Predictive Controller Based on Paraconsistent Annotated Logic for Synchronous Generator Excitation Control

Abstract

This study presents a new Model Predictive Controller (MPC), built with algorithms based on Paraconsistent Annotated Logic (PAL), with application examples in the excitation control of a synchronous generator. PAL is a non-classical evidential and propositional logic that is associated with a Hasse lattice, and which presents the property of accepting the contradiction in its foundations. In this research, the algorithm was constructed with a version of the PAL that works with two information signals in the degrees of evidence format and, therefore, is called Paraconsistent Annotated Logic with annotation of two values (PAL2v). For the validation of the algorithmic structure, the computational tool MATLAB^® Release 2012b, The MathWorks, Inc., Natick, MA, United States was used. Simulations were performed which compared the results obtained with PPC-PAL2v to those obtained in essays with the AVR (Automatic Voltage Regulator) controls in conjunction with the PSS (Power System Stabilizer) and the conventional MPC of fixed weights. The comparative results showed the PPC-PAL2v to display superior performance in the action of the excitation control of the synchronous generator, with a great efficiency in response to small signals.

1. Introduction

An Electrical Power System (EPS) is eventually subject to disturbances that can cause stability problems in the generation and distribution of energy [1]. When an EPS changes from one operating point to another, the voltage and speed of the synchronous generator can consequently enter oscillatory states at low frequency. For small-scale disturbances, such as load and voltage variations in the electrical network, effective excitation circuits and speed controllers are required to keep the system within the proper conditions for its operation [1].

The excitation control, in addition to providing voltage control at the generator terminals and reactive power control, plays an important role in maintaining stability, and response to small signals [1,2,3]. In order to mitigate these problems of instability under these conditions, network analyzers and special devices have been developed to aid in the analysis of system behavior. For example, one of the most important circuits of this type is the AVR, or Automatic Voltage Regulator, which allows the analysis of the interconnection in order to minimize the oscillations. However, with the expansions of the industrial parks and consequent demand for electricity, the electricity transmission networks have expanded, and thus, there is a great variety of parameters of power systems with extreme operating conditions [3,4,5]. To avoid the risk of power failure, the EPSs had to be adapted to these new situations in which. Despite the good performance of the AVR—Automatic Voltage Regulator—in the SEPs, it was verified that, in some situations, when switching for example large blocks of load, this type of regulator does not provide a sufficient response in magnitude and speed so as to dampen the oscillations in the system effectively. Therefore, in order to alleviate these problems, a supplementary excitation controller called the Power System Stabilizer (PSS) has been added to reduce the electromechanical oscillations of the generator [3,4,5,6,7,8,9,10].

As described in [1,2,3,4], the stabilizing action of the PSS in the control mesh is in the feedback receiving the addition of the effects of the inertia of the generator, with the final result of a faster damping of the frequency and output voltage. In this way, with the addition of the PSS (Power System Stabilizer) to the AVR (Automatic Voltage Regulator), any negative effect can be eliminated to damping the oscillations after a disturbance. However, it has been found that the combination of these two controls can generate inconvenient control situations [8,9,10]. On these aspects, it can be verified in their applications that AVR and the PSS are normally projected separately, since both have different objectives in the process, to soften the excitation failures [5].

The difference between the AVR and PSS is that the AVR is developed to meet requirements for voltage regulation and the PSS is designed to damp electromechanical oscillations [8,9]. Therefore, the coordination of the two objectives in joint actions can present problems when there is a change in the operating conditions of the system [1,3,4]. Recently, studies have been developed to discover new techniques with which to improve the performance of these types of control, and thus reduce the risk of failure in the EPS. For instance, a linear optimum control used for excitation control is highlighted in [3]. This control was developed to minimize the variations in system state. In order to work with voltage and velocity deviations, a cost function was formulated [4,6,7,8,9,10,11,12]. In Ref. [4] a controller with actions was presented, minimizing the cost function, and an optimized control was obtained. In this approach, the results were compared to those obtained by the PSS, and a wide frequency band could be obtained with a good response. However, because it is based on a specific operating point, this controller did not perform well when subjected to large variations in the system operating point. In order to achieve this, several methods of analysis and stability control using artificial intelligence (AI) and techniques with non-classical logic have been developed for these power systems, such as: Backstepping, Fuzzy Control, Direct Feedback Linearization (DFL), Artificial Neural Network (ANN), LgV, among others [5,13,14,15,16].

The adaptive controllers use Integrated Circuits (IC microcontrollers) with special algorithms based on nonlinear models to improve control performance [17,18]. In some cases, the algorithms use the precision of the predictions of the model making a common approach to consider the uncertainty in the parameters. For this, it adds an extra term in the cost function of a Minimum Variance controller, which penalizes the uncertainty in the parameters of the nonlinear approximation. Another similar approach has been proposed in [19], which is based on minimizing two separate cost functions. The first minimization is used to improve the parameter estimation and the second to generate the system output in accordance with a given reference signal [20]. Thus, adaptive controls can change their parameters according to the new operating point through preprogrammed rules by using neural networks or other non-classical techniques. However, depending on the number of instructions and the hardware of the microcontroller, the processing may not meet the speed required by the controlled system [21].

With the help of microcontrollers, special algorithms that consider both past states and future states, estimates can be used together with the adaptive control. In the interval, or step of actuation, a finite horizon of time is given in which these estimates present the minimum of possible uncertainties. This gives an excitation controller capable of defining a voltage-optimized value in a one-step prediction horizon. This type of Predictive Controller, despite the good performance in control, has a high computational cost since the algorithm needs an adjustment strategy for the cost matrix variables of the quadratic cost function [18,19].

In order to extract a better performance of this control method, the techniques involving predictive control can be improved using algorithms based on non-classical logics that are capable of responding well to uncertainties and with a greater speed. In this case, theories based on paraconsistent logics, which do not ignore contradictions and, instead, extract information from them, can present researchers with a good choice in adaptive and predictive control [17,19,20,21].

The Paraconsistent Annotated Logic (PAL) used in this work is one of the non-classical logics [22,23] suitable for reasoning with data that can bring inconsistent information. Therefore, it is possible to extract from its foundation efficient algorithms of predictive control. These can be applied in the control of excitation of a generator installed in an EPS [24,25,26,27,28]. With these considerations, this paper demonstrates the development and analysis of an excitation control for a synchronous generator, integrating the foundations of the Paraconsistent Logic Annotated in its special form of two values, PAL2v [24], into a Model Predictive Control (MPC). The application of the PAL2v will be performed in order to obtain a mechanism of intelligent self-adjustment to the classic MPC, one that can offer greater efficiency in the maintenance of the stability of the EPS when submitted to variations in load and in tension.

In addition to this introduction, the text of this work is presented as follows: in Section 2, the main foundations of Paraconsistent Logic and Paraconsistent Annotated Logic are presented. At the end of Section 2, the algorithms of PAL2v and the Paraconsistent Analysis Network (PANnet), used in the proposed Model Predictive Control, are presented. In Section 3, we present initial information about Predictive Control, based on MPC (Model Predictive Control), and the cost function equations used in the mathematical method. In Section 4 (Materials and Methods), we present the configuration of the Predictive Control, based on MPC, with the algorithms of PAL2v and its mathematical logic configuration in control of the excitation of a synchronous generator installed in an EPS. Additionally, in Section 4, the details of the computational tool MATLAB^® Release 2012b implementation of the Predictive Controller, based on MPC built with the algorithms of PAL2v (PPC-PAL2v), are presented. In Section 5, the results obtained by the simulation of the PPC-PAL2v Controller and the comparative graphical results with the AVR and PSS are presented. In Section 6, discussions about the results, obtained in the tests performed, are presented. In Section 7, the final considerations on the application of the PPC-PAL2v controller which was applied to the excitation control of an EPS synchronous generator are presented.

