A Multi-Period Optimal Reactive Power Dispatch Approach Considering Multiple Operative Goals

Abstract

The optimal reactive power dispatch (ORPD) problem plays a key role in daily power system operations. This paper presents a novel multi-period approach for the ORPD that takes into account three operative goals. These consist of minimizing total voltage deviations from set point values of pilot nodes and maneuvers on transformers taps and reactive power compensators. The ORPD is formulated in GAMS (General Algebraic Modeling System) software as a mixed integer nonlinear programming problem, comprising both continuous and discrete control variables, and is solved using the BONMIN solver. The most outstanding benefit of the proposed ORPD model is the fact that it allows optimal reactive power control throughout a multi-period horizon, guaranteeing compliance with the programmed active power dispatch. Additionally, the minimization of maneuvers on reactors and capacitor banks contributes to preserving the useful life of these devices. Furthermore, the selection of pilot nodes for voltage control reduces the computational burden and allows the algorithm to provide fast solutions. The results of the IEEE 118 bus test system show the applicability and effectiveness of the proposed approach.

1. Introduction

The ORPD can be seen as an AC optimal power flow problem with a particular set of control variables, which include shunt compensations, transformer tap changers, and voltage generator set points. In this case, the active power outputs of all generators, with the exception of the slack bus, are kept constant [1]. The ORPD problem plays an important role in the economic and secure operation of power systems, since it is one of the ways to solve the problems of reactive power management and voltage control [2]. The main idea behind the ORPD problem is finding an optimal schedule for the reactive power control devices within a network in such a way that operational constraints are met while also optimizing a given objective function [3]. Numerous ORPD formulations and solution methods have been reported in the literature to address specific instances of the problem [4]. These include different objective functions, controls, and system constraints, as well as distinct mathematical characteristics and computational requirements [5]. The resulting optimization problems go by many names, depending on the particular objective function being addressed and the constraints under consideration.

Given the nonlinear and nonconvex nature of the ORPD problem, numerous metaheuristic techniques have also been proposed for its solution. In [6], the authors proposed a particle swarm optimization (PSO) approach to deal with the ORPD problem. PSO is inspired by the sociological behavior associated with schools of fish and flocks of birds. Some modifications have been proposed to this technique to better handle the ORPD problem, such as parallel vector-evaluated PSO [7], coordinated aggregation PSO [8], fractional PSO [9], and turbulent crazy PSO [10]. Genetic algorithms (GAs) are metaheuristic algorithms that mimic the process of Darwinian natural selection. In this case, a set of individuals or candidate solutions must go through stages of selection, crossover, and mutation in order to improve a given fitness function. In [11], the authors proposed a GA to solve the ORPD problem, seeking loss minimization as well as the maximization of the voltage stability margin. For this last objective, the L-index of the system was used. A self-adaptive real coded GA was also implemented in [12] to approach the ORPD problem. Differential evolution (DE) is also a population-based technique that iteratively improves a candidate solution based on an evolutionary process. In [13,14,15], the authors used DE to solve the ORPD problem. The mean–variance mapping optimization algorithm has also been applied to tackle the ORPD problem [16]. This metaheuristic algorithm is based on a mapping function used to generate new candidate solutions based on the mean and variance of a set that contains the current best solutions of the algorithm. Other metaheuristic techniques applied to the ORPD problem include the seeker optimization algorithm [17], gravitational search algorithm [18], harmony search algorithm [19], teaching–learning-based optimization [20], krill herd algorithm [20], chaotic bat algorithm [21], evolutionary algorithm [22], heap-based optimizer [23], hierarchical distributed optimization [24], and gray wolf optimizer algorithm [25,26]. Hybrid approaches combine characteristics of two or more metaheuristic techniques to take advantage of their strengths. In [27,28,29], the authors proposed hybrid versions of PSO and the firefly algorithm to solve the ORPD problem. A detailed description of the techniques applied to the ORPD problem is outside the scope of this paper; nonetheless, a detailed classification and comparison can be found in [30].

The common feature of the aforementioned methodologies is the fact that they approach the static version of the ORPD problem; that is to say, the optimization is carried out over a single time period. Real-life ORPD problems must consider a multi-period approach and are, therefore, often significantly more challenging than the classically considered problems [31]. Under real-time circumstances, solutions provided by traditional ORPD models may not be practical because the number of control actions would be too large to be executed in actual power system operation [32]; furthermore, the computing time plays a key role in such applications. The multi-period ORPD (MP-ORPD) was originally proposed in [33]; its main focus is dealing with different loading conditions for a given future time interval instead of a single snapshot of the power network. The set of desirable features for real implementations of the MP-ORPD programs is extensive and can make the solution of the model a complex task. In this case, the control actions must be valid within a given time horizon, the improvement of the voltage profile can be achieved by resorting to pilot nodes, and the number of control actions must not be excessive. A compilation of such features is presented in [34]. Compared with its traditional or static counterpart, the MP-ORPD is a more complex optimization problem involving multiple local minima as well as nonlinear and discontinuous constraints [32].

