Multi-Stage Operation Optimization of PV-Rich Low-Voltage Distribution Networks

Abstract

The high expansion of a variable and intermittent nature of distributed generation, such as photovoltaics (PV), can cause technical issues in existing distribution networks (DN). In addition to producing electrical energy, PVs are inverter-based sources, and can help conventional control mechanisms in mitigating technical issues. This paper proposes a multi-stage optimal power flow (OPF)-based mixed-integer non-linear programming (MINLP) model for improving an operation state in LV PV-rich DN. A conventional control mechanism such as on load tap changer (OLTC) is used in the first stage to mitigate overvoltage caused by PVs. The second stage is related to reducing losses in DN using reactive power capabilities from PVs, which defines the optimization problem as a fully centralized observed from the distribution system operator’s (DSO) point of view. The optimization problem is realized under the co-simulation approach in which the power system analyzer and computational intelligence (CI) optimization method interact through an interface. This approach allows keeping the original MINLP model without approximations and using any computational intelligence method. OpenDSS is used as a power system analyzer, while particle swarm optimization (PSO) is used as a CI optimization method in this paper. Detailed case studies are performed and analyzed over a single-day period. To study validation and feasibility, the proposed model is evaluated on the IEEE LV European distribution feeder. The obtained results suggest that a combination of conventional control mechanisms (OLTC) and inverter-based sources (PVs) represent a promising solution for DSO and can serve as an alternative control method in active distribution networks.

1. Introduction

Global climate changes have led to increasing concerns worldwide, therefore governments introduce policy support. One is related to financial support for installing clean RESs, such as PVs in DNs [1,2]. More PV integration can result in technical issues which the DSO has to deal with. Such issues arise due to the variable and intermittent nature of PVs, and they are related to frequent voltage violations and higher losses [3]. The traditional DN control devices, including OLTC, CBs and VRs, preset slow response mechanisms with a limited lifespan due to mechanical parts [4]. However, combined with quick response mechanisms such as PV inverter’s reactive power support can achieve promising results in improving DN technical conditions [5,6,7,8,9].

The main aim of [10] is to keep voltage conditions under different constraints, minimize distribution network losses and PV-curtailed active power, as well as reduce switching operations of OLTC and CBs. MINLP optimization problems including DistFlow power flows, decision variables constraints, and other network constraints are tested on IEEE-33 balanced network and NSGA-II algorithm with a fast decision-making algorithm, which is proposed to find the most appropriate solution from the Pareto set. As the main limit of the proposed model, the authors emphasized the dependence of such a model on the communication system to ensure coordination between controllable units. MISOCP model with DistFlow power flows for improving voltage conditions in ADN is proposed in [11]. Control curves of DG obtained from local measurements and OLTC, as well as CB switching operations, are optimally operated in day-ahead scheduling, and their settings are sent to corresponding DG inverters which adjust their active and reactive powers in intra-day control for an optimal operation of ADN. A multi-objective problem considering losses, voltage deviation, and APC minimization is turned into a single-objective problem using the weighted sum method. Due to local control measurements, the authors concluded that, in this case, a decentralized strategy can represent a promising solution as well as the development of communication between control devices. As DG, except PVs, WTs are considered in [11,12]. The authors in [12], presented an optimization scheme to minimize network and neutral wire losses and voltage deviations. The optimization model is tested on an unbalanced IEEE-13 bus network with connected load types. The optimal solution is obtained by the SQP method and suggests that an optimal DG operation can be used in reducing network unbalance. Furthermore, ref. [13] presents a two-layer stochastic optimization model for voltage control in ADNs. The main objective of the upper layer is to minimize OLTC and CB operation, loss, and APC. Decision variables in the upper layer are OLTC and CB taps and PV active and reactive powers and the control period is 1 h. According to obtained tap positions from the upper layer, only PV active and reactive power is controlled in the second layer with a 5 min time period and the objectives are losses and APC minimization. Power flow equations are transformed into linearized DistFlow equations. A combination of centralized and local PV inverter active and reactive power control is presented in [14]. The proposed OPF-based model is divided into two phases. The first phase presents PV output optimization considering the minimization of network loss and APC of 15 min periods. The full AC OPF problem is converted into the SOCP model, and DistFlow power flow is used. Local control is related to system stability after short-term fluctuations and tracks. The main objective in [15] is to minimize active and reactive powers at each node to mitigate voltage violations in the unbalanced distribution network. Different control schemes, including Volt-Var, Volt-Watt and their combination, are presented and compared; however, the main goal is to determine reactive power support and its effect on voltages, network losses and APC. The authors conclude that this approach may lead to economic benefits from PV inverters as well as their improvement with communication systems and intelligent monitoring infrastructures. In [16], reactive power from the PV inverter and APC are studied in terms of enhancing operation performance in unbalanced ADNs. OPF-based problem is proposed with multiple objectives: voltage deviations and unbalance, network losses, PV generation and APC costs aggregated into a single-objective problem and solved by a SQP solution method. According to the obtained results, the authors highlight some factors for the future: the financial benefit of reactive power support, better communication architecture and control procedure. Remote voltage estimation scheme in ADNs that applies OLTC is proposed in [17]. First, this scheme identifies the number of loads in each phase and after that estimates voltages according to measurements in the transformer station. This approach to the use of OLTC represents a promising solution for future DNs in terms of their more cost-effective operation.

