A Two-Stage Algorithm for the Design of Wide-Area Damping Controllers

Abstract

Low-frequency oscillation modes are studied in small-signal angular stability because, if not adequately damped, they can cause power system instability in the event of a contingency. The interconnection and expansion of large power systems has led to the emergence of multiple local and inter-area modes and required new damping control strategies for these modes. The expansion of the use of Phasor Measurement Units in power systems has led to the development of new control strategies such as Wide-Area Damping Controllers (WADCs) that use data from PMUs to dampen low-frequency oscillations. Although the benefits of WADCs are promising, there are challenges in designing a WADC. This paper proposes a two-stage algorithm for the robust design of a WADC for modern power systems. The first stage consists of solving an optimization model and finding the WADC parameters that maximize the damping ratios of all modes of the linearized system model for a set of operating points. The second stage consists of refining the WADC parameters through an iterative algorithm. Cases were studied for a set of IEEE 68-bus operating points through modal analysis and time-domain simulations. The results obtained demonstrated the good performance of the proposed two-stage algorithm compared with an existing WADC design method based on a Linear Quadratic Regulator.

1. Introduction

Power systems often undergo modifications to meet consumer demand with adequate levels of quality and performance [1]. The increase in demand has required the construction of new generation sources to meet the power balance, and in recent years, the participation of wind and photovoltaic generation sources has grown and has been a concern for operating centers due to the intermittency of power generation [2]. The interconnection of large systems has exposed the system to faults, such as three-phase, two-phase, and single-phase faults. In recent years, a high number of blackouts have occurred in different countries and have been worrying operation centers due to the difficulty of operating large power systems [3]. In addition, the interconnection of large systems has in recent years caused the emergence of multiple oscillation modes related to the electromechanical quantities of power systems and presented low damping rates that can compromise the stability of power systems [4].

Advances in power systems have also provided benefits for the operation of modern power systems. One such benefit has been the development, implementation and improvement of Wide-Area Measurement Systems (WAMSs) that have Phasor Measurement Units (PMUs) as their main components [5]. PMUs collect voltage and current phasor data from any part of the power system where they are installed with high sampling rates and time synchronization using GPS. This sampled data is sent to the Phasor Data Concentrator and can be used in different applications [6,7,8,9]. The fact that the operation center has a picture of the current dynamics of the system with several measurements allows the application of several effective control measures when necessary [10].

Over the years, a number of researchers have proposed several useful applications in modern power systems equipped with PMUs. We can currently find applications in monitoring [11], control [12] and protection [13] of systems. In [14], the authors present an approach based on principal component analysis and PMU measurements to monitor a power system. In [15], the authors propose to apply a control strategy using data from PMUs in battery energy storage systems. In [16], the authors proposed the use of data from PMUs and the SCADA system for a correct estimation of states in modern power systems. In [17], the authors proposed a method using PMU data for the detection, classification and identification of faults in distribution systems.

Small-signal angular stability studies must be conducted in power systems to evaluate mechanisms that can induce system instability [18]. One of these studies is to ensure that low-frequency oscillation modes, usually between 0.1 and 2 Hz and associated with electromechanical quantities, have high damping ratios which must be identified appropriately [19]. Current power systems present multiple oscillation modes that require damping control strategies to increase the damping ratios of the system. The most effective control strategy accepted in the community is the design and installation of a Power System Stabilizer (PSS) in the local control loop where the Automatic Voltage Regulator (AVR) of the synchronous generator is located [20]. In recent times, different PSS design techniques have been developed, such as Linear Matrix Inequality (LMI) [21], Linear Quadratic Regulator (LQR) [22], metaheuristics such as Genetic Algorithm (GA) [23], Bat Algorithm (BA) [24], Particle Swarm Optimization (PSO) [25], and Gravitational Search Algorithm (GSA) [26]. Although PSSs are effective for local modes in the frequency range between 0.8 and 2 Hz with good damping ratio results, PSSs have limited effect on inter-area oscillation modes in the frequency range between 0.1 and 0.8 Hz with not very high damping ratios [27]. The expansion of power systems and the interconnection of large systems have caused an increase in multiple inter-area modes with low damping ratios, and whose PSSs have difficulty in operating because they do not present adequate system observability.

The advancement and expansion of PMU installations provide greater system observability and have stimulated researchers to develop new control strategies such as Wide-Area Damping Controllers (WADCs). WADCs can receive signals from remote locations and send control signals to the AVRs of synchronous generators at remote locations. Different techniques have been presented for the design of a WADC such as techniques based on LMIs [28], LQR [29], Reinforcement Learning Technique [30,31,32,33,34,35], Deep Neural Networks (DNNs) and Reinforcement Learning [36], Fuzzy theory [37], Classification and Regression Trees [38] and metaheuristics such as GA [39,40], PSO [41,42,43], Firefly Algorithm [44], Grey Wolf Optimizer [45], BA [46], Greylag Goose Optimization [47]. In recent years, there have been proposals for designing a WADC based on adaptive control to deal with system operating uncertainties [48,49,50,51,52,53,54]. Thus, there is a relevant set of different techniques for a good WADC design. These methods show different performance for the different dynamic characteristics of the power system. Although the results achieved so far are satisfactory, challenges remain in assessing the adequate design of a WADC.

The main challenges in a successful design of a WADC to properly dampen inter-area modes are (i) the choice of the WADC input and output signals, (ii) the appropriate model to represent the time delays for sending and receiving PMU data, and (iii) the appropriate control strategy to dampen the oscillation modes. In the literature different techniques for the selection of WADC signals have been presented, such as geometric measures [55], heuristic methods [56], and adaptive signal selection [57]. The results showed that these techniques are effective in choosing the best WADC signals for their control objectives. Regarding time delays in PMU data transmission, there are different models such as fixed time delay [29,58], time delay margin [59], time delay scheduling [60], and time delay compensation [61,62,63]. These different models presented promising results, and the control objectives were achieved with good values.