Relatively few studies have approached the dynamic MP-ORPD. In this case, the most common objective function corresponds to the minimization of power losses. In [35], a day-ahead reactive power dispatch was proposed with the aim of minimizing active power losses for wind power plants. In [36], the authors approached the ORPD problem by considering the time-varying demand and renewable energy uncertainty. In this case, the authors implemented the so called Rao-3 algorithm to solve the MP-ORPD problem. In [37], the MP-ORPD was approached through chance-constrained programming. In [38], a dynamic optimal reactive power flow considering several time intervals was carried out, taking into account voltage stability. In this case, the problem was solved by means of a branch and bound primal-dual interior point method and different areas were considered. In [39,40], the authors also proposed ORPD schemes for power loss reductions over a 24 h time horizon. This paper presents a novel approach to solve the MP-ORPD problem. The main features and contributions that differentiate this study from other papers reported in the literature are as follows:

• The proposed MP-ORPD is envisaged from the point of view of a power system operator; in this case, a single objective function is not pursued as such, but instead a set of goals regarding the minimization of deviations from desired operational values are considered;
• The main goal of the MP-ORPD is guaranteeing operative feasibility throughout a given time horizon, while minimizing the number of maneuvers carried out in reactive power devices and transformers taps, preserving their useful life and reducing eventual maintenance costs;
• Instead of considering all buses to enforce voltage limit constraints, only a set of pilot nodes are taken into account. Furthermore, dynamic limits are considered in these buses to mimic real-life operation. This results in low computational effort, which encourages real-life applications of the methodology.

This work is organized as follows. Section 1 presents a literature review of the ORPD problem. Section 2 describes the relevant aspects of the proposed model and other considerations taken into account. Section 3 presents the tests and results with the IEEE 57-bus and IEEE 118-bus test systems. Finally, the conclusions are presented in Section 4.

2. Proposed Mathematical Modeling

This section describes the mathematical modeling of the proposed MP-ORPD problem, the nomenclature for which can be found in Appendix A. The ORPD problem is subject to equality and inequality constraints [3]. The former take into account the definition of power flows, as well as the power balance equations derived from the Kirchhoff’s laws, while the latter impose physical limits of variables. Regarding power balance constraints, there are several formulations for AC power flow equations. The most popular are polar power-voltage (P), rectangular power-voltage (R), rectangular current-voltage (IV), and Y-bus formulation. While the Y-bus matrix formulation benefits from eliminating the line flow parameters in the bus balance constraints, the benefit is lost when the line flow variables still need to be defined to maintain the line flow limits. For this reason, standard line power (SLP) models (P, R, IV) typically outperform Y-bus models due to the additional calculations required. A detail description of these models can be found in [41]. In this case, the polar power-voltage formulation is chosen due to its fast convergence and lower number of nonlinearities compared to other formulations [42]. The proposed MP-ORPD includes additional considerations regarding the maneuvering of equipment (inequality constraints). In this case, the equipment movements are limited in each period and a maximum daily limit of movements is considered.

2.1. Objective Function

The proposed objective function (OF) given by Equation (1) is made up of three elements described in Equations (2)–(4); each term minimizes the deviations of the control variables from period to period. The first term, labeled the TVD (total voltage deviation), seeks to minimize the voltage difference of the set of pilot nodes with their reference values. These voltage references are changed to typical values along the optimization horizon. The second term, labeled the TQS (total reactive shunt deviation), seeks to reduce the movement of shunt elements between continuous periods. Finally, the third term, labeled the TQG (total reactive generator deviation), aims to maintain reactive power reserves; that is, the generation units are at a floating point of reactive power. In this case, ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ are penalty factors associated with each objective; $v_{it}$ is the voltage magnitude at pilot bus i at time t; $V_{ref,it}$ is the reference magnitude of pilot bus i at time t; $n_{kt}^{\mathcal{S}}$ and $n_{k\left(t-1\right)}^{\mathcal{S}}$ are the step numbers of shunt element $k\in \mathcal{S}$ in period $t$ and period $t-1$; $q_{gt}^{\mathcal{G}}$ is the reactive power generated by generator $g$ in period $t$; finally, $\mathcal{T}, \mathcal{S}$, and $\mathcal{N}p$ are the sets of periods, shunt elements, and pilot nodes, respectively.

$$\mathrm{Min} \left({\beta }_1\mathrm{TVD} +{\beta }_2\mathrm{TQD} +{\beta }_3\mathrm{TQG}\right)$$

(1)

$$\mathrm{TVD}={\displaystyle \sum }_{t\in \mathcal{T}}{\displaystyle \sum }_{i\in {\mathcal{N}}_p}{\left(v_{it}-V_{ref,it}\right)}^2$$