On the other hand, refs. [18,19,20,21] include APC fairness approaches to satisfy both utilities and PV owners. In [18], four OPF-based short-term APC schemes are presented. The first scheme’s objective represents PV harvesting (all PVs are curtailed equally), the second objective is related to energy export (all households are supplied equally), the third objective ensures the same financial benefit to all participants, and the last objective minimizes total APC (without considering fairness). Jain’s fairness index quantifies and compares fairness in certain schemes. The linearized OPF model is tested on realistic 22 kV and low-voltage 4500 household distribution networks. The main question is which scheme represents the most fairness for all participants due to their dependence on multiple factors. The authors in [19] propose a decentralized voltage control strategy including APC fairness in ADN. A PV inverter, according to the obtained calculation of its node voltage and injected active power, shares information with other PV inverters, which adjust their active and reactive powers as control parameters. Like [18], Jain’s fairness index is used to determine the fairness of PV inverters. Except for APC incentive schemes, ref. [20] deals with RPC incentive schemes. The main goal is to reduce power withdrawn from the external network adjusting PV inverter active and reactive power. As in [11], a multi-stage MISOCP optimization model is proposed for voltage violations in ADN mitigation. The first stage is related to the day-ahead dispatch of OLTC and CBs. The second stage differs from [11] in terms of presented two voltage control modes and a comparison of their benefits. Lastly, multi-dimensional local voltage control regulates active and reactive power output in real-time and corrects estimation from stage two. IEEE-33 bus test network and the CPLEX solution method are used.

The concept of [22] is divided into two parts. The first part refers to ADN operation optimization with objectives of DG active and reactive control costs and network loss minimization. Decision variables include BESS, OLTC and active and reactive power from DG. In the second part, DN provides ancillary services such as voltage support to the transmission network. This optimization model is tested on a real-life Swiss balanced network. In conclusion, the authors emphasize that changing existing grid codes is necessary due to DG’s reactive power potential.

Unlike [10,11], a distributed control scheme for adjusting PV inverter active and reactive power settings is proposed in [23,24]. This control scheme uses PV inverter reactive power as a priority and, if necessary, includes PV active power. This scheme is compared with the OPF-based centralized control scheme that minimizes APC as the main objective. In this case, the OPF-based centralized control scheme represents the most expensive concept for developing, as well as more complex for operating. Another open question is related to the application of distributed control schemes on LV unbalanced distribution networks. As in [23], the authors in [25] propose a distributed control scheme for reducing APC and PV power loss by charging and discharging BESSs. PV inverters locally provide voltage support and reduce APC to a minimum with optimal allocation and coordination. In comparison to the previous works in which both PV active and reactive powers are used, [26] proposes a three-phase Volt-Var control method. The Volt-Var control approach is divided into three stages in [27] that include central and local Volt-Var control for voltage deviation and network loss reduction. The first two stages have a similar principle as in [13,14]. Reactive power control in real-time is presented in the third stage. A fully linearized DistFlow is used to model the problem. The authors in [28] Volt-Var framework consists of optimal coordination of OLTC and CB day-ahead scheduling and intra-day dispatch of PV reactive power. The optimization problem is multi-objective and the authors propose four different algorithms to transform this problem into a single-objective and solve it. In [29], CB and OLTC are optimized on the hourly scale and PV-DSTATCOM are optimized on the intra-hourly scale. The main goal is to minimize active and reactive power losses. On the other hand, the authors in [30] provide an OPF-based dynamic model in which only active power is used as a control variable. The proposed optimization model presents a multi-objective problem considering PV active power output maximization voltage deviation and losses minimization, which is turned into a single objective problem and solved by analytical SQP with the trust region solution method.