2. Paraconsistent Logic

Paraconsistent Logic (PL) belongs to the class of non-classical logics whose fundamental structures differ from classical binary logic, opposing the law of non-contradiction. Due to these fundamentals, PL can be adapted acting as theoretical support for the algorithms that constitute computational systems for the treatment of uncertainties [22,23,24].

PL can be studied through an annotation concept in which its representation is considered a lattice. This can be the one of 4 vertices (Lattice FOUR), according to Figure 1a. Each annotation assigns to the proposition, P, a Paraconsistent Logical State, Ɛ_τ, that is represented at the vertex of the lattice. With an annotated logic structuring, PAL allows its algorithms to manipulate inconsistent information, where logical states are represented at the vertices of its associated lattice. Thus, the values of information signals obtained by measurements are represented by normalized degrees of evidence and considered in the annotation that gives logical connotation to the given proposition, P [24,25,26,27,28]. The four vertices lattice, τ, may be associated via the Paraconsistent Annotated Logic with the annotation composed of two degrees of evidence (μ, λ). Through performing analysis with the Paraconsistent Annotated Logic of annotation with two values (PAL2v), one can establish an improved representation of how much the annotations, or evidence, express the knowledge about a proposition, P [24,25,26,27,28]. In this case, a value, μ, represents the evidence favorable to proposition, P, and the other value, λ, represents the evidence unfavorable to proposition P. In the PAL2v, an associated lattice, τ, formed by pairs of independent degrees of evidence is used in the formalization of PAL2v, where μ is the degree of evidence favorable to P and λ is the degree of evidence unfavorable to P, such that [24,25,26,27,28]: τ = {(μ, λ)|μ, λ ∈ [0, 1] ⊂ $ℛ$}.

For better representation of an annotation, and also for the practical use of the PAL2 lattice in the treatment of uncertainties, some algebraic interpretations involving a Unitary Square in the Cartesian Plane (USCP) and the representative lattice of PAL2v are made. With the representation of the degrees of evidence in an USCP, one can be applied geometric and linear transformations to obtain points of intersection represented in the lattice τ associated with PAL2v and obtain equations resulting in algorithms for practice applications. The transformations between USPC and PAL2v–lattice are defined through three steps: scale change (T1), rotation (T2) and translation (T3) [24,25,26,27,28]. By making the composition of the three phases that generated the transformations T3, T2, T1, we obtain the final transformation represented by Equation (1) [24]:

$$T(X,Y)=(x-y,x+y-1)$$

(1)

Relating the components of the transformation according to the usual nomenclature of PAL2v: $x=\mu$→ degree of evidence favorable to proposition P and $y=\lambda$ → degree of evidence unfavorable to proposition P. The first term obtained in the pair of values of the transformation equation is: $X=x-y=\mu -\lambda$, which we call degree of certainty—Dc. Therefore, the degree of certainty [24,25,26] is obtained by Equation (2):

$$Dc=\mu -\lambda$$

(2)

Its values, which belong to the set $ℛ$, vary in the closed interval +1 and −1, and are on the horizontal axis of the lattice, which is called the “Degrees of Certainty Axis”. When Dc results in +1, it means that the logical state resulting from the paraconsistent analysis is True, and when Dc results in −1 means that the logical state resulting from the analysis is False [24,25,26,27,28]. The second term obtained in the pair of values of the transformation equation has: $Y=x+y-1=\mu +\lambda -1$, which is called the degree of contradiction—Dct. Therefore, the degree of contradiction Dct [24,25,26,27,28] is obtained Equation (3):

$$Dct=\mu +\lambda -1$$

(3)

Their values, which belong to the set $ℛ$, vary in the closed interval +1 and −1, and are on the vertical axis of the lattice, which is called the “Axis of degrees of Contradiction”. When Dct results in +1, it means that the logical state resulting from the paraconsistent analysis is Inconsistent, and when Dct is −1, the logical state resulting is either Paracomplete, or Undetermined [24].

The paraconsistent analysis that resulted in the calculations of certainty (Dc) and degree of contradiction (Dct) (Equations (2) and (3), respectively) produced values that are interpolated in the lattice at an internal point (Dc, Dct), according to Figure 1b. The distance d of the line from the point of maximum degree of certainty, t, represented at the right vertex of the lattice, to the point of interpolation, is calculated by Equation (4) [24]:

$$d=\sqrt{{(1-|Dc|)}^2+Dc{t}^2}$$

(4)

The projection of the distance, d, in the axis of certainty values gives the point whose value will be considered the degree of real certainty—D_CR. In Figure 1b, this condition is shown where the value of the real certainty degree—D_CR—is obtained [24].

If the certainty degree (Dc) calculated by Equation (2) results in a negative value, the distance, d, will be obtained from the point of certainty False, f, represented at the left vertex of the lattice, to the point of internal interpolation (-Dc, Dct). It is verified that, at any point in the lattice of values, it is possible to obtain the degree of real certainty, D_CR. The values of negative Dct do not modify the means of obtaining D_CR. Therefore, the value of the true degree of D_CR is obtained from the determination of distance, d, according to the conditions shown by Equation (5) and (6) [24,25]:

For Dc > 0

$$D_{CR}=1-\sqrt{{(1-|D_C|)}^2+D_{ct}{}^2}$$

(5)

For Dc < 0

$$D_{CR}=\sqrt{{(1-|D_C|)}^2+D_{ct}{}^2}-1$$

(6)

These D_CR values are normalized to become the actual resulting degree of evidence, as follows in Equation (7) [24,29,30]:

$${\mu }_{ER}=\frac{D_{CR}+1}{2}$$

(7)

where: μ_ER = resulting degree of evidence and D_CR = degree of real certainty.

With these fundamental equations, it is possible to construct Paraconsistent Annotated Logic (PAL2v) algorithms for applications in analysis and logical signal processing. These algorithms are capable of being interconnected to form Decision Analysis networks with different topologies [24].

2.1. Algorithms of PAL2v Used in the Paraconsistent Model

Paraconsistent Systems or Analysis Nodes are algorithms extracted from the Paraconsistent Logic, capable of forming Paraconsistent Analysis Networks for the treatment of information signals [24,26,31,32,33,34]. With their inputs being fed by the evidence degrees taken from external measurements or from the Uncertain Knowledge database, the Paraconsistent algorithms use the equations obtained from the PAL2v methodology and present the results in the form of the actual resulting evidence degrees, μ_ER. The PAL2v equations and their interpretation allow for the creation of algorithms for direct applications. In this work, three types of PAL2v algorithms are used in Paraconsistent Predictive Control; the algorithm for extracting the degree of evidence; the Paraconsistent Analysis Node (PAN) algorithm; and the Paraconsistent Logic Maximization Algorithm.