(2)

$$\mathrm{TQD}={\displaystyle \sum }_{t\in \mathcal{T}}{\displaystyle \sum }_{k\in \mathcal{S}}{\left(n_{kt}^{\mathcal{S}}-n_{k\left(t-1\right)}^{\mathcal{S}}\right)}^2$$

(3)

$$\mathrm{TQG}={\displaystyle \sum }_{t\in \mathcal{T}}{\displaystyle \sum }_{g\in {\mathcal{G}}_{ }}(q_{gt}^{\mathcal{G}}){ }^2$$

(4)

2.2. Equality Constraints

In the proposed MP-ORPD model, a set of equality constraints are taken into account regarding the power flow in branches and power balance constraints in buses. Equations (5) and (6) define the power flow in branches. In this case, $f_{ijct}^P$ and $f_{jict}^P$ represent the active power flow from bus i to bus j and vice versa; while $f_{ijct}^Q$ and $f_{jict}^Q$ in Equations (7) and (8) represent the reactive power flows from bus i to bus j and vice versa, respectively. Note that the power flow equations are different at each end of the line; this is because the tap ratio transformer ${\alpha }_{ijct}^{ }$ is taken account. $G_{ijc}^{ℒ}$, $B_{ijc}^{ℒ},B_{ijc}^C$, and ${\varphi }_{ijc}$ are the conductance, susceptance, branch charging susceptance, and the angle on line $ijc\in ℒ$, respectively.

Equations (9) and (10) respectively represent the active and reactive power balance constraints derived from Kirchhoff’s laws. Here, $P_{gt}^{\mathcal{G}}$ and $p_{gt}^{\mathcal{G}}$ are the fixed and variable active power of generator g at time t; $D_{it}^P$ and $D_{it}^Q$ are the active and reactive power demand at bus $i$ for period $t$; $G_i^{ℰ} \mathrm{and} B_i^{ℰ}$ are the shunt conductance and susceptance at bus $i\in \mathcal{N}$, respectively; finally, $B_k^{\mathcal{S}}$ is the shunt susceptance of shunt element $k\in \mathcal{S}$.

$$f_{ijct}^P=\frac{1}{{\alpha }_{ijct}^2}{G}_{ijc}^{ℒ}{v}_{it}^2-\frac{1}{ {\alpha }_{ijct}}{v}_{it}{v}_{jt}\left(G_{ijc}^{ℒ}\cos \left({\theta }_{it}-{\theta }_{jt}-{\varphi }_{ijc}\right)+B_{ijc}^{ℒ}\sin \left({\theta }_{it}-{\theta }_{jt}-{\varphi }_{ijc}\right)\right),\forall ijc\in ℒ;t\in \mathcal{T}$$

(5)

$$f_{jict}^P=G_{ijc}^{ℒ}{v}_{jt}^2-\frac{1}{ {\alpha }_{ijct}}{v}_{it}{v}_{jt}\left(G_{ijc}^{ℒ}\cos \left({\theta }_{jt}-{\theta }_{it}+{\varphi }_{ijc}\right)+B_{ijc}^{ℒ}\sin \left({\theta }_{jt}-{\theta }_{it}+{\varphi }_{ijc}\right)\right),\forall ijc\in ℒ;t\in \mathcal{T}$$

(6)

$$f_{ijct}^Q=-\frac{1}{{\alpha }_{ijct}^2}\left(B_{ijc}^{ℒ}+\frac{B_{ijc}^C}{2}\right)v_{it}^2-\frac{1}{ {\alpha }_{ijct}}{v}_{it}{v}_{jt}\left(G_{ijc}^{ℒ}\cos \left({\theta }_{it}-{\theta }_{jt}-{\varphi }_{ijc}\right)-B_{ijc}^{ℒ}\sin \left({\theta }_{it}-{\theta }_{jt}-{\varphi }_{ijc}\right)\right),\forall ijc\in ℒ;t\in \mathcal{T}$$

(7)

$$f_{jict}^Q=-\left(B_{ijc}^{ℒ}+\frac{B_{ijc}^C}{2}\right)v_{jt}^2-\frac{1}{ {\alpha }_{ijct}}{v}_{it}{v}_{jt}\left(G_{ijc}^{ℒ}\cos \left({\theta }_{jt}-{\theta }_{it}+{\varphi }_{ijc}\right)-B_{ijc}^{ℒ}\sin \left({\theta }_{jt}-{\theta }_{it}+{\varphi }_{ijc}\right)\right),\forall ijc\in ℒ;t\in \mathcal{T}$$

(8)

$${\displaystyle \sum }_{g\in {\mathcal{G}}_{i\left(i\ne slack\right)}}{P}_{gt}^{\mathcal{G}}+{\displaystyle \sum }_{g\in {\mathcal{G}}_{slack} }{p}_{gt}^{\mathcal{G}}-{\displaystyle \sum }_{\left(jc\right):ijc\in ℒ}{f}_{ijct}^P-{\displaystyle \sum }_{\left(jc\right):jic\in ℒ}{f}_{jict}^P-D_{it}^P-v_{it}{}^2{G}_i^{ℰ}=0,\forall i\in \mathcal{N};t\in \mathcal{T}$$