Most of the previously described research proposed convexified and linearized OPF-based models so that the analytical solution method can be applied. In multi-stage optimization, most of the papers propose a combination of centralized and local control; however, this research proposes a fully centralized multi-stage OPF-based model with full AC power flows and two different objective functions of each stage for different time horizons. Therefore, the following contributions are proposed:

• A proposed time series MINLP OPF-based model that coordinates conventional and cutting-edge control mechanisms in the ADN. The focus is on the existing network components and inverter-based energy resources. The model is developed to minimize two of the most common problems for DSO caused by the impact of high PV integration—voltage violations and power losses. The multi-stage optimization is introduced and performed for different time horizons.
• Since the model is MINLP without approximations and represents the most accurate and hardest-to-solve problem, a co-simulation optimization framework is proposed and used to maintain the originality of the problem. The power flow system analyzer is co-simulated with the CI optimization method. The used CI optimization method in this paper is PSO. This approach enables a full unbalanced AC power flow analysis and enhances the efficiency, accuracy and reliability of the model and obtained solutions.
• A solar irradiance profile represents real-life data obtained by one of the author’s measurements.

Ultimately, the paper is structured as follows: Section 2 presents a multi-stage control scheme and mathematical formulation of the OPF-based problem, as well as an optimal solution methodology framework. In Section 3, test network model and case studies are described and explained. Optimization framework settings for the proposed methodology are given in Section 3. Section 4 includes a presentation of the obtained results and a discussion. The conclusion is summarized and presented in Section 5.

Section 2 introduces the main idea of this paper. After that, a mathematical model of the optimization problem is formulated. Lastly, the co-simulation optimization framework is briefly described.

2. Proposed Multi-Stage Control Scheme and OPF Formulation

This paper deals with coordinated multi-stage control for improving technical conditions in PV-rich DN. As presented in Figure 1, the observed control scheme is fully centralized, i.e., all variables are controlled by DSO. The aim is to use the existing network components such as OLTC with integrated PV reactive power capabilities to diminish voltage violations and losses. The main advantage of using the existing network components is cost savings. DNs primarily consist of transformers and power lines and represent the significant sunk cost for DSO and adding new components causes additional capital cost. In the first stage, OLTC is assigned to control voltages at the hourly scale. The first stage aims to reduce the main issue caused by high PV integration—the overvoltage problem. OLTC is known as a slow-response device because it relies on mechanical change in the tap position and is best suited for steady-state voltage control. The design of OLTC is not appropriate for rapid changes and dynamic control such as in ADNs. When the overvoltage issue is solved, DSO strives to minimize power losses in DN to improve technical conditions as well as economic aspects. PV inverter RPC is controlled in the second stage with the obtained tap position from the first stage. As a response to the dynamic nature of ADN, RPC is performed at an intra-hourly scale (the typical time step for DSO is 15 min).

2.1. OPF Problem Formulation

The OPF-based problem consists of one objective in each stage. During high integration of PVs, there are significant changes in nodal voltages. The first stage’s objective function (Equation (1)) represents the minimization of voltage deviation using a transformer tap. The reference voltage value $V^{\mathrm{ref}}$ in this paper is set to $1.00\mathrm{p}.\mathrm{u}.$, while $V_{i,t}$ represents the node voltage value in time t.

$$O{F}_1=\sum _{i=1}^N\sum _{t=1}^t{(V_{i,t}-V^{\mathrm{ref}})}^2$$

(1)

The second stage’s objective function (Equation (2)) refers to loss minimization with PV reactive power as the decision variable.

$$O{F}_2=\sum _{t=1}^t\sqrt{P_t{}^{\mathrm{loss}}{}^2+Q_t^{\mathrm{loss}}{}^2}$$

(2)

where:

• $P_t^{\mathrm{loss}}$—active power losses in time t;
• $Q_t^{\mathrm{loss}}$—reactive power losses in time t.