These three algorithms will be described below [24,29,30,32]:

2.1.1. Algorithm for Extracting the Degree of Evidence

The algorithm, named the Extractor of Evidence Degree, calculates the value of the degree of evidence of the quantity when measured through a function which has been considered in a Discourse Universe or Interval of Interest. Depending on the application, a straight-line equation can be used to obtain the value of the degree of evidence. The Extractor of Evidence Degree is exposed in Algorithm 1 below.

Algorithm 1: Extractor of Evidence Degree

1. Enter the Maximum Limit Value of the magnitude in your unit of measure to form the Discourse Universe.      Maximum Limit Value (Maxvalue) = ......................      2. Enter the Minimum Limit Value of the magnitude in your unit of measure to form the Discourse Universe.        Minimum Limit Value (Minvalue) = ........................      3. Display the Measured Value of Greatness in your unit of measurement.        Value Greatness X (X value) = ..........................      4. Determine the Degree of Evidence favorable, μ, through the equations, considering the conditions: ${\mu }_1=\left\{\begin{array}{cc}1& \mathit{If} Xvalue\ge Ma{x}_{value}\\ \frac{Xvalue-Ma{x}_{value}}{Ma{x}_{value}-Mi{n}_{value}}& \mathit{If} Xvalue\in \left[Mi{n}_{value},Ma{x}_{value}\right]\\ 0& \mathit{If} Xvalue\le Mi{n}_{value}\end{array}\right.$      5. Determine the unfavorable Degree of Evidence, λ, by complementing the favorable degree of Evidence: $\lambda =1-\mu$      6. End.

The symbolic representation of extractor of Degree of Evidence is seen in Figure 2a.

2.1.2. PAN Paraconsistent Analysis Node Algorithm

The Algorithm Paraconsistent Analysis Node (PAN) receives two information signals, represented by degrees of evidence, and presents as a final result a single value of the resulting degree of evidence. The degree of evidence resulting from the output is a value that expresses a representation of the analysis, where the effect of the contradiction between the two values applied in its inputs is null [24]. In the construction of PAN, the equations of PAL2v are used, and its structure is the minimum cell of analysis of a Paraconsistent System of Treatment of Uncertainties [24,29,31,32].

The description of the PAN is shown in Algorithm 2.

Algorithm 2: Paraconsistent Analysis Node—PAN

1. Present two input values:    μ */ favorable Evidence degree 0 ≤ μ ≤1 */    λ */ unfavorable Evidence degree 0 ≤ λ ≤1 */  2. Calculate the Degree of Certainty: $Dc=\mu -\lambda$  3. Calculate the Degree of Contradiction: $Dct=(\mu +\lambda )-1$  4. Calculate the Normalized Degree of Contradiction: $\mu ct=\frac{\mu +\lambda }{2}$  5. Calculate the distance d (projection on the axis (horizontal) of the degrees of certainty on the PAL2v lattice): $d=\sqrt{{(1-|Dc|)}^2+Dc{t}^2}$  6. Determine the output signal.     If d > 1, then do: $S1=0.5$ → Consider Undefinition and go to End.     Otherwise go to the next item  7. Determine the Real Certainty Degree     If $Dc>0$ Calculate: $D_{CR}=(1-d)$     If $Dc<0$ Calculate: $D_{CR}=(d-1)$  8. Calculate the resultant Real Evidence Degree: ${\mu }_{ER}=\frac{D_{CR}+1}{2}$  9. Present the results in the output: Do $S1={\mu }_{ER}$  10. End

In the PAN the projection of the distance value d, Equation (4) on the axis of the degrees of certainty establishes the extraction of the effect of the contradiction between the degrees of evidence of the inputs. Thus, the value of the output, evidence degree, represents the value of the resulting evidence regarding the proposition analyzed without the effect of the contradiction [26,27]. In Ref. [27], details about the PAN algorithm are shown, including its data flowchart with the steps for the PAL2v analysis.

2.1.3. Paraconsistent Algorithm of Maximization Logical Connection—$APl{C}_{Max}$

The Paraconsistent Algorithm of Logical Maximization Connection has the function of establishing logical connectives between signals, representing degrees of evidence. In this work, we use the algorithm that makes a logical connection of maximization OR [24]. For maximization, a simple analysis is initially made by determining the normalized resulting degree of evidence, the result of which will inform which of the two input signals is the highest value [26]. With this information, the algorithm establishes the output signal as being the maximum value.

The equation used in the algorithm and the conditions that determine the outputs for a maximization process are presented in the description of Algorithm 3 below:

Algorithm 3: Maximization Logical Connection—$APl{C}_{Max}$

Since the input variables are: ${\mu }_{1A}$, such that: $0\le {\mu }_{1A}\le 1$, and ${\mu }_{1B}$, such that: $0\le {\mu }_{1B}\le 1$,    then     1. Do: ${\mu }_{1A}={\mu }_1$ and ${\lambda }_2=1-{\mu }_{1B}$     2. Calculate the Degree of Certainty by equation (2): $Dc={\mu }_1-{\lambda }_2$     3. Calculate the resulting normalized Evidence Degree by: ${\mu }_{ER}=\frac{({\mu }_1-{\lambda }_2)+1}{2}$     4. Select the highest value in the output S1 by making the conditionals:       If: ${\mu }_{ER}\ge 0,5$ → ${\mu }_{1A}\ge {\mu }_{1B}$ → Present in Output S1 the value of ${\mu }_{1A}$                      Indicate in the output S2→ Bit() = 1       If not: → Present in Output S1 the value of ${\mu }_{1B}$                      Indicate in the output S2→ Bit() = 0     5. End

The symbolic representation of $APl{C}_{Max}$ is seen in Figure 2c.

2.2. Paraconsistent Analysis Network (PANnet)

The paraconsistent system of treatment of uncertainties defined by the PAN can be used in several fields of knowledge. With its application, incomplete and contradictory information will receive adequate treatment through the equations of PAL2v [24,25,26].

As the output of a PAN has normalized value, this result value can be used as a degree of evidence for another PAN, and thus different configurations of analysis networks can be created. These configurations are called Paraconsistent Analysis Networks (PANnet). In this way, the computational structures of PANnet are used for the logical treatment of signals according to the purpose for which they are proposed.

Figure 2d shows a PANnet, composed of 3 PANs configured in the way it will be used in this work.

Initially, the modeling algorithms, which are called the Evidence Degree Detection Extractors, create the normalized degrees of evidence for the analyses in the PANnet [24,25,26,27] from the values extracted from measures of physical quantities. The PAN1 and PAN2 algorithms have their outputs, interconnected through the Paraconsistent Maximization Logic Algorithm which selects the highest value between the two outputs μ_Er1 and μ_Er2. A signaling Bit() indicates which of the two output values of the PANs are in the μ₀ output. This PANnet configuration still has the PAN3 that uses the degrees of evidence from information sources. To compose this annotation, it uses the favorable degree of evidence, μ_4, and the degree of unfavorable evidence, λ₂ = 1 − μ₂. The PAN3 performs the paraconsistent analyses and displays the result at its separate output, S_μ3. This PANNet configuration will be used in the treatment of the signals to control the excitation of the synchronous generator, and its operation will be explained in detail below.

