Efficient Day-Ahead Scheduling of PV-STATCOMs in Medium-Voltage Distribution Networks Using a Second-Order Cone Relaxation

Abstract

This paper utilizes convex optimization to implement a day-ahead scheduling strategy for operating a photovoltaic distribution static compensator (PV-STATCOM) in medium-voltage distribution networks. The nonlinear non-convex programming model of the day-ahead scheduling strategy is transformed into a convex optimization model using the second-order cone programming approach in the complex domain. The main goal of efficiently operating PV-STATCOMs in distribution networks is to dynamically compensate for the active and reactive power generated by renewable energy resources such as photovoltaic plants. This is achieved by controlling power electronic converters, usually voltage source converters, to manage reactive power with lagging or leading power factors. Numerical simulations were conducted to analyze the effects of different power factors on the IEEE 33- and 69-bus systems. The simulations considered operations with a unity power factor (active power injection only), a zero power factor (reactive power injection only), and a variable power factor (active and reactive power injections). The results demonstrated the benefits of dynamic, active and reactive power compensation in reducing grid power losses, voltage profile deviations, and energy purchasing costs at the substation terminals. These simulations were conducted using the CVX tool and the Gurobi solver in the MATLAB programming environment.

1. Introduction

Sustainable development is a global concern due to the harmful effects of global warming, which is mainly caused by fossil energy sources in the transportation and electricity sectors [1,2]. In the case of transportation, in recent years, the transition from fossil fuel-based methods to electrical mobility has become a reality as more and more electric vehicles (EVs) are seen on all roads worldwide [3,4]. However, this change must be coordinated with electricity production, given that the growing demand for power to charge EV batteries must be met by renewable and clean energy systems in order to effectively reduce the CO${}_2$ footprint in the atmosphere [5,6]. Otherwise, if electrical systems continue using fossil fuels, atmospheric pollution by greenhouse gas emissions will only move its focus from transportation systems to energy production without any real mitigation [7]. One of the most promising alternatives to effectively reduce the CO${}_2$ footprint is the massive integration of renewable energy resources in electrical networks at any voltage level, typically in the form of wind and solar power technologies, as they are mature and require reduced investment and operating costs while offering a wide array of possibilities regarding scalability from low- to high-voltage levels [8,9].

Electrical networks, especially distribution systems, have undergone a massive integration of distributed energy resources [10,11], which, even though they contribute to the sustainable development goals, pose significant challenges for utility companies and regulatory entities, since their operation requires new methodologies and remunerations [12,13]. The latter is needed to ensure the economic feasibility of renewable energy projects and guarantee quality, security, reliability, and efficiency in the electricity service for all end users [14].

Electrical distribution networks are one of the main links in the electricity service chain, as they directly connect large-scale power systems located at substations with medium- and low-voltage consumers in rural or urban areas [15]. These grids were initially designed while considering the end users’ passive behavior. Here, the energy flow typically goes from the primary substations to the consumers in an unidirectional way [16]. However, the massive integration of renewable energy resources has transformed these passive grids into active networks [17], which poses new challenges for their planning, operation, coordination, and control [18].

Considering the new paradigm of active distribution networks, this research seeks to study a class of distributed energy resources that can be connected to these grids, which is known as a photovoltaic distribution static compensator (PV-STATCOM) [19,20,21] and can compensate active and reactive power in electrical networks in order to improve technical, economic, and environmental indices [14,22,23]. The main characteristic of PV-STATCOMs is that they combine the use of clean energy sources to reduce CO${}_2$ emissions and fossil fuel energy production costs while improving grid efficiency (i.e., power loss reductions and voltage profile improvements).

To design efficient operation methodologies for PV-STATCOMs, it is necessary to deal with the solution of the exact nonlinear programming model that represents their operation for a given period, which typically corresponds to a day-ahead scheduling scenario [24]. However, the solution of this exact model is not an easy task due to its non-convexities [25,26]. Therefore, this research aims to propose a new convex optimization model in order to determine the day-ahead dispatch of multiple PV-STATCOMs interconnected with medium-voltage distribution networks while considering technical and economic objective functions.

The effective integration, operation, and control of PV-STATCOMs in electrical grids (transmission and distribution networks) have been widely explored in recent years. Some of the latest advances in this research area are discussed below.

The authors of [19] presented the application of the hunter-prey-based algorithm to determine the optimal location and sizing of PV-STATCOMs in electrical distribution networks, aiming to minimize the total grid power losses and improve the grid voltage profiles. Classical IEEE 33- and 69-bus grids were used for numerical validations, which included a comparative analysis with the particle swarm optimizer, the golden search optimizer, the differential evolution approach, and the artificial rabbits algorithm, among others. Numerical results showed that the energy losses were reduced by more than 57%, and voltage profiles were improved by more than 42% in both test feeders.

In [27], the authors proposed using PV-STATCOMs in medium-voltage distribution networks while considering a day-ahead operation scenario to improve grid voltage regulation and correct the power factor of a particular load application. The Bluewater Power Corporation in Sarnia, Canada, was used to validate the PV-STATCOM concept, considering dynamic active and reactive power compensation in a 10 kW PV system installed in its network.

The work by [28] enhanced the operation of an electrical distribution network under steady-state and fault conditions. In the steady-state scenario, the voltage profile was optimally regulated using dynamic active and reactive power compensation. In contrast, for the faulted operation, possible over-voltages caused by asymmetrical faults were mitigated. A 10 MW PV generation system connected to a utility company in Ontario, Canada, was simulated in the PSCAD software in order to confirm the new possible ancillary services provided by PV-STATCOMs with satisfactory numerical validations.