(9)

$${\displaystyle \sum }_{g\in {\mathcal{G}}_i}{q}_{gt}^{\mathcal{G}}+v_{it}{}^2{\displaystyle \sum }_{k\in {\mathcal{S}}_i}{B}_k^{\mathcal{S}}{n}_{kt}^{\mathcal{S}}-{\displaystyle \sum }_{\left(jc\right):ijc\in ℒ}{f}_{ijct}^Q-{\displaystyle \sum }_{\left(jc\right):jic\in ℒ}{f}_{jict}^Q-D_{it}^Q+v_{it}{}^2{B}_i^{ℰ}=0,\forall i\in \mathcal{N};t\in \mathcal{T}$$

(10)

2.3. Inequality Constraints

2.3.1. Generators Constraints

Equation (11) corresponds to the reactive power limits ($\overline{Q_g}$ maximum and $\underset{\_}{Q_g}$ minimum) on the set of online generators and FACTS (flexible AC transmission systems) devices. In this case, $U_{gt}$ represents the state of generator g, where 1 indicates being on service and 0 otherwise. Equation (12) corresponds to the active power limit ($\overline{P_g^{ }}$ maximum and $\underset{\_}{P_g^{ }}$ minimum) on the set of online slack generators (which guarantees the active power balance). The rest of the generators are considered as fixed sources of active power.

$$U_{gt}\underset{\_}{Q_g^{ }}\le q_{gt}^{\mathcal{G}} \le U_{gt}\overline{Q_g^{ }},\forall g\in \mathcal{G};t\in \mathcal{T}$$

(11)

$$U_{gt}\underset{\_}{P_g^{ }}\le p_{gt}^{\mathcal{G}} \le U_{gt}\overline{P_g^{ }},\forall g\in {\mathcal{G}}_{\mathit{slack}};t\in \mathcal{T}$$

(12)

2.3.2. Voltage Angle Constraints

Equation (13) represents is the angular difference between two connected nodes. This constraint guarantees the stable-state limits for a line power transfer. In this case, the angle values are set lower than the theoretical reference of π/2.

$$-\frac{\pi }{3}\le {\theta }_{it}-{\theta }_{jt}\le \frac{\pi }{3},\forall \left(ij\right):ijc \in ℒ; t\in \mathcal{T}$$

(13)

2.3.3. Transformer Constraints

Equation (14) indicates the maximum ($\overline{{\alpha }_{ijc}^{ }}$) and minimum ($\underset{\_}{{\alpha }_{ijc}^{ }}$) limits of the ratio transformer (${\alpha }_{ijct}$). Equation (15) is activated when the tap transformer is operating ($u_{ijct}^{\daleth }$, 1 represents a maneuver, 0 otherwise).

$$\underset{\_}{{\alpha }_{ijc}^{ }}\le {\alpha }_{ijct} \le \overline{{\alpha }_{ijc}^{ }} ,\forall ijc\in \daleth ;t\in \mathcal{T}$$

(14)

$$\left|{\alpha }_{ijct}-{\alpha }_{ijc\left(t-1\right)}\right|\le u_{ijct}^{\daleth },\forall ijc\in \daleth ;t\in \mathcal{T}$$

(15)

2.3.4. Shunt Constraints

Equation (16) sets limits for the number of steps in shunt elements, where $n_{kt}^{\mathcal{S}}$ is the step number of shunt element $k\in \mathcal{S}$ in period $t$ and $N_k^{\mathcal{S}}$ is the maximum number of steps of element $k\in \mathcal{S}$. Equation (17) is activated when the number of steps from one period to another one changes, which indicates an action control on the shunt element ($u_{kt}^{\mathcal{S}}$, 1 represents a maneuver, 0 otherwise).

$$0\le n_{kt}^{\mathcal{S}}\le N_k^{\mathcal{S}},\forall k\in {\mathcal{S}}_{ };t\in \mathcal{T}$$

(16)

$$\left|n_{kt}^{\mathcal{S}}-n_{k\left(t-1\right)}^{\mathcal{S}}\right|\le u_{kt}^{\mathcal{S}}{N}_k^{\mathcal{S}},\forall k\in {\mathcal{S}}_{ };t\in \mathcal{T}$$

(17)

2.3.5. Security Constraints

Equation (18) determines the limits of voltage magnitude $v_{it}$ for each bus at every time interval t ($\overline{V_i}$ maximum and $\underset{\_}{V_i}$ minimum). Equation (19) defines the active power limit ($\overline{F_l^{\mathscr{H}}}$) for interface l. Equation (20) indicates the maximum value of active power allowed by each transmission line ($\overline{F_{ijc}^P}$).