Thus, the design of a WADC presents differences in relation to the design of traditional local damping controllers of the PSS type, although the objective is the same: to obtain high damping ratios for local and inter-area modes of power systems. There are many proposed methods, but most of them present difficulties in achieving high damping ratios because the methods reach local optima, especially methods based on metaheuristics. Many metaheuristics, such as Particle Swarm Optimization, Firefly Algorithm, Greylag Goose Optimization, Bat Algorithm, Grey Wolf Optimizer, and Gravitational Search Algorithm, face the problem of converging to a global minimum depending on the optimization problem. Thus, strategies are needed to achieve high damping ratios and avoid local optima.

This paper proposes a two-stage algorithm for the robust design of a WADC for modern power systems. The first stage consists of solving an optimization model and finding the WADC parameters that maximize the damping ratios of all modes of the linearized system model for a set of operating points. The second stage consists of refining the WADC parameters through an iterative algorithm. Thus, the first stage consists of finding promising solutions for the control objective and the second stage aims to improve the control objectives through a refinement process. Cases are studied for a set of IEEE 68-bus operating points through modal analysis and time-domain simulations. The WADC design is carried out using linearized models and modal analysis techniques in MATLAB software [64]. The designed WADC is implemented in ANATEM software [65], and its performance is evaluated through simulations in ANATEM software. Statistical analyses are presented to demonstrate the benefits of a two-stage control strategy. The results obtained demonstrate the good performance of the proposed two-stage algorithm.

This paper is organized as follows. Section 2 describes the operating principle of the two-level control structure composed of PSSs at the first level and the WADC at the second level. Section 3 presents the system model, the time delay model, the WADC model and the closed-loop control system model. Section 4 presents the proposed two-stage algorithm in detail and through a step-by-step algorithm. Section 5 conducts case studies for the IEEE 68-bus system where results and discussions are presented. Section 6 presents the conclusions of this research with the main arguments and future research lines.

2. Two-Level Control Structure

Figure 1 presents the two-level control structure operating on a power system composed of a set of PMUs. This control structure comprises a local control and a central control, totaling two control levels, with each control sending a signal to the AVR comparator. In local control, the PSSs of each synchronous generator are responsible for sending a control signal to the AVR comparator itself where the PSS is installed. In central control, the WADC receives time-delayed speed signals from the system via PMU data and sends time-delayed control signals to the comparators of some of the generators’ AVRs. In this proposal for the control design, the parameters of the power system, AVRs and PSSs will be fixed, and the objective is to design the WADC using the proposed two-stage algorithm.

3. Modeling

The design of WADC-type damping controllers will be carried out through linearized models of the power system, the time delay, and the WADC itself. Successful WADC design will require that all eigenvalues of the closed-loop control system obtained by linearized models have high damping ratios greater than 5%. The use of linearized models in control design is very common due to their ease of design. There is a relevant group of researchers who use linearized power system models to design a WADC and the results are promising [28,40,41,44,48,49,50,51,63,66,67,68]. In addition, performance evaluations will be carried out for the closed-loop control system. Section 3.1 describes the system model with the time delay model. Section 3.2 presents the WADC model. Section 3.3 presents the closed-loop control system model.

3.1. Power System Model with Time Delays

Power systems have components such as synchronous and asynchronous generators, direct current and alternating current transmission lines, three-phase transformers, capacitor banks, three-phase motors, and different types of loads. Thus, modern power systems exhibit dynamic behavior and must be described by a set of differential–algebraic equations. Due to the nonlinearity of these equations and the difficulty of analysis, it is common to linearize these equations around an equilibrium condition and obtain the following state-space model [20]:

$$\dot{x}=Ax+Bu$$

(1)

$$y=Cx$$

(2)

where matrices A, B and C are commonly referred to as state, input and output matrices, respectively. The vector of state variables is x, the vector of input variables is u and the vector of output variables is y.

Time delays in data transmission from PMUs must be factored into the control design because data is sent from distinct and remote locations. The time delay was formulated mathematically with a second-order Padé approximation given by [29]

$$G\left(s\right)={\displaystyle \frac{6-2Ts}{6+4Ts+T^2{s}^2}}$$

(3)

where T is the delay time in the transmission of the WADC input and output signals. This model stimulates a maximum allowed time delay in the communication channels and, thus, if the data transmission delays are variable but respect this maximum limit, we will not have problems in the WADC operation. Although the time delay value is fixed, this value is actually an upper bound on the time delay. As can be seen in Figure 1, as data are sent, the “Time Delay” block acts as a buffer that holds the data until a fixed time limit. Thus, if data arrive before this time, the buffer holds the data until they are actually used by the WADC controller. If the data exceed this time limit, the signal is considered lost.

This transfer function (3) can also be reformulated as a set of state-space equations from the observable canonical representation of Jordan, and thus we can work with the matrices of the linearized model. The WADC receives signals from PMUs in remote regions and sends control signals to remote regions. Thus, we have a state-space model of the time delay for the input of the WADC given by

$${\dot{x}}_{di}=A_{di}{x}_{di}+B_{di}{u}_{di}$$

(4)

$$y_{di}=C_{di}{x}_{di}$$

(5)

where the subindex $di$ represents a time delay model of the WADC input and for the output of the WADC given by

$${\dot{x}}_{do}=A_{do}{x}_{do}+B_{do}{u}_{do}$$

(6)

$$y_{do}=C_{do}{x}_{do}$$

(7)

where the subindex $do$ represents a time delay model of the WADC output.