In [29], the authors proposed the optimal installation and sizing of PV-STATCOMs in medium-voltage distribution networks to provide ancillary services. The artificial rabbits optimization algorithm was employed as an optimization technique. Here, the objective was to reduce the total grid power losses and improve the voltage profile performance. The IEEE 33-bus grid was used for numerical validation, and a comparison of the results with those of the differential evolution and the golden search algorithms demonstrated the effectiveness of the proposed optimization approach. The authors of this research reported a reduction of about 54.36% in daily energy losses and improvements of about 43.29% in the voltage profiles’ behavior.

The study by [30] presented a multi-objective analysis considering PV-STATCOMs in transmission networks. The main goals were to simultaneously minimize the active and reactive power losses, improve grid voltage profiles, and reduce the total energy production costs. The IEEE 30-bus network was used as a test feeder. As a solution methodology, a combination of the optimal power flow formulation and the particle swarm optimization technique was implemented.

Additionally, optimization methodologies for the efficient integration of PV-STATCOMs in electrical networks include the gorilla troop optimizer [31], the gray wolf optimizer [32], the modified bat algorithm [33], the modified ant lion optimizer [34], and the fuzzy-lightning search algorithm [35], among others.

Table 1. Summary of literature reports on PV-STATCOMs.

Method	Objective Function	Year	Ref.
Different operating mode	Voltage regulation improvement and power factor correction	2011	[27]
Modified bat algorithm	Power losses minimization and voltage profile improvement	2019	[33]
Smart operating mode	Temporary over-voltages reduction and voltage regulation	2020	[28]
Gray wolf	Voltage regulation improvement	2020	[32]
Fuzzy-lightning search	Power losses minimization	2021	[35]
Particle swarm optimization	Power losses minimization and voltage profile improvement	2021	[30]
Gorilla troop optimization	Operating costs reduction	2022	[31]
Modified ant lion	Power losses minimization and voltage deviations reduction	2021	[34]
Hunter-prey optimization	Power losses minimization and voltage profile improvement	2023	[19]
Artificial rabbits optimization	Power losses minimization and voltage profile improvement	2023	[29]

summarizes the main contributions made regarding PV-STATCOMs in electrical distribution networks in recent years.

The main characteristic highlighted by the above-presented literature review (see Table 1) is that most optimization algorithms belong to the family of combinatorial optimization (i.e., metaheuristics). In addition, most objective functions focus on minimizing the total grid energy losses and improving voltage profiles. This work found that more research is required concerning the efficient day-ahead scheduling of PV-STATCOM applications for distribution grids, especially in the case of convex optimization, since no works were found on this topic, which constitutes a research opportunity to which this document aims to contribute.

In light of the above, the main contributions of this research are summarized below:

i.

The convex reformulation of the problem regarding the efficient operation of PV-STATCOMs in medium-voltage distribution networks via second-order cone programming in the complex variable domain. The product between two complex voltages was transformed using its hyperbolic equivalent, which allowed relaxing it as a convex cone equivalent formulation.

ii.

The evaluation of four different simulation scenarios, including a benchmark case without PV-STATCOM penetration and cases with unity, zero, and variable power factors. These simulations were conducted in the IEEE 33- and 69-bus grids while considering three possible objective functions: the minimization of grid energy losses, grid energy purchasing costs, and average voltage deviation. The results showed the positive effects of dynamically scheduling active and reactive power injections in PV-STATCOMs in order to improve technical or economic grid indices.

In the scope of this research, it is essential to mention that: (i) the active and reactive power demand curves, as well as the solar generation availability, correspond to inputs of our research, which were provided by the distribution company in the area of influence of the electrical distribution grid under analysis; and (ii) a previous planning stage defined the sizes and locations of the PV-STATCOMs (this research only focuses on their efficient daily operation while considering different simulation scenarios).

This work is structured as follows. Section 2 presents the general nonlinear programming formulation for the problem regarding the efficient operation of PV-STATCOMs in electrical distribution networks using a complex-domain representation. Section 3 defines the general convexification approach in the complex domain using the hyperbolic equivalent of the product between two variables, which allows obtaining a second-order cone equivalent formulation for the studied problem. Section 4 shows the main characteristics of the IEEE 33- and 69-bus grids and the characterization of the PV-STATCOMs (i.e., their sizes and locations). Section 5 presents the main numerical results obtained while considering four different simulation cases (i.e., operation with unity, zero, and variable power factors in the PV-STATCOMs, in addition to a benchmark case where the PV-STATCOMs were off). Finally, Section 6 describes the main concluding remarks derived from this work, as well as some possible future works.

2. General Formulation

The problem regarding the optimal operation of PV-STATCOMs in medium-voltage distribution networks while considering a day-ahead scheduling scenario is associated with an adequate active and reactive power compensation aiming to minimize the expected grid power losses, voltage profile deviations, and energy purchasing costs at the terminals of the substation bus [20]. This optimization problem can be formulated using a nonlinear programming model in the complex variable domain. Each of the objective functions and constraints are listed below.

2.1. Possible Objective Functions

The efficient operation of PV-STATCOMs in electrical distribution networks can be addressed with different objective functions. These can include technical, economic, and environmental indices [19].

2.1.1. Energy Losses Minimization

Minimizing the expected energy losses in a daily operation scenario is one of the most typical problems in electrical distribution networks, that is, when multiple distributed energy resources must be optimally coordinated. The objective function regarding energy losses minimization is defined in Equation (1).

$$\begin{array}{c}\hfill min{E}_{\mathrm{loss}}=\mathrm{real}\left\{\sum _{h\in \mathcal{H}}\sum _{k\in \mathcal{N}}\sum _{m\in \mathcal{N}}{\mathbb{V}}_{k,h}^{★}{\mathbb{Y}}_{km}{\mathbb{V}}_{m,h}{\Delta }_h\right\},\end{array}$$

(1)

where $E_{\mathrm{loss}}$ is the expected value of the daily energy losses; ${\mathbb{V}}_{k,h}$ and ${\mathbb{V}}_{m,h}$ correspond to the voltage values in the complex domain (i.e., magnitudes and angles) at nodes k and m in period h; ${\mathbb{Y}}_{km}$ is the complex value of the admittance matrix that relates nodes k and m; ${\Delta }_h$ denotes the period of analysis, which is typically one hour or fractions of one; and ${\left(·\right)}^{★}$ represents the complex conjugate operation of the argument. Note that $\mathcal{N}$ defines the set containing all the network nodes, and $\mathcal{H}$ is associated with the number of periods under analysis.