Recently, PAL2v algorithms, forming paraconsistent analysis networks (PANnet), have been used successfully in several applications which cover different fields of knowledge. As an example, we can mention in [26,27], where PAL2v algorithms were used to detect skin cancer by analyzing Raman spectroscopy signals. In Ref [28], the PAL2v algorithms were used to support the operation of electricity transmission networks. In Ref [31], the authors used the PAL2v algorithms to support the control of a flow loop, forming a Hybrid Proportional Integral (PI) Controller.

3. Model-Based Predictive Control—MBPC

The characteristic of Model-Based Predictive Control—MBPC—is its ability to predict the future responses of the controlled system. This feature distinguishes it from other types of controllers [1,2,3,4]. The Model Predictive Control MPC uses information signals, represented by discrete steps (k + 1) (or pulses) through a sample frame. If the prediction of the future output path of the process {ŷ (k + i), i = 1, …, P}, then the controller calculates the necessary control action {Δu (k + i), i = 0, 1, …, m − 1} so that the difference between the predicted path and the user-specified reference (Set Point) is minimized. The MPC can be described as a state space model and is calculated with the state-space equations, which are generally used for the mathematical modeling of a time-varying physical system [14,15].

3.1. State-Space Equations in Discrete Time

The state-space equations can be described as:

$$\begin{array}{l}\stackrel{.}{x}(t)={\mathrm{A}}_{t}x(t)+{\mathrm{B}}_{t}u(t)\\ y(t)=\mathrm{C}x(t)\end{array}$$

where $x(t)$ is the state vector of the system, ${\mathrm{A}}_t$ is the state matrix, ${\mathrm{B}}_t$ is the matrix related to the input, $u(t)$ is the input variable (manipulated variable), $y(t)$ is the output vector, and Cx(t) is the output matrix.

Considering a sampling interval ${\Delta }_t$, it is possible to define a value $k=1,2,3,\dots N$, such that $k=\frac{t}{{\Delta }_t}$, therefore, making the system discrete in time. Through Equation of Differences, we obtain the set of simplified discrete time state equations, represented by Equation (8) [1,2,3,4,5]:

$\stackrel{.}{x}(k)=\frac{x(k+1)-x(k)}{\Delta t}$ →$x(k+1)=\mathrm{A}x(k)+\mathrm{B}u(k+1)$ and $y(k)=\mathrm{C}x(k)$

$$y(k+1)=\mathrm{C}x(k+1)$$

(8)

where: $\mathrm{A}={\mathrm{A}}_t\Delta t-I$, $\mathrm{B}={\mathrm{B}}_t\Delta t$.

In the set of equations, it is possible to observe that the variables correspond to the future state given the conditions of the system in the present. The MBPC algorithm, at each instant of time, attempts to optimize the behavior of the controlled system by calculating the future values of the manipulated variables to achieve this goal [13]. An illustrative block diagram of a predictive controller is shown in Figure 3, where the plant is driven by a control generated in a system termed Optimizer. In this work, an Optimizer Block, with algorithms based on PAL2v, will be constructed [14,15].

3.2. Quadratic Cost Function

The MBPC makes an instantaneous time, t, measurement of the output variable of the plant and, through the mathematical model of the controlled system, makes a prediction. Based on this measurement and the reference r, a prediction horizon, N, is stipulated and future values for the manipulated variable are calculated by minimizing of an equation called the quadratic cost function [13,14,15].

The quadratic cost function is represented by Equation (9) [15]:

$$\mathrm{J}={[r-y(k+1)]}^T\mathrm{Q}[r-y(k+1)]+\Delta u^T\mathrm{R}\Delta u$$

(9)

where $\Delta u=u(k+1)-u(k)$.

Since matrices Q and R are called weights or cost matrices, they are used to determine degrees of importance to certain inputs and outputs, and can be represented in Equation (10):

$$y(k+1)=\mathrm{CA}x(k)+\mathrm{CB}u(k)+\mathrm{CB}\Delta u$$

(10)

Putting Equation (10) into (9) and applying the minimization of the function J, $\frac{\delta \mathrm{J}}{\Delta u}=0$ results in Equation (11):

$$\Delta u=\frac{{(\mathrm{CB})}^T\mathrm{QE}}{[{(\mathrm{CB})}^T\mathrm{QCB}+\mathrm{R}]}$$

(11)

where:

The matrix Q is a positive definite matrix that is related to the states.

The matrix R, is a positive semi definite matrix that is related to the input.

The matrices Q and R are usually diagonal, with positive elements on their diagonal.

4. Materials and Methods

In this work, a Paraconsistent Predictive Controller-PPC-PAL2v was developed. This was composed of a computational architecture, constructed from algorithms based on PAL2v. The PPC-PAL2v was applied to the model of a synchronous machine connected to an Electrical Power System (EPS) where for its performance, we established:

• Voltage regulation at the generator terminals and;
• Damping of the electromechanical oscillation that is directly related to the rotor speed variations.

For the study of the control model developed in this work, the EPS used in the modeling has a synchronous generator connected to the infinite bus, as represented in the circuit of Figure 4a.

The synchronous generator is connected via a voltage transformer to two parallel transmission lines which, at the other end, are connected to the infinite bus. Figure 4b shows the model of the EPS, where v_t is the voltage at the terminals of the generator and v_o is the voltage in the infinite bus. The reactance, X_TR, is the transformer and the X_LT transmission lines connected in parallel.

From a mathematical model created, a simulation was performed in the MATLAB^® software R2012b to analyze the excitation control actions for the electric field of the synchronous generator, integrating the fundamentals of the Paraconsistent Annotated Logic in its special two-valued form (PAL2v) into a Model Predictive Control (MPC).

The dynamic variations of the synchronous generator will be expressed in the simulation in the face of the variations of voltage and speed caused by the small perturbations applied to the EPS.

All the mathematical logical modeling of the PPC-PAL2v was performed based on the dynamic conditions of the EPS, where it was then applied to act on the excitation control of the synchronous generator.

Table 1. Electrical Power System data.

Electrical Power System Parameters
${\mathit{X}}_{\mathit{T}\mathit{R}}$	0.15
${\mathit{X}}_{\mathit{L}\mathit{T}{\displaystyle 1}}$	0.5
${\mathit{X}}_{\mathit{L}\mathit{T}{\displaystyle 2}}$	0.93

shows the data concerning the reactance of the EPS transformers used in this work.

Table 2. Synchronous Generator Data.

Nominal Values		Parameters				Saturation
Nominal power	555 MVA	${\mathit{X}}_{\mathit{d}}$	1.81 p.u.	${\mathit{X}}_{\mathit{l}}$	0.15 p.u.	${\mathit{A}}_{\mathit{s}\mathit{a}\mathit{t}}$	0.031
Nominal voltage	24 kV	${\mathit{X}}_{\mathit{q}}$	1.76 p.u.	${\mathit{R}}_{\mathit{a}}$	0.003 p.u.	${\mathit{B}}_{\mathit{s}\mathit{a}\mathit{t}}$	6.93
Frequency	60 Hz	${\mathit{X}}_{\mathit{d}}^{\prime }$	0.30 p.u.	${\mathit{T}}_{\mathit{d}0}^{\prime }$	8.00 s	${\mathsf{\Psi }}_{\mathit{T}1}$	0.80
Number of poles	2	${\mathit{X}}_{\mathit{q}}^{\prime }$	0.65 p.u.	${\mathit{T}}_{\mathit{q}0}^{\prime }$	1.00 s
Cte of Inertia	3.5 MWs/MVA	${\mathit{X}}_{\mathit{d}}^{″}$	0.23 p.u.	${\mathit{T}}_{\mathit{d}0}^{″}$	0.03 s
Cte of Damping	0	${\mathit{X}}_{\mathit{q}}^{″}$	0.25 p.u.	${\mathit{T}}_{\mathit{q}0}^{″}$	0.07 s

shows the data of the synchronous generator used.

