1.1. Motivation
Renewable energy capacity and electricity generation are expanding globally. There is a current emphasis on satisfying the increasing electricity demand in an environmentally sustainable manner. Renewable Energy Sources (RES), like PVG and Wind Generation (WG), are recognized as pivotal in achieving a sustainable energy future [
1,
2]. Consequently, the adoption of PVG has experienced substantial growth in recent years [
3]. According to the International Energy Agency (IEA), global energy demand is projected to rise by over 30.0% by 2040 [
4]. As a result, the gradual development of RES, including PVG, WG, and biomass, has gained momentum. Distributed Generation (DG) assumes a critical role in modern electric power grids. By generating electricity close to or at the consumption point, DG, such as PVG, reduces distribution losses, bolsters grid resilience, and facilitates the integration of renewable energy. This type of generation also provides increased flexibility, heightened reliability, and the potential for reducing peak demand in certain networks [
5].
PVG is one of the cleanest and most abundant energy resources [
6,
7]. The decrease in implementation costs, the development of new inverter technologies, low environmental impact rates, and growing incentive policies are responsible for this trend. Generally, PVG operates with a unity power factor, i.e., there is no generation or absorption of reactive power by the Photovoltaic Inverter (PVI) at the Connection Point (CP) [
8]. However, implementing reactive power control is an important measure for mitigating losses and managing grid voltage levels effectively in DNs [
9]. As reported in [
10], it is recommended that PVG must have the capability to support reactive power in the CP. In a typical PVG system, connected to DN, the conversion process follows these steps: (a) The solar module converts the sunlight into Direct Current (DC) power; (b) The inverter converts DC power to Alternating Current (AC), being the PVI responsible for making the integration between the solar modules and the DN; and (c) In energy systems without storage, the prosumer utilizes the generated energy to supply their demand, while the surplus energy is injected into the DN. Conversely, if the demand exceeds the amount of energy generated, the deficit is provided by the DN. Also, the PVI can control the power factor of the PVG, with
adjustment, and thus, it is possible to absorb or inject reactive power, considering the PVI limitations.
As illustrated in
Figure 1, the CP Power Factor (
) in a network with PVG can decrease when the PVIs work with a unity power factor, i.e.,
= 1. This happens because with the generation of active power from the PVGs there is a decrease in the flow of active energy from the DN at the CP; nonetheless, the reactive power demanded by the loads continues to be supplied by the DN. Therefore, as can be seen in
Figure 1, there will be a degradation of the
. Note that
is larger than
, making the
decrease, since
.
One way to avoid this situation, and still optimize the operation of the network in the presence of PVG, is to adjust the PVI, considering demand and generation forecasts; nonetheless, these forecasts are not within the scope of the present work.
From
Figure 1, it can be concluded that injecting reactive power into the CP improves the
. However, the PVI has limitations that must be considered for reactive power generation. In addition to the limitations that will be presented in the next sections, when the PVGs operate with
≠ 1, it is necessary to cut off part of the available active power of the inverter [
11]. Only with this power reduction, it is possible for the inverter to either inject or absorb reactive power. That action is needed to avoid an overload of the inverter since the total apparent power needs to be maintained. This is described by Equation (
1).
where:
: Active power injected at bus by inverter;
: Apparent power injected by inverter;
: Reactive power injected/absorbed by inverter.
When the is optimized, it is possible to reduce technical losses of the DN, improving the voltage profile and avoiding unnecessary increments in active power cuts of the PVG. This action leads to an economic and safe operation. Energy losses constitute one of the technical parameters considered in this paper, aiming to improve the network operating conditions.
1.2. Related Works
Numerous studies have been carried out regarding the impact of PVG in DNs [
12]. Several techniques are presented in the specialized literature to approach this problem, such as the use of voltage regulators, and optimal reactive power management in inverters. In [
13], a two-level over-voltage control strategy is proposed to deal with DNs that feature high penetration of PVGs. In this case, the optimal management of on-load tap changers and battery energy storage systems is required at the peak of PV generation. In [
14], the impact of PVG considering three penetration levels was analyzed. The analysis was performed for the steady state and evaluated the voltage levels, unbalance, and losses in a feeder. In this case, a unity
was considered. In [
15], the
control considering the communication between inverters in the system is studied. By exchanging data between inverters, it is possible to control the
of the entire network actively in real time, absorbing or inserting reactive power from one bus to another.