2.1.2. Voltage Profile Improvement

One of the distribution companies’ main interests is supplying electricity to all end users while ensuring efficiency, reliability, quality, and security [36]. To this effect, improving voltage profiles is common practice [32]. This objective function is defined in Equation (2).

$$\begin{array}{c}\hfill min{V}_{\mathrm{dev}}=\frac{100\%}{NH}\sum _{h\in \mathcal{H}}\sum _{k\in \mathcal{N}}\left|{\left|{\mathbb{V}}_{k,h}\right|}^2-V_{k,\mathrm{nom}}^2\right|,\end{array}$$

(2)

where H and N are the cardinalities (number of elements) of the sets $\mathcal{H}$ and $\mathcal{N}$, respectively; and $V_{k,\mathrm{nom}}$ is the nominal operating voltage at node k.

2.1.3. Energy Purchasing Costs

For distribution companies, one of the main issues with the multiple integration/operation of distributed energy resources (including PV-STATCOMs) is the minimization of the energy purchasing costs at the terminals of the substation bus, which corresponds to the energy bought in the spot market to provide electricity to all end users in medium- and low-voltage networks. This objective function is defined in Equation (3).

$$\begin{array}{c}\hfill min{E}_{\cos \mathrm{t}}=C_{\mathrm{kWh}}\mathrm{real}\left\{\sum _{h\in \mathcal{H}}\sum _{k\in \mathcal{N}}{\mathbb{S}}_{k,h}^s{\Delta }_h\right\},\end{array}$$

(3)

where $E_{\cos \mathrm{t}}$ represents the energy purchasing costs at the terminals of the substation bus; ${\mathbb{S}}_{k,h}^s$ is the complex power injection at the substation connected at node k in period h, and $C_{\mathrm{kWh}}$ represents the average value of the energy purchased in these terminals.

2.2. Set of Constraints

Multiple operating constraints must be observed while efficiently operating distributed energy resources in electrical distribution networks. These include the active and reactive power balance, device capabilities, and voltage regulations. Each one of these constraints is presented below.

Equation (4) defines the complex power equilibrium for each network node and period.

$$\begin{array}{c}\hfill c{\mathbb{S}}_{k,h}^s+{\mathbb{S}}_{k,h}^{p-s}-{\mathbb{S}}_{k,h}^d=\sum _{k\in \mathcal{N}}\sum _{m\in \mathcal{N}}{\mathbb{V}}_{k,h}{\mathbb{Y}}_{km}^{★}{\mathbb{V}}_{m,h}^{★},\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\}\end{array}$$

(4)

where ${\mathbb{S}}_{k,h}^{p-s}$ represents the complex power generation of the PV-STATCOM connected at node k in period h, and ${\mathbb{S}}_{k,h}^d$ denotes the apparent power consumption of the constant power load connected at node k in period h.

To ensure the correct operation of the conventional power source (i.e., the substation bus), its nominal capacities can not exceed the transformer capabilities, as defined by inequality constraint Equation (5). In addition, constraint Equation (6) is added to ensure that no active power is transferred from the distribution grid to the power system at the terminals of the substation bus.

$$\begin{array}{cc}\hfill \left|{\mathbb{S}}_{k,h}^s\right|\le S_{k,\mathrm{nom}}^s,& \left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \mathrm{real}\left\{{\mathbb{S}}_{k,h}^s\right\}\ge 0,& \left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(6)

where $S_{k,\mathrm{nom}}^s$ represents the maximum power transference capacities at the terminals of the substation bus.

For the efficient operation of PV-STATCOM devices, constraints regarding active and reactive power injection are imposed, as the nominal power transference capacities of the power electronic converter that interfaces these systems with the distribution grid must be ensured in every period. Constraints Equations (7) and (8) are therefore defined.

$$\begin{array}{cc}\hfill \left|{\mathbb{S}}_{k,h}^{p-s}\right|\le S_{k,\mathrm{nom}}^{p-s},& \left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill 0\le \mathrm{real}\left\{{\mathbb{S}}_{k,h}^{p-s}\right\}\le P_{k,h}^{pv,\mathrm{nom}},& \left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(8)

where $S_{k,\mathrm{nom}}^{p-s}$ represents the nominal power transference capacity of the converter associated with the PV-STATCOM system connected at node k, and $P_{k,h}^{pv,\mathrm{nom}}$ represents the expected solar power availability at node k for each period. This parameter provides a projection based on the expected weather conditions for a day-ahead operation.

Finally, the voltage profile must be maintained between two permissible regulation bounds due to regulatory policies applicable to medium-voltage distribution networks. This operating constraint is defined in Equation (9).

$$\begin{array}{c}\hfill V_{k,min}\le \left|{\mathbb{V}}_{k,h}\right|\le V_{k,max},\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\end{array}$$

(9)

where $V_{k,min}$ and $V_{k,max}$ represent the minimum and maximum voltage regulation bounds.

2.3. Model Characterization

The main characteristics of the general optimization model defined from Equations (1)–(9) are the following:

i.

The objective function defined in Equation (1) is a nonlinear function of the voltage magnitudes in all the network nodes and periods. However, it has a convex structure [37].

ii.