The variables selected to achieve optimum performance in voltage regulation at the generator terminals and the damping of electromechanical oscillations are electromagnetic torque, $T_e$, voltage at the generator terminals, $V_t$ and rotor speed, ${\omega }_r$. These variables, in the MBPC representation, are generated through the outputs, $y(k+1)$, predicted by the synchronous generator model, and the reference values to be applied in the equations of the reference trajectories that are processed in the optimization algorithm PAL2v. In the control of the voltage variations at the generator terminals, the variable is used $\Delta e_{vt}$, which is expressed by the difference between the voltage deviation in the step, $k$, and the deviation predicted in the step $k+1$, according to the following discrete time state Equation (12):

$$\Delta e_{vt}(k+1)={\Delta }_{vt}(k)-{\Delta }_{vt}(k+1)$$

(12)

Being the voltage reference $r_{vt}={\Delta }_{vt}(k)$.

The voltage deviation in the step $k$ is determined from the set-point of the voltage $V_0$ and the voltage in the step $k$, called $V_t(k)$, at the terminals of the generator. As for the damping characteristic of electromechanical oscillations, there are two variables that participate in this function: $\Delta e_{\omega r}$, which is related to the rotor speed, and $\Delta e_{Te}$, related to the electromagnetic torque. These two variables are expressed, respectively, by Equation (13) and Equation (14).

$$\Delta e_{\omega r}(k+1)=\Delta {\omega }_r(k+1)-\frac{1}{2H}{\Delta }_t\Delta Te(k+1)$$

(13)

Being the reference of the speed of the rotor $r_{\omega r}=\frac{1}{2H}{\Delta }_t\Delta Te(k+1)$

$$\Delta e_{Te}(k+1)=T_m-T_e(k)-\Delta T_e(k+1)$$

(14)

Being the torque reference $r_{Te}=T_m-T_e(k)$, where: $\Delta e_{\omega r}(k+1)$ is the prediction value of the model of the speed deviation of the generator and $\Delta e_{Te}(k+1)$ is the prediction of the model of the deviation of the electromagnetic torque.

4.1. General Aspects of the Paraconsistent Predictive Model

In the optimization of the control, the values of the current speed ${\omega }_r(k)$, the values of the speed in the previous instant ${\omega }_r(k-1)$, and the reference voltage (Set Point) $V_0$ are used. The variables involved in the PPC-PAL2v model applied to the excitation control of the synchronous generator are shown in Figure 5.

4.2. Paraconsistent Predictive Controller-PPC-PAL2v

In order to meet the voltage regulation and damp effects of electromechanical oscillations, the PPC-PAL2v was structured in functional blocks, where each block has specific actions to compose joint actions in the predictive control in the excitation of the synchronous generator. For better understanding, the details of the control configuration used in the PPC-PAL2v are described below.

• Structural configuration of PPC-PAL2v

The structural configuration of the model-based Paraconsistent Predictive Controller—PPC-PAL2v—will be presented in two main blocks which are interconnected by the signal flow of the variables involved. The general configuration of the PPC-PAL2v is shown in Figure 6, with its signal flows in its two main blocks: OT-PAL2v—Optimizer— and MM—mathematical model of the synchronous generator.

• OT-PAL2v—Optimizer Block

In the Predictive Controller model, the PAL2v algorithms were used in the construction of the Optimizer block (OT-PAL2v). In Figure 7, the composition of Optimizer PAL2v with three secondary blocks is shown, as follows: Paraconsistent Analysis Network (PANnet), Weight Adjustment Mechanisms (WAM) and Field Voltage Setting Determination Block (FVSD).

As shown in Figure 8, the Paraconsistent Analysis Network (PANnet), inserted into the Optimizer-PAL2v block, has the objective of performing the logical treatment of the normalized values. These values are obtained by measuring the generator speed, ${\omega }_r$ and the voltage, $V_t$.

The result of the paraconsistent logical treatment in the signals will serve to determine the weight factors used in the block of Adjustment Mechanisms.

The PANnet configuration was developed by analyzing the conditions of the quantities involved in the control, and the optimization of the signals was performed through the configurations of PANs.

Table 3. Analysis of the Variations of Voltage and Speed.

	Signal	Condition	$\mathbf{Value}\mathbf{of}{\mathit{e}}_{\mathit{f}\mathit{d}}$
$\Delta {\omega }_r$	Positive	Acceleration	Increases
$\Delta V_t$	Positive	Overvoltage	Decreases
$\Delta {\omega }_r$	Positive	Acceleration	Increases
$\Delta V_t$	Negative	Undervoltage	Increases
$\Delta {\omega }_r$	Negative	Deceleration	Decreases
$\Delta V_t$	Negative	Undervoltage	Increases
$\Delta {\omega }_r$	Negative	Deceleration	Decreases
$\Delta V_t$	Positive	Overvoltage	Decreases

shows the analysis of the field voltage variation according to the speed and voltage variation conditions at the generator terminals. Two points of contradiction are observed:

1. When there is acceleration of the rotor and the voltage is above the desired voltage $V_0$;

2. When there is deceleration of the rotor and the voltage is below the desired value $V_0$.

From the conditions indicated in Table 3, we can define the equations of the algorithms that extract the degrees of evidence that represent the values for the input signals of PPC-PAL2v.

• PPC-PAL2v Input Variables with Normalized and Selected Values

As seen in Figure 8, the PANnet, inserted into the Optimizer-PAL2v block, has algorithms for extracting the degree of evidence that normalizes and selects the signals of the input variables. These are: the speed at the present instant ${\omega }_r(k)$, speed at the previous instant ${\omega }_r(k-1)$, voltage at the generator terminals at the present instant $V_t(k)$ and the reference voltage (Set Point) $V_0$.

The first select/normalization that results in μ₁ for PAN 1 is performed by considering the acceleration condition above the desired value, so the mathematical function that extracts the degree of evidence and selects the output for this condition is:

$${\mu }_1=\left\{\begin{array}{ll}0& if[{\omega }_r(k)-{\omega }_r(k-1)]\le 0\\ \frac{[{\omega }_r(k)-{\omega }_r(k-1)]}{20.41\times {10}^{-9}}& if0<[{\omega }_r(k)-{\omega }_r(k-1)]\le 20.41\times {10}^{-9}\\ 1& if[{\omega }_r(k)-{\omega }_r(k-1)]>20.41\times {10}^{-9}\end{array}\right.$$

(15)

The second select/normalization that results in λ₂ = 1-μ₂ for PAN 1and λ₄ = 1-μ₂ for PAN 3 is made considering the maximum deviation of 5% of the voltage at the generator terminals above the desired voltage, and so the mathematical function that extracts the degree of evidence for this condition is:

$${\mu }_2=\left\{\begin{array}{ll}1& if[V_t-V_0]\le 0\\ \frac{0.05-[V_t-V_0]}{0.05}& if0<[V_t-V_0]\le 0.05\\ 0& if[V_t-V_0]>0.05\end{array}\right.$$

(16)

