in Figure 6.

4.4. Comparative Analysis of Model-Based and Model-Free Cooperative Control

This section presents the comparison between the model-free and model-based methods. This comparative study has been performed for the modified IEEE 13-bus shown in Figure 3. The model-based method is implemented for the aforesaid distribution system using MATLAB R2023a and OpenDSS (version 10.1.0.1). The same irradiance profile as shown in Figure 4 and the same cases shown in Table 2 are studied. The same learning rate is adopted. The DG and non-DG nodes considered for the study shown in Section 4.3 are selected for this analysis as well. Moreover, the MSE has been calculated for the node voltage profile as the mean square error using 1.0 as the target voltage.

4.4.1. Case 1—Minimizing the Voltage Deviation $(w_{v_{DG}}={\mathrm{w}}_{{\mathrm{v}}_{\mathrm{N}\mathrm{D}\mathrm{G}}}=1,{\mathrm{w}}_{\mathsf{\alpha }}=0)$

Figure 8 and Figure 9 show the comparison between the voltage profile obtained for DG node 692.1 and non-DG node 675.1, respectively, using model-based and model-free methods for case 1. The MSE obtained for node 692.1 for the model-based and model-free systems is 2.87 × 10⁻⁵ and 2.18 × 10⁻⁵, respectively.

The voltage profiles for case 1 resulting from both methods are very similar, indicating that the model-free method can deliver effective voltage regulation as well. The MSE obtained for node 675.1 for model-based and model-free systems is 2.51 × 10⁻⁶ and 1.82 × 10⁻⁶, respectively.

4.4.2. Case 2—Minimizing the Generation/Consumption of Reactive Power at DG Nodes $(w_{v_{DG}}={\mathrm{w}}_{{\mathrm{v}}_{\mathrm{N}\mathrm{D}\mathrm{G}}}=0,{\mathrm{w}}_{\mathsf{\alpha }}=1)$

Figure 10 and Figure 11 show the voltage profiles obtained for case 2, where the emphasis is on minimizing the supply/consumption of reactive power, leading to minimal voltage regulation. It can be observed that the voltage profiles for model-free and model-based approaches exhibit similarity with minor deviations observed across the time horizon.

The MSE obtained for node 692.1 for model-based and model-free systems is 4.50 × 10⁻⁴ and 9.80 × 10⁻⁴, respectively, and the MSE for node 675.1 is 6.76 × 10⁻⁴ and 13.00 × 10⁻⁴, respectively.

4.4.3. Case 3—Reducing Both the Reactive Power at DG Nodes and Voltage Deviation at DG and Non-DG Nodes $\left(w_{v_{DG}}={\mathrm{w}}_{{\mathrm{v}}_{\mathrm{N}\mathrm{D}\mathrm{G}}}=1,{\mathrm{w}}_{\mathsf{\alpha }}=0.1\right)$

This section presents the voltage profiles for DG and non-DG nodes for case 3, where both voltage deviation and reactive power utilization limitation are considered. The voltage profiles shown in Figure 12 and Figure 13 display a high level of similarity with minor deviations between the methods. The results again show that the model-free method can improve voltage profiles and may be used as a potential alternative method when the power grid model is not available.

The MSE obtained for node 692.1 for model-based and model-free systems is 7.05 × 10⁻⁵ and 2.65 × 10⁻⁴, respectively, and the MSE for node 675.1 is 1.62 × 10⁻⁴ and 4.68 × 10⁻⁴, respectively.

5. Conclusions

This paper presented a model-free cooperative control method for voltage regulation and to increase the lifetime of the PV inverters through reactive power optimization. A feedforward neural network is designed to capture the relationship between the node voltage and nodal power injections for a modified IEEE 13-bus system. The results obtained using model-free cooperative control have shown improved voltage profiles in comparison to those without harnessing the reactive power control capability of inverters. Comparative analysis is also performed between model-based and model-free control methods. The model-free method performs similarly to the model-based method. The results have demonstrated that the proposed method may be used as a feasible alternative for voltage control in the absence of the distribution network model. It is noted that the proposed method requires real-world measurements obtained from the metering system to train the ANN. More studies may be performed in the future to examine the requirements on the number and location of meters and the impacts of measurement errors and further study the method based on larger-sized power grids.