The objective function in Equation (2) is a nonlinear non-convex function due to the embedded structure of two $l_1$-norms [32].

iii.

The objective function defined in Equation (3) exhibits a linear structure, which implies that it is a convex-concave function [38].

iv.

The power balance constraint in Equation (4) corresponds to the main complication regarding the optimal operation of PV-STATCOM systems in distribution grids, as it implies the product between two complex variables, which are non-convex due to the equality imposition [39].

v.

The remaining box-type constraints Equations (5)–(9) correspond to a set of linear and norm constraints, which exhibit a convex structure [34].

Note that, in order to obtain an equivalent convex optimization model that represents the problem under study, it is necessary to obtain an approximation of the objective function Equation (2) and the power balance constraint in Equation (4). In the next section, the convex approximation approach is presented.

3. Convexification Proposal

This research proposes a conic approximation aimed at obtaining a convex approximation of the nonlinear programming model Equations (1)–(8). The main idea of conic programming is to approximate some hyperbolic constraints into equivalent cones [40], which allows reaching a second-order cone programming equivalent that represents the studied problem [41].

To obtain the convex equivalent, the following auxiliary variables are defined:

$$\begin{array}{rr}\hfill {\mathbb{W}}_{km,h}={\mathbb{V}}_{k,h}{\mathbb{V}}_{m,h}^{★},& \left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\mathbf{U}}_{k,h}={\mathbb{V}}_{k,h}{\mathbb{V}}_{k,h}^{★}={\mathbb{W}}_{kk,h}={\left|{\mathbb{V}}_{k,h}\right|}^2,& \left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\mathbf{U}}_{m,h}={\mathbb{V}}_{m,h}{\mathbb{V}}_{m,h}^{★}={\mathbb{W}}_{mm,h}={\left|{\mathbb{V}}_{m,h}\right|}^2,& \left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(12)

Note that Equation (10) is multiplied on both sides by ${\mathbb{W}}_{km,h}^{★}$. Thus, the following set of results is reached:

$$\begin{array}{cc}\hfill {\mathbb{W}}_{km,h}{\mathbb{W}}_{km,h}^{★}=& {\mathbb{V}}_{k,h}{\mathbb{V}}_{k,h}^{★}{\mathbb{V}}_{m,h}{\mathbb{V}}_{m,h}^{★},\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \\ \hfill {\left|{\mathbb{W}}_{km,h}\right|}^2=& {\mathbf{U}}_{k,h}{\mathbf{U}}_{m,h},\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(13)

where the right-hand side of Equation (13) can be transformed using the hyperbolic equivalent of the product between two real variables, as defined in Equation (15).

$$\begin{array}{cc}\hfill \left|{\mathbb{W}}_{km,h}\right|=& \frac{1}{4}{\left({\mathbf{U}}_{k,h}+{\mathbf{U}}_{m,h}\right)}^2-\frac{1}{4}{\left({\mathbf{U}}_{k,h}-{\mathbf{U}}_{m,h}\right)}^2,\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \\ \hfill {\left|2{\mathbb{W}}_{km,h}\right|}^2=& {\left({\mathbf{U}}_{k,h}+{\mathbf{U}}_{m,h}\right)}^2-{\left({\mathbf{U}}_{k,h}-{\mathbf{U}}_{m,h}\right)}^2,\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \\ \hfill {\left|2{\mathbb{W}}_{km,h}\right|}^2+{\left({\mathbf{U}}_{k,h}-{\mathbf{U}}_{m,h}\right)}^2=& {\left({\mathbf{U}}_{k,h}+{\mathbf{U}}_{m,h}\right)}^2,\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \\ \hfill ∥\begin{array}{c}2{\mathbb{W}}_{km,h}\\ {\mathbf{U}}_{k,h}-{\mathbf{U}}_{m,h}\end{array}∥& ={\mathbf{U}}_{k,h}+{\mathbf{U}}_{m,h},\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(14)

It is worth mentioning that constraint Equation (14) is still non-convex, as the only set of solutions is contained in the circle that fulfills the equality condition. However, as recommended by the authors of [39], this constraint can be convexified by relaxing the equality condition to a lower, equal one, as presented below.

$$\begin{array}{c}\hfill ∥\begin{array}{c}2{\mathbb{W}}_{km,h}\\ {\mathbf{U}}_{k,h}-{\mathbf{U}}_{m,h}\end{array}∥\le {\mathbf{U}}_{k,h}+{\mathbf{U}}_{m,h},\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\end{array}$$

(15)

Now, considering definitions in Equations (10)–(12), the nonlinear optimization model Equations (1)–(11) can be represented as a second-order cone approximated model:

Objective functions:

$$\begin{array}{cc}\hfill & min{E}_{\mathrm{loss}}=\mathrm{real}\left\{\sum _{h\in \mathcal{H}}\sum _{k\in \mathcal{N}}\sum _{m\in \mathcal{N}}{\mathbb{Y}}_{km}{\mathbb{W}}_{km,h}^{★}{\Delta }_h\right\},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & min{V}_{\mathrm{dev}}=\frac{100\%}{NH}\sum _{h\in \mathcal{H}}\sum _{k\in \mathcal{N}}\left|{\mathbf{U}}_{k,h}-V_{k,\mathrm{nom}}^2\right|,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & min{E}_{\cos \mathrm{t}}=C_{\mathrm{Wh}}\mathrm{real}\left\{\sum _{h\in \mathcal{H}}\sum _{k\in \mathcal{N}}{\mathbb{S}}_{k,h}^s{\Delta }_h\right\},\hfill \end{array}$$

(18)