The third select/normalization that results in μ₃ for PAN 2 is made considering a deceleration condition with a maximum tolerance of $20.41\times {10}^{-9}$ below the desired value. Therefore, the mathematical function that extracts the degree of evidence for this condition is:

$${\mu }_3=\left\{\begin{array}{ll}0& if[{\omega }_r(k-1)-{\omega }_r(k)]\le 0\\ \frac{[{\omega }_r(k-1)-{\omega }_r(k)]}{20.41\times {10}^{-9}}& if0<[{\omega }_r(k-1)-{\omega }_r(k)]\le 20.41\times {10}^{-9}\\ 1& if[{\omega }_r(k-1)-{\omega }_r(k)]>20.41\times {10}^{-9}\end{array}\right.$$

(17)

The fourth select/normalization that results in λ₃ = 1-μ₄ for PAN 2 and μ₄ for PAN 3 is made considering the maximum deviation of 5% of the voltage at the generator terminals above the desired voltage. Therefore, the mathematical function that extracts the degree of evidence for this condition is:

$${\mu }_4=\left\{\begin{array}{ll}1& if[V_0-V_t]\le 0\\ \frac{0.05-[V_0-V_t]}{0.05}& if0<[V_0-V_t]\le 0.05\\ 0& if[V_0-V_t]>0.05\end{array}\right.$$

(18)

The inputs to PAN 3 are the same normalized values ${\mu }_2$, obtained by Equation (16), and ${\mu }_4$, obtained by Equation (18).

4.3. Detail of the Paraconsistent Analysis Network (PANnet)

The PANnet is the first block of the OT-PAL2v-Optimizer and is where the values of the speed ${\omega }_r(k)$ at actual instant, the speed at the previous instant ${\omega }_r(k-1)$, the voltage at the terminals of the generator at the actual instant $V_t(k)$ and the reference voltage $V_0$ are applied. In the PANnet inputs, all these values were transformed into degrees of evidence, and from these paraconsistent variables PANnet defines the degree of real evidence for speed ${\mu }_{ER}{}_{\_{\omega }_r}$. This represents the evidence (intensity) $e_{fd}$ of the need to increase or decrease to satisfy both the voltage control and the damping of electromechanical oscillations.

The result, in terms of the degree of real evidence for the voltage ${\mu }_{ER\_}{}_V$ that is simultaneously obtained in PANnet, represents evidence (intensity) of the need for tension control. Based on these control conditions, the function of each PAN in the PANnet configuration shown in Figure 8 is described below:

• PAN 1 ⇒ Analysis of the increase condition of $e_{fd}$;
• PAN 2 ⇒ Analysis of the decrease condition of $e_{fd}$;
• PAN 3 ⇒ Analysis of the variation of $K_v$.

After the analysis performed by PANs 1 and 2, the resultant evidence degrees, ${\mu }_{ER1}$ and ${\mu }_{ER2}$, corresponding to the need to increase $e_{fd}$ and the need to decrease $e_{fd}$, are passed through a Paraconsistent Logic Maximization Algorithm $APl{C}_{Max}$. The $APl{C}_{Max}$ acts on only the greater value between the two, and sends the resulting signal directly to the Weighting Mechanism Block. Thus, dominant degree of evidence that is represented by ${\mu }_{ER\_\omega }$ will provide the increase or decrease in $e_{fd}$ to satisfy both the voltage control and the damping of electromechanical oscillations. Simultaneously, after the analysis performed by the PAN3, the degree of evidence resultant of its output is represented by ${\mu }_{ER\_V}$. This will act in the Weights Adjustment Mechanism (WAM) Block for the control of the tension.

The WAM Block is described below.

4.4. Weights Adjustment Mechanism Block (WAM)

In the Mechanism of Adjustment of Weights, an analysis of the degrees of evidence, received from the PANs that compose the PANnet, is made to decide how to modify the weights in order to achieve optimal control. In the outputs of this block are:

$K_{\omega }$($p_{\omega }$) which represents the weight relative to the speed adjustment.

$K_V$($P_V$) which is the weight relative to the voltage.

$K_{Te}$($p_{Te}$) which is the weight relative to the electromagnetic torque.

Thus, in the PAL2v-Optimizer block, the resulting signals from the Mechanism of Adjustment of Weights block are obtained through the equations of the multiplier factors for the Q-Matrix, as shown below.

• Equations of the multiplier factors for the Q-Matrix

From the resultant degrees of evidence provided by PANnet, we calculate the multiplier factors ${\rho }_{\omega }$, ${\rho }_V$ and ${\rho }_{Te}$ which, respectively, change the weights, $K_{\omega }=8$, $K_V=1$, $K_{Te}=0.05$. The dominant degree of evidence ${\mu }_{ER\_\omega }$ varies in the interval $[0,1]$, and when the output of Paraconsistent Logic Maximization Algorithm-$APl{C}_{Max}$ provides zero (or one) value bit, the multiplier factors vary according to Equation (19):

$${\rho }_{\omega }={\rho }_{Te}=2{\mu }_{ER\_\omega }+1$$

(19)

Likewise, since the value of the resultant evidence degree ${\mu }_{ER\_V}$ referring to the voltage is within the range, then the multiplier of the weight relative to the voltage is expressed by Equation (20):

$${\rho }_V=-{\mu }_{ER-V}+2$$

(20)

Once the weights and factors have been defined, it is possible to express the Q matrix in its complete form through Equation (21):

$$\mathrm{Q}=\left[\begin{array}{ccc}{\rho }_{\omega }{K}_{\omega }& 0& 0\\ 0& {\rho }_V{K}_{\omega }& 0\\ 0& 0& {\rho }_{Te}{K}_{Te}\end{array}\right]$$

(21)

• Field Voltage Setting Determination block (FVSD)

The Field Voltage Setting Determination Block (FVSD) operates using a matrix equation in which the field voltage variation is effectively obtained to achieve the required values of the controlled variables.

The selected variables, which are required to reach the optimum performance in voltage regulation at the generator terminals and the damping of electromechanical oscillations, are: (a) electromagnetic torque $T_e$; (b) voltage at the generator terminals $V_t$ and; (c) rotor speed ${\omega }_r$. In the representation of the PPC-PAL2v, these variables generate, through the outputs, $y(k+1)$, predicted by the synchronous generator model and the reference values, the reference trajectories that are processed in the optimization algorithm PAL2v. The constants, represented in the equations of the trajectories, are dependent on the initial conditions and the inductances of the system and are obtained in the small signal stability study.

4.5. MM-Mathematical Model of the Synchronous Generator

The PPC-PAL2v model must act to regulate the voltage at the generator terminals and damp the electromechanical oscillation directly related to the rotor speed variations. These two premises can be described mathematically through the output vector of the state space formulation, as shown Equation (22) [18]:

$$y(k)=\left[\begin{array}{l}{w}_r(k)\\ V_t(k)\end{array}\right]$$

(22)

where: $w_r$ is the rotor speed and $V_t$ is the voltage at the terminals of the synchronous generator.