Network operational constraints are given below.

Active and reactive power flow (balance equations) are expressed in Equations (3) and (4):

$$P_{i,t}=\sum _{k=i}^N|V_{i,t}\left|\right|V_{k,t}\left|\right|Y_{ik}|cos({\delta }_{i,t}-{\delta }_{k,t}-{\theta }_{ik})$$

(3)

$$Q_{i,t}=\sum _{k=i}^N|V_{i,t}\left|\right|V_{k,t}\left|\right|Y_{ik}|sin({\delta }_{i,t}-{\delta }_{k,t}-{\theta }_{ik})$$

(4)

where:

• $P_{i,t}$—active power at bus i in time t;
• $Q_{i,t}$—reactive power at bus i in time t;
• $V_{i,t}$—voltage magnitude at bus i in time t;
• $V_{i,t}$—voltage magnitude at bus k in time t;
• $Y_{ik}$—$i,k^{\mathrm{th}}$ element of bus admittance matrix ${\mathbf{Y}}_{bus}$;
• ${\delta }_{i,t}$—voltage phase angle at ith bus in time t;
• ${\delta }_{k,t}$—voltage phase angle at kth bus in time t;
• ${\theta }_{ik}$—phase angle of $i,k$th element of bus admittance matrix ${\mathbf{Y}}_{bus}$.

Voltage constraints are defined between upper $V_i^{\max }$ and lower limits $V_i^{\min }$ (generally defined as $\pm 10\%{\mathrm{U}}_{\mathrm{n}}$ according to [31]) Equation (5):

$$V^{\min }\le V_{i,t}\le V^{\max },\forall i\in N,\forall t\in T$$

(5)

The PV inverter power capability curve is defined in Equation (6):

$$S_{pv,t}\le \sqrt{P_{pv,t}^2+Q_{pv,t}^2},\forall pv\in PV,\forall t\in T$$

(6)

where:

• $S_{pv,t}$—apparent power of PV in time t;
• $P_{pv,t}$—active power of PV in time t;
• $Q_{pv,t}$—reactive power of PV in time t.

The decision variable’s bounds are presented below. The decision variable in the first stage is the transformer tap setting (Equation (7)). A tap changer mechanism is set in the high-voltage winding side. Distribution MV/LV transformers generally have five tap positions with a voltage range from $0.95\mathrm{p}.\mathrm{u}.$ to $1.05\mathrm{p}.\mathrm{u}.$ with the step of $2.50\%$. The physical transformer tap mechanism presents a discrete variable that defines the optimization problem as mixed integer nonlinear programming (MINLP).

Transformer tap bounds:

$$Ta{p}^{\min }\le Ta{p}_t^{\mathrm{num}}\le Ta{p}^{\max },\forall Ta{p}_t^{\mathrm{num}}\in \mathbb{Z},\forall t\in T$$

(7)

where:

• $Ta{p}^{\min }$—minimal transformer tap position;
• $Ta{p}^{\max }$—maximum transformer tap position;
• $Ta{p}_t^{\mathrm{num}}$—transformer tap position in time t.

The transformer physically has a mechanical tap changer. A frequent changing of the tap position can lead to wear and reduce the operation lifespan of the transformer. Therefore, it is important to prevent the mechanical stress on transformers. In this study, the tap changes ($\alpha$) should be less than five times in the observed period. Two indicators are introduced. If the tap position is changed compared to the previous tap position, the indicator is set to 1. Otherwise, the indicator is set to 0 (Equation (8)).

$$\begin{array}{c}\hfill \mathrm{If}:|Ta{p}_t-Ta{p}_{t-1}|>0,\alpha =1,\forall t\in T\\ \hfill \mathrm{If}:|Ta{p}_t-Ta{p}_{t-1}|=0,\alpha =0,\forall t\in T\end{array}$$