Time delays are variable, but we can work with a maximum allowable time delay and set this value as T and use the above model. Thus, by setting this value T and having knowledge of the operating points and the power system model, we can work with a single state-space model that includes the system model and the time delay model. This single model is given by

$$\dot{\overline{x}}=\overline{A}\overline{x}+\overline{B}\overline{u}$$

(8)

$$\overline{y}=\overline{C}\overline{x}$$

(9)

and the matrices $\overline{A}$, $\overline{B}$ and $\overline{C}$ are

$$\overline{A}=\left[\begin{array}{ccc}A& B{C}_{di}& 0\\ 0& A_{di}& 0\\ B_{do}C& 0& A_{do}\end{array}\right]$$

(10)

$$\overline{B}=\left[\begin{array}{c}0\\ B_{di}\\ 0\end{array}\right]$$

(11)

$$\overline{C}=\left[\begin{array}{ccc}0& 0& C_{do}\end{array}\right]$$

(12)

In the control design of this research, the matrices $\overline{A}$, $\overline{B}$ and $\overline{C}$ of (8) and (9) will be known and evaluated so that the WADC is able to provide good performance for the closed-loop system.

3.2. WADC Model

The purpose of the WADC is to achieve high damping ratios for the closed-loop system, especially for inter-area modes where PSSs have difficulty and limited effects. WADC is advantageous because it works with signals from PMUs of the system as a whole and thus has high system observability. PMU data can be used to estimate system states and also to obtain estimates of the speed signals of the system generators that will be the input signals to the WADC. The number of PMUs in the system is important to ensure complete system observability and to facilitate state estimation. The focus of this research is not to determine the number and location of PMUs in the system. It is assumed that this number of PMUs is known and sufficient for a correct determination of the speed signals of the generators. To ensure the benefits of good system observability, the WADC worked on in this research will be a controller with multiple inputs and multiple outputs and can be mathematically formulated as a matrix of transfer functions.

$$W\left(s\right)=\left[w_{k,m}\left(s\right)\right]=\left[\begin{array}{cccc}{w}_{1,1}\left(s\right)& w_{1,2}\left(s\right)& \cdots & w_{1,p}\left(s\right)\\ w_{2,1}\left(s\right)& w_{2,2}\left(s\right)& \cdots & w_{2,p}\left(s\right)\\ ⋮& ⋮& \ddots & ⋮\\ w_{p,1}\left(s\right)& w_{p,2}\left(s\right)& \cdots & w_{p,p}\left(s\right)\end{array}\right]$$

(13)

where each of the above elements $w_{k,m}$, $k=1,...,p$, $m=1,...,p$ can be represented as

$$w_{k,m}\left(s\right)=K_{k,m}·{\displaystyle \frac{T{1}_{k,m}s+1}{T{2}_{k,m}s+1}}·{\displaystyle \frac{T{3}_{k,m}s+1}{T{4}_{k,m}s+1}}$$

(14)

where $T{1}_{k,m}$, $T{2}_{k,m}$, $T{3}_{k,m}$, $T{4}_{k,m}$ are time constants between 0 and 1, and $K_{k,m}$ is a gain that must respect the minimum $\left(K_{min}\right)$ and maximum limits $\left(K_{max}\right)$. In this research, we are interested in determining the values of $K_{k,m}$, $T{1}_{k,m}$, $T{2}_{k,m}$, $T{3}_{k,m}$, $T{4}_{k,m}$ that achieve certain control objectives.

The transfer function matrix $W\left(s\right)$ in (13) will also be formulated in state space and presents the following model:

$${\dot{x}}_c=A_c{x}_c+B_c{u}_c$$

(15)

$$y_c=C_c{x}_c+D_c{u}_c$$

(16)

where the matrices $A_c$, $B_c$, $C_c$ and $D_c$ can be obtained through the Jordan representation [20].

3.3. Closed-Loop Control System

The design of the WADC will respect certain performance criteria for the closed-loop control system and thus the following model is formulated:

$$\dot{\widehat{x}}=\widehat{A}\widehat{x}$$

(17)

where $\widehat{x}={\left[\begin{array}{cc}{\overline{x}}^T& x_c^T\end{array}\right]}^T$. This model includes the power system model, the time delay model and the WADC model, and matrix $\widehat{A}$ is

$$\widehat{A}=\left[\begin{array}{cc}\overline{A}+\overline{B}{D}_c\overline{C}& \overline{B}{C}_c\\ B_c\overline{C}& A_c\end{array}\right]$$

(18)

One of the system stability requirements is to ensure that all eigenvalues have a negative real part. However, in evaluating the small-signal stability of power systems, it is common for all eigenvalues to also have a minimum damping ratio to ensure well-damped angular responses [20]. The higher the value of this ratio, the better the degree of small-signal stability of the system. In practical terms, a 5% damping ratio is acceptable and the goal of many control designs [20]. The small-signal stability margin then refers to the lowest damping ratio value found for the system’s oscillation modes.

4. Proposed Method

The proposed algorithm for the design of a WADC has two stages and both must solve an optimization model. Each stage presents distinct operators. The first stage consists of revolving an optimization model to find regions that provide good damping ratios. This stage involves applying a bio-inspired algorithm technique called Particle Swarm Optimization (PSO). The second stage involves applying a heuristic and refining the values to achieve high damping ratio values. Section 4.1 describes the optimization model that must be solved for the design of the WADC. Section 4.2 presents the details of the first stage of the proposed algorithm while Section 4.3 presents the details of the second stage. Section 4.4 presents the step-by-step algorithm of the proposed method composed of two stages. Section 4.5 describes the flowchart steps of the proposed method for determining the WADC parameters.