Subject to:

$$\begin{array}{cc}\hfill & {\mathbb{S}}_{k,h}^s+{\mathbb{S}}_{k,h}^{p-s}-{\mathbb{S}}_{k,h}^d=\sum _{k\in \mathcal{N}}\sum _{m\in \mathcal{N}}{\mathbb{Y}}_{km}^{★}{\mathbb{W}}_{km,h}^{★},\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & \left|{\mathbb{S}}_{k,h}^s\right|\le S_{k,\mathrm{nom}}^s,\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & \mathrm{real}\left\{{\mathbb{S}}_{k,h}^s\right\}\ge 0,\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & \left|{\mathbb{S}}_{k,h}^{p-s}\right|\le S_{k,\mathrm{nom}}^{p-s},\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & 0\le \mathrm{real}\left\{{\mathbb{S}}_{k,h}^{p-s}\right\}\le P_{k,h}^{pv,\mathrm{nom}},\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & V_{k,min}^2\le {\mathbf{U}}_{k,h}\le V_{k,max}^2,\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & ∥\begin{array}{c}2{\mathbb{W}}_{km,h}\\ {\mathbf{U}}_{k,h}-{\mathbf{U}}_{m,h}\end{array}∥\le {\mathbf{U}}_{k,h}+{\mathbf{U}}_{m,h},\left\{\forall k\in \mathcal{N},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & {\mathbf{U}}_{j,h}={\mathbb{W}}_{jj,h}=V_{\mathrm{nom}}^2,\left\{\forall j=\mathrm{slack},\forall h\in \mathcal{H}\right\},\hfill \end{array}$$

(26)

where $V_{\mathrm{nom}}$ is a constant parameter that defines the voltage magnitude at the substation as a real constant.

Note that, as previously mentioned, the approximation of the objective function Equation (2) regarding the average voltage deviation of the system is non-convex due to the presence of two embedded norms. Therefore, in Equation (17), a convex approximation of this function is proposed. To illustrate the convexification effect, consider a system with two voltage variables and one period ($N=2$ and $H=1$), which implies that Equation (2) takes the form $f\left(x,y\right)=50\left(|x^2-1| + |y^2-1|\right)$ and Equation (17) can be defined as $f\left(x,y\right)=50\left(|x-1| + |y-1|\right)$. Figure 1 shows the 3D plot of both functions. The scale was exaggerated to show that $f\left(x,y\right)$ is non-convex and $f\left(x,y\right)$ is convex.

It is worth mentioning that the range of application of the convex approximation for the voltage deviation function Equation (2) using the approximation in Equation (17) lies between the minimum and maximum voltage regulation bounds, which, in medium-voltage networks, implies that variables x and y will be contained within the interval between $90\%$ and $110\%$ of the nominal voltage, i.e., $0.90$ and $1.10$ using the per-unit representation.

A flowchart of the day-ahead scheduling of PV-STATCOMs using the proposed convex model is illustrated in Algorithm 1.

Algorithm 1: SOC relaxation for day-ahead scheduling of PV-STATCOMs.

4. Test Feeders

To validate the effectiveness of the day-ahead optimization model to operate PV-STATCOM systems in medium-voltage distribution networks, conventional grids of this type were used, i.e., the IEEE 33- and 69-bus systems. The electrical configuration of these networks and their electrical parameters are presented in Figure 2 and Figure 3 and

Table 2. Electrical parameters of the IEEE 33-bus grid.

Node i	Node j	${\mathit{R}}_{\mathbf{ij}}$ ($\Omega$)	${\mathit{X}}_{\mathbf{ij}}$ ($\Omega$)	${\mathit{P}}_{\mathit{j}}$ (kW)	${\mathit{Q}}_{\mathit{j}}$ (kvar)	Node i	Node j	${\mathit{R}}_{\mathbf{ij}}$ ($\Omega$)	${\mathit{X}}_{\mathbf{ij}}$ ($\Omega$)	${\mathit{P}}_{\mathit{j}}$ (kW)	${\mathit{Q}}_{\mathit{j}}$ (kvar)
1	2	0.0922	0.0477	100	60	17	18	0.7320	0.5740	90	40
2	3	0.4930	0.2511	90	40	2	19	0.1640	0.1565	90	40
3	4	0.3660	0.1864	120	80	19	20	1.5042	1.3554	90	40
4	5	0.3811	0.1941	60	30	20	21	0.4095	0.4784	90	40
5	6	0.8190	0.7070	60	20	21	22	0.7089	0.9373	90	40
6	7	0.1872	0.6188	200	100	3	23	0.4512	0.3083	90	50
7	8	1.7114	1.2351	200	100	23	24	0.8980	0.7091	420	200
8	9	1.0300	0.7400	60	20	24	25	0.8960	0.7011	420	200
9	10	1.0400	0.7400	60	20	6	26	0.2030	0.1034	60	25
10	11	0.1966	0.0650	45	30	26	27	0.2842	0.1447	60	25
11	12	0.3744	0.1238	60	35	27	28	1.0590	0.9337	60	20
12	13	1.4680	1.1550	60	35	28	29	0.8042	0.7006	120	70
13	14	0.5416	0.7129	120	80	29	30	0.5075	0.2585	200	600
14	15	0.5910	0.5260	60	10	30	31	0.9744	0.9630	150	70
15	16	0.7463	0.5450	60	20	31	32	0.3105	0.3619	210	100
16	17	1.2860	1.7210	60	20	32	33	0.3410	0.5302	60	40

and

Table 3. Electrical parameters of the IEEE 69-bus grid.