$$\underset{\_}{V_i^{ }}\le v_{it} \le \overline{V_i} ,\forall i\in \mathcal{N};t\in \mathcal{T}$$

(18)

$${\displaystyle \sum }_{ijc,jic\in {ℒ}_l}{f}_{ijct}^P\le \overline{F_l^{\mathscr{H}}},\forall l\in \mathscr{H};t\in \mathcal{T}$$

(19)

$$-\overline{F_{ijc}^P}\le f_{ijct}^P\le \overline{F_{ijc}^P},t\in \mathcal{T}; ijc\in ℒ$$

(20)

2.3.6. Operating Times Constraints

Equation (21) indicates that the total number of maneuvers on the capacitors, reactors, and transformer taps must not exceed the maximum set in one day of operation. In this case, $u_{ijct}^{\daleth }$ is the maneuver on tap transformer $ijc\in \daleth$ in period $t$, $u_{kt}^{\mathcal{S}}$ is the maneuver on shunt element $k\in \mathcal{S}$ in period $t$, and $\overline{M}$ indicates the maximum number of maneuvers allowed.

$${\displaystyle \sum }_{t\in \mathcal{T}}\left( {\displaystyle \sum }_{ijc\in \daleth }{u}_{ijct}^{\daleth }+{\displaystyle \sum }_{k\in \mathcal{S}}{u}_{kt}^{\mathcal{S}}\right)\le \overline{M} ,\forall t\in \mathcal{T};k\in \mathcal{S}; ijc\in \daleth$$

(21)

The MP-ORPD model presented in this paper is a nonlinear, nonconvex optimization problem, classified as a mixed-integer nonlinear programming (MINLP) problem, as it handles continuous, integer, and discrete control variables. A MINLP problem is said to be convex if its continuous relaxation, i.e., the problem obtained by dropping the integrality constraints, is a convex optimization problem; otherwise, it is said to be nonconvex. Note that in this case, equality constraints include the multiplication of variables, as well as sine and cosine functions of voltage angles. Additionally, absolute values are included in Equations (15) and (17).

3. Tests and Results

3.1. Pilot Nodes

Real power systems are required to keep an appropriate voltage profile throughout the transmission network in the face of the hourly evolution of the load and topological changes. In real-life applications, direct voltage optimization of every bus within the entire power system is impractical [43]. This is why system-wide information is used to identify pilot nodes. A pilot node can be defined as one that represents a set of nodes and their behavior in terms of voltage sensitivity against reactive power sources [44]. A characteristic of a pilot node is its poor relationship with other pilot nodes, so that independence between voltage control areas can be approximated. To find pilot nodes, different authors have developed techniques based on optimization, statistics, metaheuristics, empirical, unsupervised classification, and combinatorial problems, among others [45]. A detailed description of these methodologies is outside of the scope of the present study; therefore, the pilot nodes reported in [44] were used, involving the identification of voltage control areas based on voltage stability for the IEEE test system.

3.2. Test Systems

Several simulations were performed on the IEEE 57-bus and 118-bus test systems. The data for these systems can be consulted in [46]. All tests were carried out on a personal computer equipped with an Intel Core i5 (Quadcore) 1.8 GHz processor and 8 GB of RAM memory. The proposed MP-ORPD model was implemented in General Algebraic Modeling System (GAMS, version 24.8.5), using the BONMIN (Basic Open-Source Nonlinear Mixed Integer Programming) solver [47].

The MP-ORPD contemplates time-varying loads. In this paper, load conditions are featured through a load curve of a spring weekday available in [48]. The behavior of the 24 h load curve for both test systems is illustrated in Figure 1, where each of the four colors represents a load condition with which the reference voltage value was chosen for the pilot nodes; these are purple (minimum load), blue (medium load), orange (high load), and gray (low load). Figure 1 illustrates the time-varying load of the IEEE 118-bus test system. A scale factor of 0.2947 was used to obtain the load of the IEEE 57-bus test system.

3.2.1. IEEE 57-Bus Test System

The IEEE 57-bus test system consists of 62 branches and 26 control variables; these include 7 generation units, 16 transformers, and 3 capacitors. A single-line diagram of the system is depicted in Figure 2, whereby each color represents a different voltage control area with a single pilot node. The total active and reactive load demand are 1250 MW and 336 MVAr for period 11 on the 100 MVA base. The initial output of active and reactive power from the generators come from the solution of the unit commitment (UC) problem reported in [41]. Additionally, the initial settings of the tap transformers are in the nominal ratio and all shunt elements are connected.

The minimum and maximum limits of control variables for the IEEE 57-bus test system are as follows. Each transformer ratio varies from 0.9 to 1.1 per-unit (p.u) in equal steps of 0.01. Shunt elements are between 0 and its MVAr parameter, which have no steps. Voltage set points of generators vary in the range of [0.85, 1.1] p.u (except for lower voltage limits of the pilot nodes, which are reported in

Table 1. Lower voltage limits of pilot nodes and MVAr parameters of shunt elements (IEEE 57-bus test system).