To obtain the equations of the model in the MM (Mathematical Model of the synchronous Generator), the simplification of a synchronous machine with the inclusion of the dynamic effect of the field circuit is performed [10,12,18]. The state space matrix of the corresponding model is represented by Equation (23):

$$\left[\begin{array}{l}\Delta \stackrel{.}{{\omega }_r}\\ \Delta \stackrel{.}{\delta }\\ \Delta \stackrel{.}{{\mathsf{\Psi }}_{fd}}\end{array}\right]=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& 0& 0\\ 0& a_{32}& a_{33}\end{array}\right]\left[\begin{array}{l}\Delta \omega \\ \Delta \delta \\ \Delta \mathsf{\Psi }\end{array}\right]+\left[\begin{array}{ll}{b}_{11}& 0\\ 0& 0\\ 0& {\mathrm{b}}_{33}\end{array}\right]\left[\begin{array}{l}\Delta T_m\\ \Delta E_{fd}\end{array}\right]$$

(23)

where: ${\omega }_r$ is the rotor speed, $\delta$ is the power angle and ${\mathsf{\Psi }}_{fd}$ is the field flow.

The elements of the state matrix are function of the system parameters and the following equations show how they are obtained.

$$\begin{array}{l}{a}_{11}=\frac{K_D}{2H}{a}_{12}=\frac{K_1}{2H}{a}_{13}=\frac{-K_3}{2H}\\ a_{21}={\omega }_0{a}_{32}=-\frac{{\omega }_0{\mathrm{R}}_{fd}}{L_{fd}}{m}_1{L}_{ads}\\ a_{33}=-\frac{{\omega }_0{\mathrm{R}}_{fd}}{L_{fd}}\left[1-\frac{L_{ads}^{\prime }}{L_{fd}}+m_2{L}_{ads}^{\prime }\right]\\ b_{11}=\frac{1}{2H}{b}_{32}=\frac{{\omega }_0{\mathrm{R}}_{fd}}{L_{adu}}\end{array}$$

(24)

where: $K_1$, $K_2$, $m_1$ and $m_2$ are constants that depend on the parameters of the system and its initial conditions of operation. $K_D$ is the damping coefficient and $H$ is the inertia constant.

• Dynamic analysis of the synchronous generator

In the aspect of the dynamic analysis of the synchronous generator, the state variables are represented by the state space matrix. Therefore, the dynamics of the synchronous generator through the differential equations from these state variables is described as:

$$\mathrm{For}\mathrm{rotor}\mathrm{speed}\to {\omega }_r\to {\stackrel{.}{\omega }}_r(k)=\frac{1}{2H}\left(T_m(k)-T_e(k)\right)$$

$$\mathrm{For}\mathrm{power}\mathrm{angle}\to \delta \to \stackrel{.}{\delta }(k)={\omega }_0\Delta {\omega }_r(k)$$

$$\mathrm{For}\mathrm{field}\mathrm{flow}\to {\mathsf{\Psi }}_{fd}\to {\stackrel{.}{\mathsf{\Psi }}}_{fd}(k)={\omega }_0\Delta e_{fd}(k)$$

where $T_m$ is the mechanical torque, $T_e$ is the electromagnetic torque and $H$ is the inertia constant. Therefore, the state of the system can be described at any instant t, or step $k$, by the matrix represented by Equation (25):

$$x(k)=\left[\begin{array}{l}\delta (k)\\ {\omega }_r(k)\\ {\mathsf{\Psi }}_{fd}(k)\end{array}\right]$$

(25)

This modeling is performed through the equations of the reference trajectories, as explained below.

• Equations of the reference trajectories

The control signal, that is, the input $u(k)$ of the state space equations, is represented in the dynamic analysis of the synchronous generator by the voltage applied to the field circuit $e_{\mathrm{fd}}$.

The equations of the reference trajectories are expressed by measurable quantities of the synchronous generator. For the reference trajectory related to the voltage at the terminals, we have: $\Delta e_{Vt}(k+1)=\Delta V_t(k)-K_3\Delta \delta (k)-K_4\Delta {\mathsf{\Psi }}_{fd}(k)$.

Where the final analysis is performed by Equation (26):

$$\Delta e_{Vt}(k+1)=(-K_3{\omega }_0{\Delta }_t)\Delta {\omega }_r(k)+\Delta V_t(k)+(-K_4{\omega }_0{\Delta }_t)\Delta e_{fd}(k)$$

(26)

For the reference trajectory related to the speed of the synchronous generator, we have: $\Delta e_{\omega r}(k+1)=2\Delta \omega (k)-\Delta \omega (k-1)-\frac{{\Delta }_t}{2H}(K_1\Delta \delta (k)+K_2\Delta {\mathsf{\Psi }}_{fd}(k))$.

Where the final analysis is performed by Equation (27):

$$\Delta e_{\omega r}(k+1)=(-1)\Delta \omega (k-1)+\left(2-\frac{K_1{\omega }_0{\Delta }_t{}^2}{2H}\right)\Delta {\omega }_r(k)+\left(-\frac{K_2{\omega }_0{\Delta }_t{}^2}{2H}\right)\Delta e_{fd}(k)$$

(27)

Additionally, for the reference trajectory related to the electromagnetic torque of the generator, we have: $\Delta e_{Te}(k+1)=-\frac{2H}{{\Delta }_t}\Delta \omega (k-1)+\frac{2H}{{\Delta }_t}\Delta \omega (k)-K_1\Delta \delta (k)-K_2\Delta {\mathsf{\Psi }}_{fd}(k)$.

Where the final analysis is performed by Equation (28):

$$\Delta e_{Te}(k+1)=\left(-\frac{2H}{{\Delta }_t}\right)\Delta \omega (k-1)+\left(\frac{2H}{{\Delta }_t}-K_1{\omega }_0{\Delta }_t\right)\Delta \omega (k)+\left(-K_2{\omega }_0{\Delta }_t\right)\Delta e_{fd}(k)$$

(28)

Therefore, the reference trajectory in its matrix form is represented by Equation (29):

$$\Delta e_{Te}(k+1)={\mathrm{A}}_1\Delta \omega (k-1)+{\mathrm{A}}_2\Delta \omega (k)+{\mathrm{A}}_3\Delta V_t(k)+{\mathrm{E}}_2\Delta e_{fd}(k)$$

(29)

where the multiplier matrices used in the simplification are:

$$\begin{array}{l}{E}_2=\left[\begin{array}{c}-{\mathrm{K}}_4{\omega }_0\Delta t\\ \frac{-{\mathrm{K}}_2{\omega }_0\Delta t^2}{2H}\\ -{\mathrm{K}}_2{\omega }_0\Delta t\end{array}\right]\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{A}}_1=\left[\begin{array}{c}0\\ -1\\ \frac{-2H}{\Delta t}\end{array}\right]\\ {\mathrm{A}}_2=\left[\begin{array}{l}-{\mathrm{K}}_3{\omega }_0\Delta t\\ 2-\frac{{\mathrm{K}}_1{\omega }_0\Delta t^2}{2H}\\ \frac{2H}{\Delta t}-{\mathrm{K}}_1{\omega }_0\Delta t\end{array}\right]\hspace{1em}\hspace{1em}{\mathrm{A}}_3=\left[\begin{array}{l}1\\ 0\\ 0\end{array}\right]\end{array}$$

(30)

Reducing the matrix equation even more by joining the first three terms, results in Equation (31):

$${\mathrm{E}}_1={\mathrm{A}}_1\Delta \omega (k-1)+{\mathrm{A}}_2\Delta \omega (k)+{\mathrm{A}}_3\Delta V_t(k)$$

(31)