(8)

where:

• $Ta{p}_t$—current tap position;
• $Ta{p}_{t-1}$—previous tap position;

The sum of $\alpha$ needs to be in line with the following constraint (Equation (9)):

$$\sum _{t=1}^t{\alpha }_t\le 5,\forall t\in T$$

(9)

The second stage’s decision variable is PV reactive power so-called PV RPC. PV reactive power is defined by PV apparent and active power (Equation (10)). PV has the capability of absorbing and injecting reactive power. The amount of reactive power can range from lagging to leading.

$$-\sqrt{S_{pv,t}^2-P_{pv,t}^2}\le Q_{pv,t}\le \sqrt{S_{pv,t}^2-P_{pv,t}^2},\forall pv\in PV,\forall t\in T$$

(10)

where:

• $S_{pv,t}$—apparent power of PV in time t;
• $P_{pv,t}$—active power of PV in time t;
• $Q_{pv,t}$—reactive power of PV in time t.

2.2. General Co-Simulation Optimization Description

The co-simulation optimization framework represents the interaction between power system analysis software and the CI optimization method realized through interfaces. This approach enables the use of a more realistic and accurate model of DN without approximations and transformations, since full AC power flows are analyzed and, consequently, lead to obtaining a more valid and reliable solution. Decision variables obtained by the CI optimization method present input for power systems analysis software whose output represents an objective function value, which is input for the CI optimization method. The co-simulation approach is called “black-box” optimization, which means that it receives input parameters and gives output at the end. This approach does not provide any internal structure or mathematical computation. The flowchart diagram with an indicated co-simulation optimization framework is presented in Figure 2.

3. Test Network Model, Case Study and Optimization Settings Description

This section contains a test network model and case study scenarios, as well as an optimization settings description.

3.1. Test Network Model

For testing the validation and feasibility of the proposed control scheme, the IEEE European Low Voltage Test Feeder [32] is used. The network is well-suitable for testing the time-series solutions and dynamic behavior of ADN. Substation $11/0.416\mathrm{kV}$ supplies the observed radial feeder. The feeder consists of 55 consumers and provides load shapes of each consumer for a single day with a minute resolution encompassing a total of 1440 data points. This results in uneven distribution over phases and unbalanced power flows are analyzed during the simulation. One-line diagram of the test network is presented in Figure 3. In this paper, the original test network model is modified, i.e., PV is integrated on each load node.

Solar irradiance profile for one summer day (1440 min) is presented in Figure 4. This profile is attached to each PV as an input parameter scaled to the observed time period.

3.2. Case Study Description

The proposed multi-stage OPF-based model is evaluated over several case studies presented in

Table 1. Case studies description.

Case Studies	Without Control	OLTC	PV RPC
Base case	√	×	×
Case study 1	×	√	×
Case study 2	×	×	√
Case study 3	×	√	√

√ include, × exclude.

. The base case includes 55 single-phase connected PVs with a rated power of $3.68\mathrm{kW}$ and without any control mechanism. Case study 1 uses OLTC as a control mechanism over a single day (1 h resolution) to minimize voltage deviation, while transformer tap settings are constrained to a maximum of five changes over a single-day period. In this case study, stage 1 is performed separately. PV RPC capability is used as a control mechanism in case study 2 to minimize power losses. Case study 2 performs stage 2 separately. OPF-based multi-stage optimization model that combines case study 1 and case study 2 is evaluated in case study 3. Case study 1 is performed with a 1 h resolution, and case study 2 is performed with an intra-hourly (15 min) resolution over a single day.

3.3. Optimization Settings

The co-simulation approach is realized by interfacing OpenDSS [33] as a power system analyzer and PyGMO [34] optimization package with embedded CI optimization methods. The interaction is established by the API interface. The code is implemented in Python programming language. Although PyGMO features a wide range of CI optimization methods and any of these methods can be used, PSO [35] is applied in this paper. The aim of this paper is not to improve the CI optimization method; it is only used as a tool for solving optimization the problem. The PSO parameters are fine-tuned by the trial method to achieve the best solution as soon as possible.