4.1. Optimization Model

The optimization model applied in this research for the design of WADCs is formulated in (19). The vector of variables $\left(x\right)$ of this model consists of the WADC parameters $K_{k,m}$, $T{1}_{k,m}$ and $T{3}_{k,m}$, for $k=1,...,p$ and $m=1,...,p$. The time constants $T{2}_{k,m}$ and $T{4}_{k,m}$ associated with the poles of each of the WADC transfer functions will be fixed to facilitate the design. The objective function $(f_{obj}())$ informs the lowest damping ratio $\left({\zeta }_{min}\right)$ among all the operating points for the candidate to solve the problem. The WADC parameters are subject to minimum and maximum values in this problem.

$$\begin{array}{cc}\mathrm{Find}\hfill & x=\left[\begin{array}{ccc}{K}_{k,m}& T{1}_{k,m}& T{3}_{k,m}\end{array}\right]\hfill \\ \mathrm{Maximize}\hfill & f_{obj}\left(x\right)={\zeta }_{min}\hfill \\ \mathrm{Subject}\mathrm{to}\hfill & K_{min}<K_{k,m}<K_{max}\hfill \\ \hfill & 0<T{1}_{k,m}<T_{max}\hfill \\ \hfill & 0<T{3}_{k,m}<T_{max}\hfill \end{array}$$

(19)

for $k=1,...,p$ and $m=1,...,p$.

4.2. First Stage

PSO is a bio-inspired algorithm that aims to imitate the behavior of a flock of birds in nature [69]. It is an algorithm widely used in solving optimization problems and has presented several improvements over the years [70,71,72,73,74]. PSO is also widely applied in control projects due to its ability to find control parameter values with good objective function values.

To apply PSO, we must first define the population size $\left(N_p\right)$, the size of the variable vector $\left(N_d\right)$, the minimum $\left(L_{min}\right)$ and maximum $\left(L_{max}\right)$ limits of each variable, and the number of epochs $\left(N_e\right)$ of the algorithm. Initially, the particle vector x and the velocity vector v have their initial values randomly chosen, respecting the required limits. At each iteration of PSO, the velocity vector and position vector are updated according to the following mathematical formulations:

$$v_{d,t+1}^p=\omega ·v_{d,t}^p+c_1·rand·(x_{d,L}^p-x_{d,t})+c_2·rand·(x_{d,G}-x_{d,t})$$

(20)

$$x_{d,t+1}^p=x_{d,t}^p+v_{d,t+1}^p$$

(21)

where $p=1,...,N_p$, $d=1,...,N_d$, $t=1,...,N_e$, $\omega$ is a parameter usually with a value of 0.7, $c_1$ and $c_2$ are two parameters usually between 1.5 and 2, $rand$ is a function that returns a random value from a uniform distribution between 0 and 1, $x_{d,G}$ is the position with the best objective function among all particles among all iterations so far, and $x_{d,L}^p$ is the best position of each particle so far.

At each iteration, the vectors of positions $\left(x\right)$ and velocities $\left(v\right)$ are updated according to Formulations (20) and (21) up to the limit of the number of epochs $\left(N_e\right)$. At the end of the number of epochs, the vector $x_{d,G}$, $d=1,...,N_d$ is the final solution of the problem. Although PSO is widely applied in different optimization problems, it tends to stagnate in solutions far from the optimum. Thus, a second stage was developed to improve this solution, and this second stage is described in the following section.

4.3. Second Stage

This second stage aims to improve the solution from the first stage. The algorithm of this second stage is a heuristic with an elitist characteristic where only the best solution goes to the next iteration. The algorithm starts with the best position from the first stage ($x^p=x_G$, $p=1,...,N_p$) and at each iteration the vector is updated according to the following formulation:

$$x_d^p=x_d^p·fator$$

(22)

where $d=1,...,N_d$, and $factor$ is a random value between 0.7 and 1.3 chosen at each iteration for a single position d of the vector $x_{d,G}$. Thus, since the factor is a value between 0.7 and 1.3, this second stage aims to change the value of each position of the vector by small amounts from the optimal value found so far. This second stage can be interpreted as a small refinement of the best value of the variable vector from the first stage.

4.4. Two-Stage Algorithm

The step-by-step algorithm of the proposed method composed of two stages is given by

• First Stage
  Step 1: Define $N_p$, $N_d$, $L_{min}$, $L_{max}$, $N_e$.
  Step 2: Initialize the vectors x and v with random values respecting the limits.
  Step 3: Set $t=1$.
  Step 4: Calculate

  $$v_{d,t+1}^p=\omega ·v_{d,t}^p+c_1·rand·(x_{d,L}^p-x_{d,t})+c_2·rand·(x_{d,G}-x_{d,t})$$

  (23)

  $$x_{d,t+1}^p=x_{d,t}^p+v_{d,t+1}^p$$

  (24)

  for $p=1,...,N_p$, $d=1,...,N_d$.
  Step 5: Calculate the objective function $(f_{obj}())$ for each position $\left(x_{t+1}^p\right)$ and update the vectors $x_L^p$ and $x_G$ if the solutions present better objective function values.
  Step 6: If $t=N_e$, go to Step 7; otherwise, $t=t+1$ and go to Step 3.
  Step 7: The solution for this first stage is $x_G$.
• Second Stage
  Step 1: Set $t=1$.
  Step 2: All $N_p$ particles will be initialized with the value $x_G$

  $$x^p=x_G$$

  (25)

  for $p=1,...,N_p$.
  Step 3: For each particle p $(p=1,...,N_p)$ and a single value of d $(d=1,...,N_d)$, calculate the new position value according to the formulation

  $$x_{d,t}^p=x_{d,t}^p·fator$$

  (26)

  for $p=1,...,N_p$ and $factor$ is a random value from a uniform distribution between 0.7 and 1.3.
  Step 4: Calculate the objective function $(f_{obj}())$ for each position $\left(x_t^p\right)$, and $x_G$ is the solution with the best objective function.
  Step 5: If $t=N_e$, go to Step 6; otherwise, $t=t+1$ and go to Step 2.
  Step 6: The solution for this second stage is $x_G$.