Node i	Node j	${\mathit{R}}_{\mathbf{ij}}$ ($\Omega$)	${\mathit{X}}_{\mathbf{ij}}$ ($\Omega$)	${\mathit{P}}_{\mathit{j}}$ (kW)	${\mathit{Q}}_{\mathit{j}}$ (kvar)	Node i	Node j	${\mathit{R}}_{\mathbf{ij}}$ ($\Omega$)	${\mathit{X}}_{\mathbf{ij}}$ ($\Omega$)	${\mathit{P}}_{\mathit{j}}$ (kW)	${\mathit{Q}}_{\mathit{j}}$ (kvar)
1	2	0.0005	000012	0.00	0.00	3	36	0.0044	0.0108	26.00	18.55
2	3	0.0005	0.0012	0.00	0.00	36	37	0.0640	0.1565	26.00	18.55
3	4	0.0015	0.0036	0.00	0.00	37	38	0.1053	0.1230	0.00	0.00
4	5	0.0251	0.0294	0.00	0.00	38	39	0.0304	0.0355	24.00	17.00
5	6	0.3660	0.1864	2.60	2.20	39	40	0.0018	0.0021	24.00	17.00
6	7	0.3810	0.1941	40.40	30.00	40	41	0.7283	0.8509	1.20	1.00
7	8	0.0922	0.0470	75.00	54.00	41	42	0.3100	0.3623	0.00	0.00
8	9	0.0493	0.0251	30.00	22.00	42	43	0.0410	0.0478	6.00	4.30
9	10	0.8190	0.2707	28.00	19.00	43	44	0.0092	0.0116	0.00	0.00
10	11	0.1872	0.0619	145.00	104.00	44	45	0.1089	0.1373	39.22	26.30
11	12	0.7114	0.2351	145.00	104.00	45	46	0.0009	0.0012	29.22	26.30
12	13	1.0300	0.3400	8.00	5.00	4	47	0.0034	0.0084	0.00	0.00
13	14	1.0440	0.3450	8.00	5.50	47	48	0.0851	0.2083	79.00	56.40
14	15	1.0580	0.3496	0.00	0.00	48	49	0.2898	0.7091	384.70	274.50
15	16	0.1966	0.0650	45.50	30.00	49	50	0.0822	0.2011	384.70	274.50
16	17	0.3744	0.1238	60.00	35.00	8	51	0.0928	0.0473	40.50	28.30
17	18	0.0047	0.0016	60.00	35.00	51	52	0.3319	0.1114	3.60	2.70
18	19	0.3276	0.1083	0.00	0.00	9	53	0.1740	0.0886	4.35	3.50
19	20	0.2106	0.0690	1.00	0.60	53	54	0.2030	0.1034	26.40	19.00
20	21	0.3416	0.1129	114.00	81.00	54	55	0.2842	0.1447	24.00	17.20
21	22	0.0140	0.0046	5.00	3.50	55	56	0.2813	0.1433	0.00	0.00
22	23	0.1591	0.0526	0.00	0.00	56	57	1.5900	0.5337	0.00	0.00
23	24	0.3463	0.1145	28.00	20.00	57	58	0.7837	0.2630	0.00	0.00
24	25	0.7488	0.2475	0.00	0.00	58	59	0.3042	0.1006	100.00	72.00
25	26	0.3089	0.1021	14.00	10.00	59	60	0.3861	0.1172	0.00	0.00
26	27	0.1732	0.0572	14.00	10.00	60	61	0.5075	0.2585	1244.00	888.00
3	28	0.0044	0.0108	26.00	18.60	61	62	0.0974	0.0496	32.00	23.00
28	29	0.0640	0.1565	26.00	18.60	62	63	0.1450	0.0738	0.00	0.00
29	30	0.3978	0.1315	0.00	0.00	63	64	0.7105	0.3619	227.00	162.00
30	31	0.0702	0.0232	0.00	0.00	64	65	1.0410	0.5302	59.00	42.00
31	32	0.3510	0.1160	0.00	0.00	11	66	0.2012	0.0611	18.00	13.00
32	33	0.8390	0.2816	14.00	10.00	66	67	0.0470	0.0140	18.00	13.00
33	34	1.7080	0.5646	19.50	14.00	12	68	0.7394	0.2444	28.00	20.00
34	35	1.4740	0.4873	6.00	4.00	68	69	0.0047	0.0016	28.00	20.00

, respectively. The main characteristics of these test feeders are the following: (i) the voltage supplied at the terminals of the substation is 12.66 kV, and (ii) both networks exhibit a radial structure.

To determine the electrical behavior of the networks in a day-ahead operation environment, the active and reactive power demand curves, as well as the solar availability curve, are depicted in Figure 4 [24].

The following aspects were considered for the numerical simulations carried out in the IEEE 33- and 69-bus grids.

i.

The average cost of the energy purchasing costs at the terminals of the substation (i.e., $C_{\mathrm{kWh}}$) was assigned as USD/kWh $=0.1390$. Note that H is equal to 48 periods (see Figure 1), which means that ${\Delta }_h=0.5$ h and N is 33 or 69 depending on the test feeder under analysis.

ii.

The locations of the PV-STATCOMs in the IEEE 33-bus grid were nodes 13, 24, and 30, with sizes of 800 kW, 1090 kW, and 1050 kW, respectively.

iii.

The locations of the PV-STATCOMs in the IEEE 69-bus grid were nodes 17, 50, and 61, with sizes of 510 kW, 720 kW, and 1810 kW, respectively.

iv.

The voltage regulation bounds (i.e., $V_{k,min}$ and $V_{k,max}$) were set between $90\%$ and $110\%$ of the nominal voltage, which is 12,660 V for both test feeders.

Note that the information regarding the nominal sizes of the PV systems was obtained from [42], and these were set as the nominal rates for operating the PV-STATCOMs. In addition, the voltage relation bounds for the IEEE 33- and 69-bus grids were set as $\pm 10\%$.