4. Results

This section encompasses the outcomes derived from the optimization process. The voltage values in the base scenario and with the proposed OPF-based model are presented and compared, as well as losses.

4.1. Voltage Values

The following section presents voltage values for each case study.

4.1.1. Base Case

Voltage profiles for each node are presented in Figure 5, where L1, L2, and L3 represent phases of three-phase DN. This nomenclature will be used further in the paper. Figure 5 is partitioned into three distinct subfigures, each illustrating the voltage profiles for a particular phase with different levels of PV integration. PVs are connected to the node where the load has been previously integrated. As can be seen from the figure, phase 1 contains the highest number of loads, while phase 3 is the least loaded. This results in a high voltage rise in phase 3, as well as exceeding the upper node voltage permissible value. Figure 5 shows an indicative representation, the node whose voltages will be displayed is red marked shown in details in the next figure.

According to voltage profiles from Figure 5, the highest difference between PV production and loads is in phase 3 at node 336 with $1.13\mathrm{p}.\mathrm{u}.$ Therefore, the voltages at this node will be presented and compared with the voltages at the same node obtained through the optimization procedure. Figure 6 represents the voltage profile of node 336 over 24 h. It can be observed that in phase 3 the voltage exceeds the permissible limit at 10 a.m. and it continues to about 4 p.m. In this period, the highest voltage value occurs. In phase 2, the voltage limit is exceeded from 1 and 2 p.m.

4.1.2. Case Study 1

A comparison of voltages at node 336 in the base case and case study 2 is presented in Figure 7. Voltage rise caused by PVs in phase 2 and phase 3 is reduced within the permissible limits. Since the transformer tap equally affects all three phases, the voltages are also reduced in phase 1, although that phase has the lowest voltage deviation compared to the other two phases. From 8 p.m., the voltages in the base case and case study 2 overlap, and the secondary transformer voltage is set to $1.00\mathrm{p}.\mathrm{u}.$, which is actually the reference voltage value.

The secondary transformer voltage values are presented in Figure 8. Although the number of the transformer tap changes is limited to the maximum of 5, only three transformer tap changes appear over 24 h period. The first is at 10 a.m., when PV production rises, and this position continues until 6 p.m., when PV production is low. The last transformer tap change is at 9 p.m.

4.1.3. Case Study 2

Figure 9 shows voltage profiles at the 336th node before optimization and after optimization in case study 3. Significant voltage rise occurs in phase 2 and phase 3 during high PV production. Voltages after optimization deviate according to $1.02\mathrm{p}.\mathrm{u}.$, and they are significantly reduced in phase 2 and phase 3. However, in phase 1, in which voltages are the lowest, they are slightly increased according to $1.02\mathrm{p}.\mathrm{u}.$

4.1.4. Case Study 3

Figure 10 compares voltage values at node 336 in the base case and case study 3. Transformer tap and PV reactive power impacts on voltages can be observed.

4.2. Losses

Table 2. Energy losses by active and reactive powers in each case study.

Case Study	Base Case	1	2	3
Energy losses by active power [kWh]	$645.53$	$652.52$	$181.94$	$187.09$
Energy losses by active power reduction [%]	-	$-1.08$	$71.82$	$71.01$
Energy losses by reactive power [kVARh]	$332.21$	$335.22$	$189.64$	$194.80$
Energy losses by reactive power reduction [%]	-	$-0.91$	$42.91$	$41.36$

presents energy losses by active and reactive powers for the conducted case studies. It can be observed that the transformer tap shows sensitivity only to voltage deviations. Voltages are reduced in the first stage; however, there is a slight rise in energy losses by active and reactive powers. DSO does not produce reactive power for loss minimization; therefore, reactive power needs to be imported from an external source such as a PV inverter. PV RPC significantly reduces both energy losses by active and reactive powers and introduces a promising solution for DSO. It also brings benefits for PV owners in the provision of auxiliary services. Although these services are currently not clearly defined in the near future such tariff systems need to be introduced to achieve compromise between the DSO and PV owner for exploiting such potential.

4.3. Performances

The described optimization procedure is performed on a PC with an Intel i5-11400 CPU and 16 GB of DDR4 RAM. Figure 11 represents the flow of the objective function value depending on the number of generations. Convergence ensues when the curve is parallel with a generation number axis. Objective functions converge differently for different case studies. The number of generations in case study 1 and case study 2 is set to 200, whereas the number of generations in case study 3 is set to 400.