In this research, the WADC parameters $K_{k,m}$, $T{1}_{k,m}$ and $T{3}_{k,m}$ are the design variables and will thus be the position vector x.

The two-stage algorithm for the WADC design was implemented in MatLab software version R2016a [64]. Dynamic simulations of the system with the integrated WADC were performed in ANATEM software version 12.5.1 [65], a commercial-grade software available in Brazil to evaluate electromechanical transient stability studies of power systems.

4.5. Flowchart of the Proposed Method

Figure 2 describes the flowchart of the proposed method to determine the WADC parameters and can further be described by the following step-by-step algorithm:

• Step 1: Define the test system with all its components and obtain the complete and detailed differential–algebraic equations (DAEs) for all the system operating points.
• Step 2: Linearize the DAEs for each operating point and obtain the state, output and input matrices.
• Step 3: Apply modal analysis techniques and identify local and inter-area modes with the lowest damping ratios.
• Step 4: Identify the best input and output signals through geometric measures of observability and controllability, respectively.
• Step 5: Define the parameters of the proposed algorithm composed of two stages.
• Step 6: Execute the first stage of the proposed algorithm until it reaches the maximum number of epochs, and the best solution will go to the next stage.
• Step 7: Execute the second stage of the proposed algorithm until it reaches the maximum number of epochs.
• Step 8: The result of the convergence of the previous step is the WADC parameters that can be implemented in the test system to achieve high damping ratios.

5. Case Studies and Discussion

The evaluation of the two-stage algorithm was performed on the IEEE 68-bus system, a large system for small-signal stability studies available in [75]. The single-line diagram of the IEEE 68-bus power system is available from reference [75]. The complete and detailed equation for this test system is present in the Technical Report PES-TR18 of the Power & Energy Society [76]. It is a smart power system of 68 buses and 16 generators where only generators 1 to 12 are equipped with well-designed AVRs. The case studies considered the nominal operating point (BC0) available at [76] and four other operating points: case BC1 with a 1% increase in active load power, case BC2 with a 2% increase in active load power, and case BC3 with a 3% increase in active load power.

Table 1. Oscillation modes with lower damping ratios for each operating case.

Case	Modes (Eigenvalues)	Frequency [Hz]	Damping Ratio [%]
BC0	$-0.16568\pm j4.89173$	0.77854	3.38505
	$-0.11844\pm j3.26647$	0.51988	3.62364
BC1	$-0.10587\pm j3.23828$	0.51539	3.26759
	$-0.16502\pm j4.88944$	0.77818	3.37304
BC2	$-0.09268\pm j3.20363$	0.50987	2.89186
	$-0.16432\pm j4.88547$	0.77755	3.36151
BC3	$-0.07966\pm j3.16135$	0.50314	2.51887
	$-0.16357\pm j4.87966$	0.77662	3.35016

gives the oscillation modes (eigenvalues), frequencies, and damping ratios for cases BC0, BC1, BC2, and BC3. In all these four operating cases, the damping rates are less than 5% for two conjugate pairs of eigenvalues (oscillation modes). All oscillation modes are low-frequency and are in the range between 0.1 and 2 Hz.

The first step in the successful design of a WADC consists of choosing the appropriate input and output signals for the WADC design. This test system presents 16 generators whose speed signals could be input signals to the WADC. However, only generators 1 to 12 are equipped with AVRs and could receive control signals sent by the WADC. Applying geometric controllability measures on the modes of cases BC0, BC1, BC2 and BC3 reported in Table 1, generators 5, 9, 10, 11 and 12 present the highest values, indicating that they are the best options to receive the WADC control signals in their respective AVRs for these operation cases. Applying geometric observability measures to the modes of cases BC0, BC1, BC2 and BC3 reported in Table 1, generators 12, 13, 14, 15 and 16 present the highest values, indicating that they are the best options to be WADC inputs for these operation cases. Thus, the WADC to be designed is described in (27) and is composed of a set of transfer functions. The control action is described in (28), where $\Delta {\omega }_{12}$, $\Delta {\omega }_{13}$, $\Delta {\omega }_{14}$, $\Delta {\omega }_{15}$ and $\Delta {\omega }_{16}$ represent the speed signals estimated by PMUs and $V{C}_5$, $V{C}_9$, $V{C}_{10}$, $V{C}_{11}$ and $V{C}_{12}$ represent the control signals provided by the WADC that will go to the AVR of the synchronous generators.

$$W\left(s\right)=\left[w_{k,m}\left(s\right)\right]=\left[\begin{array}{ccccc}{w}_{5,12}\left(s\right)& w_{5,13}\left(s\right)& w_{5,14}\left(s\right)& w_{5,15}\left(s\right)& w_{5,16}\left(s\right)\\ w_{9,12}\left(s\right)& w_{9,13}\left(s\right)& w_{9,14}\left(s\right)& w_{9,15}\left(s\right)& w_{9,16}\left(s\right)\\ w_{10,12}\left(s\right)& w_{10,13}\left(s\right)& w_{10,14}\left(s\right)& w_{10,15}\left(s\right)& w_{10,16}\left(s\right)\\ w_{11,12}\left(s\right)& w_{11,13}\left(s\right)& w_{11,14}\left(s\right)& w_{11,15}\left(s\right)& w_{11,16}\left(s\right)\\ w_{12,12}\left(s\right)& w_{12,13}\left(s\right)& w_{12,14}\left(s\right)& w_{12,15}\left(s\right)& w_{12,16}\left(s\right)\end{array}\right]$$