Pilot Node	Lower Voltage Limit (p.u)	Shunt	MVAr
${\mathrm{V}}_{\mathrm{np}1}$	0.9097	${\mathrm{Q}}_{\mathrm{C}18}$	10
${\mathrm{V}}_{\mathrm{np}4}$	0.9177	${\mathrm{Q}}_{\mathrm{C}25}$	25
${\mathrm{V}}_{\mathrm{np}10}$	0.9244	${\mathrm{Q}}_{\mathrm{C}53}$	53
${\mathrm{V}}_{\mathrm{np}12}$	0.9252
${\mathrm{V}}_{\mathrm{np}13}$	0.9039
${\mathrm{V}}_{\mathrm{np}22}$	0.8927
${\mathrm{V}}_{\mathrm{np}29}$	0.9274
${\mathrm{V}}_{\mathrm{np}31}$	0.8500
${\mathrm{V}}_{\mathrm{np}36}$	0.8640
${\mathrm{V}}_{\mathrm{np}41}$	0.9119
${\mathrm{V}}_{\mathrm{np}48}$	0.9038

). Table 1 shows the shunt MVAr parameter as well as the lower voltage limits for pilot nodes taken from [44]. In this case, the maximum number of maneuvers allowed in a day ($\overline{M})$ is set to 150.

3.2.2. IEEE 118-Bus Test Case

Several simulations were performed on the IEEE 118-bus test system, which has 186 branches and 77 control variables; these consist of 54 generation units, 9 transformers, 12 capacitors, and 2 reactors. The single-line diagram of the system is depicted in Figure 3, where each color represents a different voltage control area with a single pilot node. The total active and reactive base loads are 4242 MW and 1438 MVAr for period 11 on the 100 MVA base. The initial output of active and reactive power from the generators comes from the solution of the UC problem reported in [41]. Additionally, the initial settings of the tap transformers are in the nominal ratio and the states of the shunt elements are all connected.

The minimum and maximum limits of control variables for the IEEE 118-bus test system are as follows. Each transformer ratio varies from 0.9 to 1.1 p.u in equal steps of 0.01. The shunt elements are between 0 and its MVAr parameter, which have no steps. The voltage set points of generators vary in the range of [0.9, 1.1] p.u (except for lower voltage limits of pilot nodes, which are reported in

Table 2. Lower voltage limits of pilot nodes and MVAr parameters of shunt elements (IEEE 118-bus test system).

Pilot Node	Lower Voltage Limit (p.u)	Shunt	MVAr
${\mathrm{V}}_{\mathrm{np}69}$	0.935	${\mathrm{Q}}_{\mathrm{L}5}$	−40
${\mathrm{V}}_{\mathrm{np}5}$	0.9481	${\mathrm{Q}}_{\mathrm{C}34}$	14
${\mathrm{V}}_{\mathrm{np}37}$	0.9276	${\mathrm{Q}}_{\mathrm{L}37}$	−25
${\mathrm{V}}_{\mathrm{np}56}$	0.9427	${\mathrm{Q}}_{\mathrm{C}44}$	10
${\mathrm{V}}_{\mathrm{np}77}$	0.9349	${\mathrm{Q}}_{\mathrm{C}45}$	10
${\mathrm{V}}_{\mathrm{np}66}$	0.90	${\mathrm{Q}}_{\mathrm{C}46}$	10
${\mathrm{V}}_{\mathrm{np}46}$	0.9341	${\mathrm{Q}}_{\mathrm{C}48}$	15
${\mathrm{V}}_{\mathrm{np}23}$	0.90	${\mathrm{Q}}_{\mathrm{C}74}$	12
${\mathrm{V}}_{\mathrm{np}12}$	0.932	${\mathrm{Q}}_{\mathrm{C}79}$	20
${\mathrm{V}}_{\mathrm{np}70}$	0.9434	${\mathrm{Q}}_{\mathrm{C}82}$	20
${\mathrm{V}}_{\mathrm{np}17}$	0.90	${\mathrm{Q}}_{\mathrm{C}83}$	10
${\mathrm{V}}_{\mathrm{np}63}$	0.90	${\mathrm{Q}}_{\mathrm{C}105}$	20
${\mathrm{V}}_{\mathrm{np}80}$	0.954	${\mathrm{Q}}_{\mathrm{C}107}$	6
${\mathrm{V}}_{\mathrm{np}8}$	0.9655	${\mathrm{Q}}_{\mathrm{C}110}$	6
${\mathrm{V}}_{\mathrm{np}49}$	0.9542
${\mathrm{V}}_{\mathrm{np}32}$	0.90

). Table 2 shows the shunt MVAr parameter and lower voltage limit for pilot nodes taken from [44]. In this case, $\overline{M}$ is set to 100 for all operation days.