The authors in [
16] study the integration of PVG in DN and the problems associated with it when there is no adequate planning. The article presents an implementation of Volt-Var control to reduce voltage fluctuation resulting from high PV penetration. The effect of this control on the system power losses was also studied. An algorithm was implemented to minimize system losses, keeping bus voltages within allowed limits. The study considers that in some cases, the PV active power can be reduced for greater reactive power availability. However, the paper does not consider the problem of cutting off the active power of the photovoltaic system. The algorithm was tested on the IEEE 13-bus system and on a larger Electric Power Research Institute (EPRI) distribution system known as the J1 feeder. The authors concluded that the methodology effectively minimized system losses and maintained the voltage profile within acceptable limits.
The coordination between voltage regulators and PVG is studied in [
17]; the main goal is to minimize the voltage deviation in the bus considering the minimization of voltage regulator operations. In [
18], the losses in a medium voltage distribution network are analyzed considering two distinct situations of PVG penetration. Initially, a Photovoltaic (PV) source was allocated in only one feeder, and then 17 sources of PV generation, one per feeder, were also allocated. A fixed value of PVG was used along with three levels of load variation (low, medium, and heavy); it was also considered that the inverters could operate with
different from the unity. Furthermore, the losses in the inverters were modeled for the case of them being operating at a power factor different from the unity. The authors concluded that injecting reactive power into the grid through the inverters is a viable energy-saving solution. In [
19], the authors discuss the energy management of smart DNs that include WG and PVG. The model aims at minimizing DN operating costs stated as an objective function in relation to the constraints of an optimal AC power flow. In addition, stochastic programming is adopted to model the likely behavior of loads, renewable generations, and energy market prices. The problem was tested on a 33-bus test system using General Algebraic Modeling System (GAMS). The results attained evidenced the potential of the proposed approach for minimizing power losses, voltage deviations, and energy costs compared to traditional load flow studies.
In [
20], the authors presented a hybrid approach associating a mathematical programming technique with a metaheuristic approach, to optimize the electricity distribution network with PVG penetration. The optimization considered Distribution Network Reconfiguration (DNR) and the adjustment of the
on the reactive power, controlling the voltage profile and minimizing the losses in the network. The methodology used a Particle Swarm Optimization (PSO) algorithm for DNR and an Optimal Power Flow (OPF) to adjust the value of the photovoltaic inverters and maintain the limits of grid operation; however, the work did not consider the global optimization of the
.
A MOPF was presented in [
21] to find the optimal allocation of DG seeking to minimize energy losses. The algorithm considers the voltage levels and DG power factor, aiming to increase its efficiency with extra loss reduction. The power factor range studied was 0.98 (capacitive and inductive) and 1.0. The MOPF technique presented significant benefits in terms of loss reduction.
In [
22], the authors studied the impacts of high PVG penetration in DNs. To reduce the negative impact of PVG, an OPF was modeled with the objective of minimizing the technical losses and energy consumption of the network. To reach the objective function, voltage and reactive control levels at the buses are used. The power at the inverter terminals was considered with a time and load variation curve modeled as light and heavy loads. The goal was achieved through the reactive control of the inverter. A 5.0 MW PV was modeled at a randomly chosen bus. The proposed control aims to optimize the injection of reactive power in order to minimize technical losses and control the voltage amplitude so that consumption is minimized. It was verified that the technique is effective in mitigating voltage fluctuation problems and decreasing technical losses.
Drawing upon prior research, as summarized in
Table 1, it is clear that optimizing the operation of DNs is particularly crucial when confronted with a large penetration of PVG. Considering the research works consulted in the literature review, the necessity and applicability of the proposed approach in DN planning become evident since this approach encompasses both a global and hour-by-hour adjustment of the
. It is worth mentioning that this type of
adjustment has not been explored in the specialized literature, which constitutes the main contribution of this research.
The goal of this work is to develop a MOPF to minimize active power losses in DNs. This goal is achieved by optimizing the
adjustment to be set in the PVIs, thus it is possible to control the reactive power in the DN. The nature of these types of problems motivates the use of metaheuristics [
27]. Nevertheless, the emergence of powerful commercial tools brings the possibility of using commercial solvers, which is more attractive for the energy industries. In this sense, we propose to solve the problem in question using A Modeling Language for Mathematical Programming (AMPL) through the Knitro [
28] solver.
Hourly PVG and load curves were considered in the optimization process, which can be done in two ways. The first one is hourly, so every hour, the algorithm optimizes the values of the in the inverters. This stage is performed by running an OPF every hour. In the second one, the algorithm optimizes the for a single value that leads to the best solution for the time horizon under consideration. This second approach is more attractive because it requires only one adjustment according to the algorithm response. This proposal can be extended to longer periods, such as months and years, by simply using the load and generation curves forecast for the period of time under consideration. To implement the first option, it would be necessary to invest in technology capable of dynamically changing the of the PVG on an hourly basis. However, this technology is far from the reality of many energy companies and may require a significant investment. Therefore, the possibility of a single and optimal adjustment may be more feasible.