(27)

$$\left[\begin{array}{c}V{C}_5\\ V{C}_9\\ V{C}_{10}\\ V{C}_{11}\\ V{C}_{12}\end{array}\right]=\left[\begin{array}{ccccc}{w}_{5,12}\left(s\right)& w_{5,13}\left(s\right)& w_{5,14}\left(s\right)& w_{5,15}\left(s\right)& w_{5,16}\left(s\right)\\ w_{9,12}\left(s\right)& w_{9,13}\left(s\right)& w_{9,14}\left(s\right)& w_{9,15}\left(s\right)& w_{9,16}\left(s\right)\\ w_{10,12}\left(s\right)& w_{10,13}\left(s\right)& w_{10,14}\left(s\right)& w_{10,15}\left(s\right)& w_{10,16}\left(s\right)\\ w_{11,12}\left(s\right)& w_{11,13}\left(s\right)& w_{11,14}\left(s\right)& w_{11,15}\left(s\right)& w_{11,16}\left(s\right)\\ w_{12,12}\left(s\right)& w_{12,13}\left(s\right)& w_{12,14}\left(s\right)& w_{12,15}\left(s\right)& w_{12,16}\left(s\right)\end{array}\right]·\left[\begin{array}{c}\Delta {\omega }_{12}\\ \Delta {\omega }_{13}\\ \Delta {\omega }_{14}\\ \Delta {\omega }_{15}\\ \Delta {\omega }_{16}\end{array}\right]$$

(28)

The next step in the WADC design is the application of the proposed two-stage algorithm. The two-stage algorithm for the WADC design was implemented in MATLAB software version R2016a [64]. Dynamic simulations of the system with the integrated WADC were performed in ANATEM software version 12.5.1 [65], a commercial-grade software available in Brazil to evaluate electromechanical transient stability studies of power systems. The parameter values of the first-stage algorithm were chosen as $N_p=20$, $N_e=200$, $c_1=1.5$, $c_2=2$, $\omega =0.7$, $K_{min}=-20$, $K_{max}=20$, and $T=0.100$ ms. The parameter values of the second-stage algorithm were chosen as $N_p=20$ and $N_e=200$. The time constants $T{2}_{k,m}$ and $T{4}_{k,m}$ of the WADC were fixed at 0.04 and associated with the poles of the WADC, that is, $T{2}_{k,m}=0.04$ and $T{4}_{k,m}=0.04$. The two stages of the proposed algorithm present heuristic operations and the initial conditions of the variables are random. Thus, each execution of the proposed two-stage algorithm may present different results. Since the convergence results may be different for each execution of the proposed two-stage algorithm, the proposed algorithm was executed 200 times, resulting in 200 WADC designs, and the results were evaluated by the minimum, average, maximum, and standard deviation values in

Table 2. Objective function results for 200 simulations of the proposed two-stage algorithm.

Stage	Minimum Damping Ratio [%]	Average Damping Ratio [%]	Maximum Damping Ratio [%]	Standard Deviation
First Stage	3.6233	5.3396	6.9965	0.0084
Second Stage	8.5239	9.7255	12.0206	0.0077

.

Table 3. Execution times for 200 simulations of the proposed two-stage algorithm.

Stage	Minimum Time [s]	Average Time [s]	Maximum Time [s]	Standard Deviation
First Stage	1800.9268	1899.5448	1999.2269	58.8396
Second Stage	1672.9270	1762.8701	1858.8262	53.4117

presents the minimum, average, maximum and standard deviation times of each execution of the proposed algorithm for each stage.

These detailed results of the two-stage algorithms provide the following discussions:

• The first stage provided a WADC whose closed-loop control system modes have damping rates between 3.6233% and 6.9965%. The second stage provided a WADC whose closed-loop control system modes have damping rates between 8.5239% and 12.0206%.
• The second stage provided high damping ratio values for all cases, indicating that a refinement of the results from the first stage is beneficial in the successful design of the WADC.
• The average execution time of the first stage of the two-stage algorithm was 1899.5448 s while the second stage had an average execution time of 1762.8701 s.

A comparative analysis was performed with the LQR-based method for WADC design proposed in [29]. The LQR-based method guarantees optimality only for the nominal operating point and presents difficulties with multiple operating points. Furthermore, LQR-based techniques do not guarantee high damping ratio values, as the goal of this technique is to reduce control effort. The WADC parameters after the convergence of the LQR-based method are presented in

Table 4. WADC parameters of the LQR method (WADC(LQR)).

${\mathit{w}}_{\mathit{k},\mathit{m}}$	num (s)	den (s)
$w_{5,12}$	$-14.105s^2-439.16s-2599.5$	$s^2+50s+625$
$w_{5,13}$	$1.28s^2+171.43s+3744.2$	$s^2+50s+625$
$w_{5,14}$	$149.17s^2+1381.7s+3149.8$	$s^2+50s+625$
$w_{5,15}$	$24.798s^2+1518.7s+6248.4$	$s^2+50s+625$
$w_{5,16}$	$61.748s^2+1103.4s+2623.5$	$s^2+50s+625$
$w_{9,12}$	$119.27s^2+1490.1s+4653$	$s^2+50s+625$
$w_{9,13}$	$-64.625s^2-1504.6s-6825.9$	$s^2+50s+625$
$w_{9,14}$	$532.42s^2+5086.1s+9251.6$	$s^2+50s+625$
$w_{9,15}$	$51.555s^2+4255.2s+9357.3$	$s^2+50s+625$
$w_{9,16}$	$855.9s^2+3743.6s+4071.2$	$s^2+50s+625$
$w_{10,12}$	$23.252s^2+810.83s+6636.8$	$s^2+50s+625$
$w_{10,13}$	$1073.9s^2+4726.4s+5189.8$	$s^2+50s+625$
$w_{10,14}$	$-154s^2-867.55s-1218.4$	$s^2+50s+625$
$w_{10,15}$	$245.09s^2+3206.2s+7780.8$	$s^2+50s+625$
$w_{10,16}$	$-169.08s^2-4532.3s-9291.5$	$s^2+50s+625$
$w_{11,12}$	$-1138.8s^2-6419.2s-8851.2$	$s^2+50s+625$
$w_{11,13}$	$195.16s^2+1679.4s+2881.3$	$s^2+50s+625$
$w_{11,14}$	$172.09s^2+2320.7s+6469$	$s^2+50s+625$
$w_{11,15}$	$-0.409s^2-28.765s-393.75$	$s^2+50s+625$
$w_{11,16}$	$2162.7s^2+8966.7s+9285.3$	$s^2+50s+625$
$w_{12,12}$	$-803.16s^2-5811.9s-9239.1$	$s^2+50s+625$
$w_{12,13}$	$-7.3114s^2-427.01s-2240.4$	$s^2+50s+625$
$w_{12,14}$	$1558.8s^2+7624.8s+9183.8$	$s^2+50s+625$
$w_{12,15}$	$-313.07s^2-4507s-8638.5$	$s^2+50s+625$
$w_{12,16}$	$888.52s^2+4233.1s+5038.8$	$s^2+50s+625$