3.3. Results

The schedule of control variables for the MP-ORPD (on IEEE 57-bus and 118-bus test systems) is reported in Appendix B, which indicates the settings for capacitors, reactors, transformers, and generation voltages from period to period. It was possible to verify that all the variables were kept within their operative ranges. All results in the below subsections correspond to the complete OF (TQG+ TVD+TQD) given by Equation (1).

3.3.1. IEEE 57-Bus Test System

Figure 4 depicts the voltage profile curves of four pilot nodes. Note that the voltage control implemented in this system fits the reference curve proposed for each pilot node. The shape of the voltage profiles of the pilot nodes is in accordance with the operational strategy, which increases the reference values of these nodes at peak demand in order to guarantee the voltage stability of the system.

The proposed system has a high reactive power demand and must meet the voltage profiles required in the pilot nodes, as shown in Figure 4. In addition, of the 26 control elements, only 10 have the capacity to provide reactive power, and of these, 7 are generation units. This shows that the IEEE 57-bus test system is quite limited for an adequate reactive power dispatch. Figure 5 shows the reactive power outputs of all generators of the system for each period. In periods of low demand, in order to obtain the desired voltage in the pilot nodes, it is necessary to increase the amount of reactive power injected, while for periods of medium and high demand, a smaller increase in reactive power injection is observed. This behavior is an indicator of the depletion of reactive power reserves and provides signals for the expansion of reactive power sources.

Figure 6 shows the operation of the taps for four randomly selected transformers. They behave according to the demand of the control area of each pilot node and the availability of shunt elements and generators. Transformer T10-51 is the only control element in the area it controls; therefore, its behavior features a greater number of maneuvers, while transformers T24-26 and T7-29 have internal or nearby control elements, such as shunt compensation, shunt generation, and the transformers themselves, which is why the numbers of maneuvers for these elements are lower.

3.3.2. IEEE 118-Bus Test System

One of the objectives within the MP-ORPD is to maintain adequate voltage profiles for the set of pilot nodes. In this paper, four settings of voltage values were defined for an operation day, namely minimum, low, medium, and high voltages, which ware characterized by the demand behavior and user requirements. Figure 7 shows the reference voltage profiles and the voltage after optimization for a sample of pilot nodes (5, 63, 46, and 8). Additionally, as a particular case, bus 5 was set to have a higher voltage profile in the peak periods (0.058 p.u with respect to its minimum reference and a 138 kV voltage base). The results show that even with an “aggressive” voltage profile, it is possible to follow the voltage reference value.

The maximum error between the reference and the optimized voltage is less than 0.95 kV, as shown in Figure 8, where the indicated period is not the same in each node, although it is the period with the greatest difference. In Figure 8, pilot node 49 in period 11 has an error of −0.94 kV; this difference is not constant in the other periods, which have lower absolute values.

The sum of the reactive power in terms of the absolute value of the set of generators at each period is a way of representing the use of the dynamic reactive power reserve, which in a power system is necessary to manage uncertainty and contingencies [49]. Figure 9 presents the dynamic reserve of the reactive power in the base case (orange) and its subsequent optimization (blue). In this case, reactive power reserves after optimization represent only 31.3% of the base case. This is important, since it allows the system operator to better react in cases of disturbance and uncertainty, while maintaining sufficient reactive reserves in the most quick and effective VAR sources, which is an intrinsic requirement for proper corrective control actions.

In this case, compared to the 57-bus system, there are about three times as many elements that can control the reactive dispatch, so the system is able to follow the voltage profile of the pilot nodes.

Figure 10 presents the reactive power output for each generator, representing a sample set of generators in a random period (in this case: generators 4, 10, 59, 69, 77, and 90; period 11). A significant reduction in reactive power can be observed in the optimized case (blue) versus the base case (orange).

Figure 11 shows the variations in tap positions (expressed in terms of a ratio) for each transformer throughout a day of operation. It can be observed that the numbers of maneuvers on the tap are greater during periods with high demand and with high-voltage profiles; nonetheless, the maximum number of maneuvers is kept the same, as indicated in Equation (21).

These results show the relationship between the number of transformer operations and the availability of additional elements for reactive power control. When there are other elements that can be used for reactive power control in a given area in addition to the transformers, the number of maneuvers in these elements is reduced. This is the case for transformers T81-80 and T8-5, which are located in small control areas with other elements that provide reactive power control, resulting in lower numbers of transformer maneuvers.

3.4. Sensibility of Objective Function Components

Table 3. Sensibility of the objective function components for the IEEE 57-bus system.