5. Computational Validation

The proposed day-ahead scheduling model to operate PV-STATCOMs in medium-voltage distribution networks via second-order cone programming (see optimization model Equations (16)–(26)) was implemented in the Yalmip toolbox [43], using the Gurobi solver [44] of the MATLAB 2021a interface. A Dell Inspiron 15 7000 Series (Intel Quad-Core i7-7700HQ @2.80 GHz) PC with 16 GB RAM and 64-bit Windows 10 Home Single Language was used to carry out the simulations. The following scenarios were considered in order to validate the effectiveness of the proposed solution methodology.

i.

S1: each objective function was evaluated considering that the PV-STATCOM systems were disconnected from the main grid. This simulation scenario corresponds to the benchmark case.

ii.

S2: each objective function was minimized considering that the PV-STATCOM systems operated with a zero power factor, i.e., they could not inject active power.

iii.

S3: each objective function was minimized considering that the PV-STATCOM systems operated with a unity power factor, i.e., they could not inject reactive power.

iv.

S4: each objective function was minimized considering that the PV-STATCOM operated with a variable power factor, which means that active and reactive power could be simultaneously injected.

It is worth mentioning that all the simulation scenarios have a variable power factor with regard to the load behavior, as the load curves depicted in Figure 4 cause the power factors to change at each load per period. However, in the simulation scenarios presented above, the variable power factor case only refers to when the PV-STATCOMs vary each active and reactive power output at each period as a function of the grid requirements.

5.1. Numerical Results in the IEEE 33-Bus Grid

Table 4. Numerical results in the IEEE 33-bus grid for all the tested simulation cases.

Scenario	${\mathit{E}}_{\mathbf{loss}}$ (kWh/day)	${\mathit{E}}_{\mathbf{cost}}$ (USD/day)	${\mathit{V}}_{\mathbf{dev}}$ (%)
S1	2222.15 (0%)	8343.83 (0%)	6.38 (0%)
S2	1641.01 (26.15%)	8263.05 (0.99%)	2.66 (58.31%)
S3	1248.78 (43.80%)	4870.75 (41.62%)	4.07 (36.21%)
S4	723.02 (67.46%)	4803.84 (42.43%)	1.27 (80.09%)

presents the numerical results of each simulation scenario in the IEEE 33-bus grid. Note that the percentages of improvement with respect to the benchmark case defined by S1 are included in parentheses.

These results show that:

i.

Using PV-STATCOMs with different power factors affects the reduction of energy losses and energy purchasing costs at the substation terminals. The reductions in energy losses for S2 to S4 were $26.15\%$, $43.80\%$, and $67.46\%$ with respect to the benchmark case. As for the energy purchasing costs at the terminals of the substation, the reductions for S2 to S4 were $0.97\%$, $41.62\%$, and $42.43\%$, respectively.

ii.

As expected, the injection of reactive power in S2 shows that it is possible to reduce the power losses with respect to the benchmark case. However, the expected reductions in energy purchasing costs are minimal, as the main substation still supplies all of the energy consumed by constant power loads. In this scenario, the reduced energy losses were $581.14$ kWh/day, with an average cost of about USD/day $80.78$. This value is the difference between the energy purchasing costs of S2 and S1.

iii.

In all the simulation scenarios where active and reactive power was injected, it was observed that the average voltage profile improved with respect to the benchmark case. However, it was noted that reactive power has a greater influence on the voltage profile. For example, in S2 and S4, the improvements were $58.31\%$ and $80.09\%$, respectively, whereas, in the case of active power injection, this improvement was about $32.21\%$.

To illustrate the positive effect of the efficient scheduling of PV-STATCOMs in distribution networks, the active power generation at the terminals of the substation bus is reported in Figure 5 for each simulation scenario.

From the behavior of the energy generated at the substation terminals depicted in Figure 5, three prominent aspects can be highlighted. (i) The main effect of using PV-STATCOMs with unity or variable power factors (see the curve for S1 and compare it against the curves for S3 and S4) is that, at the terminals of the substation bus, the power generation is reduced as much as possible. From periods 12 to 40 (6:00 to 20:00), the power injection in the slack source is reduced. During these times, there is active power available in the PV-STATCOMs (Figure 4). (ii) As observed in Figure 5, the amount of renewable energy resources is not enough to cover all the electricity demand of the network; during all periods, conventional sources (substation) generate power to ensure the power equilibrium. (iii) The effect of reactive power on the total grid generation cost is minimal, as this allows reducing only some part of the total grid power losses, which results in a slight reduction of the total grid operating costs.

5.2. Numerical Results in the IEEE 69-Bus Grid

Table 5. Numerical results in the IEEE 69-bus grid for all the tested simulation cases.

Scenario	${\mathit{E}}_{\mathbf{loss}}$ (kWh/Day)	${\mathit{E}}_{\mathbf{cost}}$ (USD/Day)	${\mathit{V}}_{\mathbf{dev}}$ (%)
S1	2359.63 (0%)	8742.93 (0%)	3.17 (0%)
S2	1726.19 (26.84%)	8654.88 (1.01%)	1.79 (43.53%)
S3	1271.95 (40.10%)	5140.61 (41.20%)	1.91 (39.75%)
S4	700.17 (70.33%)	5067.32 (42.04%)	1.02 (67.82%)

presents the numerical results of each simulation scenario in the IEEE 69-bus grid. Note that the percentages of improvement with respect to the benchmark case defined by S1 are included in parentheses.

These results show that:

i.

The efficient operation of PV-STATCOMs in medium-voltage distribution networks has a positive effect on each one of the objective functions. In the case of the total grid energy losses, between S1 and S4, the total reduction was about $70.33\%$. Meanwhile, in the case of the total grid operating costs at the terminals of the substation, this reduction was about $42.04\%$. As for the voltage profile improvements, the best reductions were reported by the scenarios where reactive power was injected. Comparing S2 and S4 to S1, the improvements in the average voltage profile were about $43.53\%$ and $67.82\%$, respectively.

ii.