and the oscillation modes with the lowest damping ratios for each of the operating points are presented in

Table 5. Oscillation modes with lower damping ratios for each operating case for WADC(LQR).

Case	Modes (Eigenvalues)	Frequency [Hz]	Damping Ratio [%]
BC0	$-0.2493\pm j4.9397$	0.7862	5.0409
BC1	$-0.2566\pm j4.9460$	0.7872	5.1807
BC2	$-0.2841\pm j5.5330$	0.8806	5.1279
BC3	$-0.2795\pm j5.5364$	0.8811	5.0423

. The design of the WADC controller was conducted using linearized models, but power systems are nonlinear dynamic systems. Thus, the WADC needs to be evaluated in the nonlinear dynamic model of the system. The WADC that provided the highest damping ratio in the second stage of the proposed two-stage algorithm was chosen for the time-domain analysis. The parameters of these WADCs of the first and second stages are described in

Table 6. WADC parameters of the first stage (WADC1).

${\mathit{w}}_{\mathit{k},\mathit{m}}$	${\mathit{K}}_{\mathit{k},\mathit{m}}$	$\mathit{T}{1}_{\mathit{k},\mathit{m}}$	$\mathit{T}{3}_{\mathit{k},\mathit{m}}$
$w_{5,12}$	7.2091	0.2175	0.4671
$w_{5,13}$	2.5528	0.8775	0.9683
$w_{5,14}$	9.1121	0.6137	0.6599
$w_{5,15}$	2.3351	0.0522	0.1226
$w_{5,16}$	8.3029	0.0297	0.0224
$w_{9,12}$	3.5462	0.0907	0.9222
$w_{9,13}$	14.8500	0.0236	0.0025
$w_{9,14}$	19.1232	0.9772	0.1645
$w_{9,15}$	19.5556	0.1049	0.7173
$w_{9,16}$	9.7557	0.4802	0.1814
$w_{10,12}$	15.8200	0.7322	0.0165
$w_{10,13}$	12.0232	0.6555	0.4052
$w_{10,14}$	12.6001	0.8601	0.8961
$w_{10,15}$	15.9413	0.6434	0.0925
$w_{10,16}$	12.2471	0.3667	0.4601
$w_{11,12}$	13.9584	0.5771	0.1596
$w_{11,13}$	15.3611	0.6937	0.7519
$w_{11,14}$	3.6694	0.0260	0.2226
$w_{11,15}$	0.7514	0.4078	0.0267
$w_{11,16}$	19.1326	0.6434	0.6513
$w_{12,12}$	13.6843	0.2888	0.3188
$w_{12,13}$	12.1438	0.0788	0.7430
$w_{12,14}$	19.3695	0.4765	0.0445
$w_{12,15}$	1.1856	0.6842	0.5200
$w_{12,16}$	10.8726	0.1856	0.2671

and

Table 7. WADC parameters of the second stage (WADC2).

${\mathit{w}}_{\mathit{k},\mathit{m}}$	${\mathit{K}}_{\mathit{k},\mathit{m}}$	$\mathit{T}{1}_{\mathit{k},\mathit{m}}$	$\mathit{T}{3}_{\mathit{k},\mathit{m}}$
$w_{5,12}$	19.3257	0.0691	0.4276
$w_{5,13}$	1.5901	0.6911	0.5498
$w_{5,14}$	19.0629	0.9907	0.9854
$w_{5,15}$	1.9377	0.0686	0.1719
$w_{5,16}$	13.3286	0.0170	0.0230
$w_{9,12}$	2.7647	0.0599	0.2148
$w_{9,13}$	15.1080	0.0117	0.0034
$w_{9,14}$	19.6429	0.9772	0.2159
$w_{9,15}$	15.9990	0.0782	0.7116
$w_{9,16}$	2.0962	0.3488	0.1437
$w_{10,12}$	13.4645	0.7421	0.0165
$w_{10,13}$	6.5231	0.7380	0.2823
$w_{10,14}$	14.0638	0.7134	0.8666
$w_{10,15}$	10.7047	0.5251	0.0766
$w_{10,16}$	9.1882	0.3689	0.6829
$w_{11,12}$	14.0206	0.6790	0.4785
$w_{11,13}$	16.7241	0.6937	0.8199
$w_{11,14}$	3.4396	0.0201	0.3445
$w_{11,15}$	0.8279	0.3680	0.0300
$w_{11,16}$	19.1326	0.6966	0.8145
$w_{12,12}$	5.4536	0.3722	0.2740
$w_{12,13}$	2.3466	0.0796	0.7261
$w_{12,14}$	17.8214	0.7688	0.0319
$w_{12,15}$	1.1156	0.6383	0.7505
$w_{12,16}$	6.5582	0.1337	0.1412

, respectively. In addition,

Table 8. Oscillation modes with lower damping ratios for each operating case for WADC1.