4.4. Discussion

It can be observed from Figure 4 that the highest PV production occurs in the middle of the day. As the observed network encompasses loads with different profiles, voltage unbalance is inevitable. Such conditions bear relevance to real-world distribution networks. Uneven load distribution can be noticed on voltage profiles in Figure 5. Since voltage violation is addressed as one of the main issues during high PV integration, voltage behavior is analyzed. In the base case, PV integration causes voltage rise in all three phases. The first phase is the most loaded, thus results in the lowest voltage rise. On the other hand, the third phase is the least loaded, which results in the highest voltage rise. At some nodes, voltage exceeds the upper permissible value of $10\%$ of nominal voltage. The nodes with the highest difference between PV production and load have the most significant voltage increase, i.e., node 336. To enable easier analysis of the acquired results, comparison of the voltages at node 336 obtained in each case study are given in Figure 12. In case study 1, the implementation of a conventional voltage control device (OLTC) effectively mitigates voltage rise at the observed node, ensuring that the voltage is maintained within permissible limits. In case study 2, PV RPC strives to ensure voltages around the nominal value aiming to establish a stable and effective voltage profile. Multi-stage optimization (case study 3) reduces the voltage rise in the first stage. In the second stage, it is only necessary to keep voltages within permissible limits, which is achieved due to OLTC positions representing input for the second stage. Furthermore, in case study 1, voltages are unnecessarily reduced in the middle of the day due to OLTC’s equal performance. One notable advantage of the PV RPC is its capability to operate bi-directionally, allowing for a transition from leading to lagging reactive power conditions.

As can be noticed, OLTC exhibits a lack of sensitivity to losses, and its application results in an increase in both energy losses by active and reactive powers. A combination of OLTC and reactive power sources such as CB and FACTS devices has the potential in yield to improve losses. Since this paper refrains from additional device implementation, their application would represent a future step for this paper. PV inverter as a reactive power source successfully decreases losses in ADN.

5. Conclusions

This paper tends to solve multi-stage OPF-based problems in PV-rich DN; therefore, the MINLP model is introduced. The concept of the paper comprises two main stages. The first stage is related to overvoltage problems, which occur with high PV integration. An approach to resolving this problem involves the existing voltage control components such as OLTC, without additional investment in new ones. After solving the overvoltage problem, the second stage strives to minimize losses using PV RPC. To preserve the original MINLP problem and not perform any transformation such as convexification and linearization, a co-simulation approach is used to obtain a solution. OLTC can effectively reduce overvoltages caused by PVs, but there is simultaneously a slight increase in energy losses by active power of $1.08\%$ and energy losses by reactive power of $0.91\%$. The problem of OLTC occurs in unbalanced DN due to OLTC equally performing in all three phases. However, the shortcomings of OLTC can be solved with inverter-based sources such as PVs. PV RPC represents a promising solution for both voltage control and reducing losses. In the second stage, energy losses by active and reactive powers are reduced to $71.82\%$ and $42.91\%$, respectively. Multi-stage optimization reduces energy losses by active power of $71.01\%$ and energy losses by reactive power of $41.36\%$. Therefore, the question arises as to why to use OLTC when conditions can be achieved with a PV inverter. When a problem in the network occurs, DSO strives to solve it with its own resources, so OLTC imposes as a logical solution. DSO is not a reactive power producer, so a PV inverter can compensate for the lack of reactive power needed for loss minimization. According to the findings, the proposed model increases the flexibility of DN and represents an alternative control method in ADN.

One disadvantage of this optimization model is the assumption that all PV inverters can control reactive power and engage in the optimization process. This assumption may not align with the practical constraints present in the real-world scenario. Also, the development of the communication infrastructure would improve the use of PV inverters. The central controller receives information about network issues and determines the specific inverter that should supply reactive power to address the problem. Lastly, except for technical conditions, future research would include an economic aspect of DN operation to satisfy both DSO and PV owners. Future expansion of this methodology would involve the problem of unbalance in LV DN.