The energy losses reduction observed when comparing S1 to S2 was $633.44$ kWh/day, which, considering the average cost of USD/kWh $0.1390$, corresponds to savings of $88.05$ dollars per day of operation, with the main characteristic that it is the exact difference between the energy purchasing costs at the terminals of the substation in both simulation cases. This confirms that the impact of using reactive power to reduce energy purchasing costs is only associated with the possible savings with regard to the total grid power losses.

iii.

The use of PV-STATCOMs while considering reactive power injection has a direct effect on the average voltage profile behavior (see S2 and S4) since, for both cases, the most significant improvements in this objective function were observed. These results confirm the direct relationship between voltage profiles and reactive power, which is due to the fact that PV-STATCOMs work as a variable reactive power source, injecting this type of power in the nodes where they are installed. This allows improving the lagging power factor at the terminals of the substation.

The active power generation at the terminals of the substation bus is reported in Figure 6 for each simulation scenario.

Figure 6 shows that: (i) the energy generation reduction at the terminals of the substation is directly related to the ability to inject active power with PV-STATCOMs (i.e., S2 and S4); (ii) the contribution of reactive power injection to the total grid operating costs is minimal since its contribution corresponds to the reduction of the total grid power losses, which is small when compared to the energy consumed by all the loads during a day of operation; and (iii) the available solar energy for day-ahead scheduling is not enough to replace the substation bus in any period, which is attributable to the initial sizes assigned for the IEEE 69-bus grid, which are small in comparison with the total demand.

5.3. Complementary Analysis

To illustrate the effect of the variable power factor in minimizing each objective function under analysis, the $E_{\mathrm{loss}}$ function was selected for g minimized in S2 to S4 for the IEEE 33-, and 69-bus grids. For the sake of compactness, all the active or reactive power in the PV-STATCOM systems was added in order to observe the cumulative effect of these variables as a function of the operation scenario, as can be seen in Figure 7.

The behavior of the active and reactive power injection in Figure 7 shows that:

i.

The amount of reactive power injections in the IEEE 33-bus grid exhibits essential variations in the IEEE 33-bus grid between periods 17 and 46, being higher in S2, as these reactive injections are required to minimize the power losses. However, in the case of S4, this reactive power injection is lower, which is attributable to the fact that active power injection in S4 allows for better energy losses reductions when compared to the zero power factor case (S2). Note that, in S3 and S4, the available active power is fully injected into the network. This may imply that the PV plants for the IEEE 33-bus grid have been subsized and higher active power injection capabilities may help with better objective function improvements.

ii.

The results for the IEEE 33-bus grid showed a more promising performance since the total active power injection capabilities for this system were about 3040 kW. However, the active power injection in S3 and S4 is about 2825 kW, which implies that a full capacity is not required to minimize the total grid energy losses. In addition, the reactive power behavior has similar, which means that the active and reactive power injections are complementary variables that allows for better objective function values when working simultaneously, i.e., when comparing S4 to S2 and S3.

The numerical results in Figure 7 suggest that more research is required to define the optimal sizes and location of PV-STATCOMs in medium-voltage distribution networks, which can be considered as an opportunity for future works.

6. Conclusions

An efficient day-ahead scheduling strategy for operating PV-STATCOMs in medium-voltage distribution networks via convex optimization was proposed in this study. The exact nonlinear programming model representing this problem was transformed using a second-order cone approximation in the complex domain. This approximation was performed by transforming the product of two complex variables into its equivalent hyperbolic representation, which was then relaxed to obtain a convex cone. Four simulation scenarios, including a benchmark case, were considered to test the impact of optimal PV-STATCOM scheduling on electrical networks. These scenarios assumed operations with unity, zero, and variable power factors. Numerical simulations in the IEEE 33- and 69-bus grids confirmed that:

i.

The use of PV-STATCOMs with zero or variable power factors had a significant impact on the average voltage profile behavior. For both test feeders, the expected improvements were greater than $58\%$ (IEEE 33-bus grid) and $43\%$ (IEEE 69-bus grid) when considering a zero power factor (as shown in S2). Furthermore, when a variable power factor was employed (S4), the improvements reached 80% and 67% for the each grid.

ii.

The most significant benefit of using variable active power injection with PV-STATCOMs was observed in the form of a substantial decrease in grid energy purchasing costs at the substation terminals. This reduction was achieved by injecting active power from renewable energy sources, which directly resulted in a decrease in the total power injection at the substation terminals. For both test grids in S3 and S4, the reductions in energy purchasing costs exceeded $41\%$ when PV-STATCOMs were optimally scheduled while considering both unity and variable power factors.

iii.

In the case of daily energy losses minimization, all operating scenarios exhibited significant reductions. For the IEEE 33-bus grid, the minimum and maximum reductions in the total grid power losses were between $26.15\%$ and $67.35\%$ (as could be seen in S2 and S4), whereas, for the IEEE 69-bus grid, these were $26.84\%$ and $70.33\%$.

As a general conclusion based on the numerical results, using PV-STATCOMs with a zero power factor (i.e., purely dynamic reactive power compensation) is a suitable method for minimizing grid power losses or improving grid voltage profiles. However, in the case of energy purchasing costs minimization, the only contribution of this type of operation is associated with a reduction in the costs of energy losses, which is a small contribution when compared to the aggregated energy consumption costs in all of the demand nodes.

As for future work, the following studies can be conducted: (i) combining the efficient operation of PV-STATCOMs with battery energy storage systems in order to obtain an efficient scheduling methodology that allows providing active power in periods where renewable generation resources are not available; (ii) transforming the proposed second-order cone optimization model into a mixed-integer convex equivalent for simultaneously determining the daily scheduling and optimal placement and sizing of PV-STATCOMs; and (iii) comparing the proposed convex methodology against combinatorial approaches or semi-definite programming models in meshed distribution networks.