Case	Modes (Eigenvalues)	Frequency [Hz]	Damping Ratio [%]
BC0	$-0.1939\pm j3.2496$	0.5172	5.9566
BC1	$-0.1889\pm j3.2289$	0.5139	5.8417
BC2	$-0.1819\pm j3.2024$	0.5097	5.6705
BC3	$-0.1728\pm j3.1692$	0.5044	5.4452

and

Table 9. Oscillation modes with lower damping ratios for each operating case for WADC2.

Case	Modes (Eigenvalues)	Frequency [Hz]	Damping Ratio [%]
BC0	$-0.6063\pm j5.0070$	0.7969	12.0206
BC1	$-0.9937\pm j8.0878$	1.2872	12.1942
BC2	$-0.9915\pm j8.0853$	1.2868	12.1722
BC3	$-0.3853\pm j3.1520$	0.5017	12.1349

provide the oscillation modes (eigenvalues), frequencies and damping ratios for each of the operating cases BC0, BC1, BC2 and BC3 with the chosen WADC in the first and second stages. The WADC of the second stage provides the highest damping ratios for the four operating cases and thus the design achieved its purpose. The WADC obtained by the LQR-based method provided damping rates very close to 5%, far from the damping rates achieved by the proposed method, which were above 12%.

Now, a time-domain analysis will be conducted by applying a contingency using the ANATEM software [65]. A three-phase fault of 100 ms duration was applied to bus 40 in the cases BC0, BC1, BC2 and BC3. Figure 3, Figure 4, Figure 5 and Figure 6 show the angular responses of generator 14 for cases BC0, BC1, BC2 and BC3, respectively, without WADC, with the first-stage WADC and with the second-stage WADC of the proposed two-stage algorithm and the WADC designed by the LQR-based method. A second contingency analysis was applied and analyzed. A temporary 100 ms fault was also applied to bus 40 in the cases BC0, BC1, BC2 and BC3, and transmission lines 31–53 of the test system were permanently disconnected. Figure 7, Figure 8, Figure 9 and Figure 10 show the angular responses of generator 14 for cases BC0, BC1, BC2 and BC3, respectively, without WADC, with the first-stage WADC and with the second-stage WADC of the proposed two-stage algorithm and the WADC designed by the LQR-based method.

These results provide the following discussions:

• Table 8 and Table 9 show that the system modes with the first- and second-stage WADCs presented high damping ratios for all operating points and higher than the cases without WADC reported in Table 1. The second-stage WADC modes provided the best damping ratios. The oscillation modes of the system with the WADC(LQR) described in Table 5 have damping rates very close to 5% and far from the values achieved by the WADC designed by the proposed method.
• The angular responses of the system with WADC described in Figure 3, Figure 4, Figure 5 and Figure 6 are more damped and thus better than the system without WADC for all operating cases. The second-stage WADC provided the most damped angular response. Thus, the design was successful in achieving high system damping ratios.
• The angular responses of the system with WADC described in Figure 7, Figure 8, Figure 9 and Figure 10 are more damped and thus better than the system without WADC for all operating cases and disconnection of transmission lines 31–53. The second-stage WADC provided the most damped angular response. Thus, the design was successful in achieving high system damping ratios.
• The modal and time-domain analyses show that the second stage of the proposed algorithm was able to provide better WADC parameters and thus achieve higher damping ratios for the closed-loop control system modes.
• The incorporation of a second stage increased the design time of a WADC, but benefited the closed-loop control system with a WADC robust to multiple operating conditions with a higher objective function value.

6. Conclusions

This paper presents a proposal and analysis of a two-stage algorithm for the design of WADCs to increase the damping rates of the system modes. The first stage is based on the PSO algorithm and aims to obtain the WADC parameters that provide the highest damping ratios. The second stage is based on a heuristic and uses the WADC parameters from the convergence of the first stage to obtain new WADC parameters that provide the highest damping ratios.

The results showed that the first stage of the proposed algorithm provided damping ratios between 3.6233% and 6.9965%, with an average value of 5.3396%. The second stage of the proposed algorithm provided damping ratios between 8.5239% and 12.0206%, with an average value of 9.7255%. The LQR-based method already available in the literature achieved a value very close to 5%, a damping ratio value far from that achieved by the proposed method.

The analyses conducted in the time domain with contingency applications showed well-damped angular responses for the system with the designed WADCs. The second-stage WADC presented much more damped responses. All the operation cases under study presented this behavior and evidence the benefits of the proposed two-stage algorithm.

The main advantages of the proposed method are as follows: (i) easy implementation; (ii) few parameters need to be defined; (iii) the method achieved the control objectives for all 200 simulations; (iv) the objectives were achieved even though the initialization of the design variables was random; (v) the dynamic analysis showed good performance of the WADC designed using linearized models; (vi) the WADC design time was relatively satisfactory. The disadvantages of the proposed method are as follows: (i) the WADC design time can be reduced with parallel programming; (ii) the more power system operating points considered, the longer the WADC design time; (iii) the model requires correct and accurate linearization of power systems.

Future work will consider simulations with hardware-in-the-loop (HIL) validation and assessment under real-world disturbances. Multi-objective functions will also be incorporated into the proposed algorithm to meet new system performance requirements. Adaptive control techniques and machine learning-based techniques for WADC design will also be evaluated in the future. Real-time simulation platforms such as RTDS will be evaluated in the future with the proposed WADC. Furthermore, control strategies against cyber attacks will be studied and incorporated into the WADC project.