	Case Base	TQG	TQD	TVD	TQG+ TQD	TQG+ TVD	TVD+ TQD	TQG+ TVD + TQD
${\displaystyle \sum }{u}_{ijt}^{\daleth }$	0	117	51	141	118	118	135	118
${\displaystyle \sum }{u}_{it}^{\mathcal{S}}$	0	0	0	4	0	1	0	1
$\sqrt{TQG}$	4757	3928	4588	6965	3928	5101	7080	5101
$\sqrt{TVD}$	10.56	29.94	10.97	0.18	29.9	1.69	0.2	1.69

shows a sensitivity analysis of the results for the IEEE 57-bus test system. The first row shows the total number of taps operations. The second row corresponds to the total number of operations on shunt elements, while the third and fourth rows indicate the reactive deviation of the generators and the voltage deviation with respect to the pilot nodes. When evaluating a single objective, the proposed approach manages to reduce the objective function considerably. For the reactive deviation, the formulation that only considers the TQG is the lowest of all cases analyzed, while the lowest number of maneuvers is achieved with the TQD formulation, with a total number of 51 maneuvers; similarly, it can be observed that the lowest voltage deviation, with a value of 0.18, is obtained with the TVD formulation. It was already mentioned that this test system requires a higher reactive power support. This fact can be seen when comparing the maximum and minimum values of the reactive power deviations in generators and the voltage deviations of the pilot nodes. The last column in Table 3 (TQG+ TVD+TQD) indicates the best trade-off in the use of elements for reactive power control; the results are shown in Figure 4, Figure 5 and Figure 6.

Table 4. Sensibility of the objective function components for IEEE 118-bus system.

	Case Base	TQG	TQD	TVD	TQG+ TQD	TQG+ TVD	TVD+ TQD	TQG+ TVD + TQD
${\displaystyle \sum }{u}_{ijt}^{\daleth }$	0	85	20	51	72	71	54	78
${\displaystyle \sum }{u}_{it}^{\mathcal{S}}$	0	13	0	20	14	19	0	17
$\sqrt{TQG}$	33037	4493	35997	36509	4494	10337	37000	10341
$\sqrt{TVD}$	18.22	20.98	7.37	8E-06	20.98	0.66	4E-12	0.66

presents a comparison of each component of the objective function and their combinations in order to evaluate the individual and joint sensitivity of the objective function. The number of maneuvers carried out on the transformer taps, the number of maneuvers carried out on the shunt elements, the absolute value of the reactive power used by the generators, and the voltage deviation in the pilot nodes with respect to the reference value were evaluated; all information corresponds to an operation period of 24 h.

For this system, there is a large reactive power reserve, which is evident when comparing the maximum and minimum values of the formulations illustrated in Table 4. For this case, the compared values have a greater range of variation with respect to the IEEE 57-bus test system. The last column in Table 4 (TQG + TVD + TQD) indicates the best trade-off in the use of elements for reactive power control; these results are shown in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. In this case, the simulation time was 121 s. Note that the use of dynamic reactive power reserves was reduced to more than one-third of the base case; in addition, the results followed the voltage profiles successfully.

The simulation times obtained with the objective function given by Equation (1) for the IEEE 57-bus and IEEE 118-bus test systems were 17.5 s and 100 s, respectively. Additionally, an experiment was performed in which the whole set of nodes was included in the optimization problem. In this case, the objective voltage profiles of the nodes correspond to the values previously defined in each control area and are represented by the pilot nodes. The computation times in this experiment were 128 s and 239 s for the IEEE 57-bus and IEEE 118-bus, respectively. This shows that selecting a reduced set of nodes (pilot nodes) improves the computational time of the algorithm.

4. Conclusions

This paper presented a novel approach to the MP-ORPD problem. A new formulation considering three operative goals was presented and implemented. The first goal under consideration was to maintain adequate voltage profiles in all power systems using a reduced set of buses (pilot nodes), whereby the values of the voltage set points were adjusted dynamically, changing every period according to load conditions. The second goal was to avoid excessive maneuvers on shunt devices, since a large number of these lead to lower life expectancy for the devices and more maintenance under real-life circumstances. The third goal was to maintain the dynamic reactive power reserves of the generators, guaranteeing a fast response to control voltages during the contingency scenario. The main feature of the proposed MP-ORPD model lies in its applicability in real power systems, since it does not follow conventional objective functions, such as power loss reduction; instead, it guarantees operative feasibility throughout a given time horizon, while minimizing the number of maneuvers on reactive power devices. Several tests carried out on the IEEE 57-bus and 118-bus test systems proved the effectiveness and applicability of the proposed approach. It was found that the weighting method allows the priority of the objectives to be adjusted, providing flexibility to meet the needs of each system under analysis. The decision to choose a set of pilot nodes significantly improved the computation times with respect to the optimization of the system nodes. The computation times for the optimization process were 17.5 and 100 s for IEEE 57-bus and 118-bus test systems, respectively, showing that the model can be used in coordinated voltage control strategies for day-ahead and very short-term operation planning. Although dynamic reactive power reserves are considered, voltage stability issues are not explicitly integrated in the model. This could be a topic of future research, along with the implementation of a multi-area MP-ORPD model and parallelization process in order to reduce computation times.