Equation (29) can be rewritten as Equation (32):

$$\Delta e_{Te}(k+1)={\mathrm{E}}_1+{\mathrm{E}}_2\Delta e_{fd}(k)$$

(32)

The quadratic cost function can be written as Equation (33):

$$\mathrm{J}=\Delta e^T\mathrm{Q}\Delta e+\Delta e_{fd}{}^T\mathrm{R}\Delta e_{fd}$$

(33)

By doing (32) in (33), we get the quadratic cost function represented as Equation (34):

$$\mathrm{J}={\left({\mathrm{E}}_1+{\mathrm{E}}_2\Delta e_{fd}(k)\right)}^T\mathrm{Q}\left({\mathrm{E}}_1+{\mathrm{E}}_2\Delta e_{fd}(k)\right)+\Delta e_{fd}{}^T\mathrm{R}\Delta e_{fd}$$

(34)

Rearranging the terms, it remains:

$$\mathrm{J}={\mathrm{E}}_1{}^T{\mathrm{QE}}_1+({\mathrm{E}}_1{}^T{\mathrm{QE}}_2+{\mathrm{E}}_2{}^T{\mathrm{QE}}_1)\Delta e_{fd}+({\mathrm{E}}_2{}^T{\mathrm{QE}}_2+\mathrm{R})\Delta e_{fd}{}^2$$

(35)

Applying the minimization of the function J, that is: $\frac{\delta \mathrm{J}}{\Delta e}=0$

$\left({\mathrm{E}}_1{}^T{\mathrm{QE}}_2+{\mathrm{E}}_2{}^T{\mathrm{QE}}_1\right)+2\left({\mathrm{E}}_2{}^T{\mathrm{QE}}_2+\mathrm{R}\right)\Delta e_{fd}=0$

The final equation is represented as Equation (36):

$$\Delta e_{fd}=\frac{-\left({\mathrm{E}}_1{}^T{\mathrm{QE}}_2+{\mathrm{E}}_2{}^T{\mathrm{QE}}_1\right)}{2\left({\mathrm{E}}_2{}^T{\mathrm{QE}}_2+\mathrm{R}\right)}$$

(36)

The Q matrix from Equation (21) is included in Equation (36) and was developed with the weights extracted from the analysis and modeling of PAL2v. In the model of the PPC-PAL2v, the factors obtained from the characteristics of the EPS are applied in Equation (36) to optimize the value for the field excitation control of the synchronous generator according to the variables involved in the process.

5. Results

In this work, two particular cases are presented for the performance evaluation of the PPC-PAL2v:

Case 1—Variation of 5% of the mechanical power of the generator.

Case 2—Variation of 2% in the reference voltage at the generator terminals.

Comparisons of results are made in relation to those obtained with the AVR and PSS, acting together and using the conventional MPC (Model Predictive Controller).

The results which are obtained for these two conditions with a prediction horizon N = 1 are shown below.

Results in case 1: At the 10 s instant, the synchronous generator begins to suffer a decrease in the mechanical torque of 0.018 pu/min, which, in nominal values of the generator, is equivalent to 10 MW/min. This causes an initial imbalance between mechanical torque and electric torque, and the machine rotor begins to decelerate.

Figure 9a shows the comparisons of velocity deviation, resulting in the 3 control methods: AVR+PSS, conventional MPC and the PPC-PAL2v.

The graphs show the behavior of the speed at the moment when the mechanical torque reaches its final value of −5% of its initial value.

Figure 9b shows the comparisons of values resulting from voltage at the generator terminals in the 3 control methods: AVR+PSS, conventional MPC and the PPC-PAL2v.

The graphs show the behavior of voltage values at the generator terminals at the moment when the mechanical torque applied reaches the stabilization.

Figure 10a–c show the degrees of evidence of the outputs of the Paraconsistent Analysis Network algorithms.

Figure 10d,e show the behaviors of the multiplier factors of weights that act in the Q Matrix in this case 1.

Results in case 2: The performance of the voltage at the generator terminals, in case 2, when a decrease of 2% of the reference voltage in the generator bus is realized at the instant of time 10 s.

Figure 11a shows the comparisons of velocity deviation resulting in the 3 control methods: AVR+PSS, conventional MPC and the PPC-PAL2v.

The graphs show the behavior of the speed in the moment when the mechanical torque reaches its final value of −2% of its initial value.

Figure 11b shows the comparisons of values resulting from voltage at the generator terminals in the 3 control methods: AVR+PSS, conventional MPC and the PPC-PAL2v.

The graphs show the behavior of voltage values at the generator terminals in the moment when the mechanical torque applied reaches the stabilization.

Figure 12a–c show the degrees of evidence of the outputs of the Paraconsistent Analysis Network algorithms.

Figure 12d,e show the behaviors of the multiplier factors of weights that act in the Q Matrix in this case 2.

6. Discussion

Case 1: In case 1, at the initial moment the velocity deviation, the PPC-PAL2v and conventional MPC controllers presented better performance than the AVR+PSS in terms of the peak values of the oscillations and the speed in the oscillation damping. As can be seen in Figure 9, the PPC-PAL2v and conventional MPC controllers reach a practically constant value after 12 s, where the speed is lower than the nominal value, while the AVR+PSS controller tends to have an accommodation with a lower speed than the other two.

The performance of the PPC-PAL2v cushions the electromechanical oscillation faster compared to the conventional MPC. In Figure 9a, in 160.4 s the mechanical torque reaches the final value, but electric torque continues to decrease in a short time, causing the acceleration of the rotor. Thus, its velocity increases, reaching stability in 162 s for PPC-PAL2v, 162.25 s for MPC and 163 s for AVR+PSS.

In Figure 9b, when the synchronous generator initiates the descent of the mechanical torque, the voltage at the terminals of the generator decreases and soon after an ascending ramp begins. The conventional MPC controller starts its rise by a value above the PPC-PAL2v, and the AVR+PSS controller has a higher slope than the other two controllers. When the mechanical torque reaches its final value, with increasing rotor speed, the voltage tends to rise in a small step and stabilizes within the 5% variation band determined by the controllers, with the value for PPC-PAL2v a little above the accommodation value for the conventional MPC, and below the value for the AVR+PSS.

Figure 10 shows the values resulting from the actions of the Paraconsistent Algorithms to perform the control in the circuit under study. We can verify that, for the optimized control obtained by PPC-PAL2v in case 1, there was no saturation of any output values of the PANs nor of the multiplier factors of the weight.

Case 2: It is observed in Figure 11a that, for the deviation in velocity, the PPC-PAL2v and MPC controllers presented better performance than the AVR+PSS for damping the oscillation. It is found that the MPC achieves stability after 12 s, while the PPC-PAL2v controller approaches the desired value faster. Regarding the voltage performance at the generator terminals in case 2, shown in Figure 11b, until the instant 10.3 s, the AVR+PSS controller has a more efficient response in the voltage adjustment than the PPC-PAL2v and MPC controllers, while the MPC controller walks faster to 0.98 p.u. than the PPC-PAL2v at the initial time. After 10.6 s, the PPC-PAL2v crosses 0.98 p.u., and accommodates quickly compared to AVR+PSS and MPC controllers.

Figure 12 shows the values resulting from the actions of the Paraconsistent Algorithms to perform the control in the circuit under study. We can verify that, for the optimized control obtained by PPC-PAL2v in case 2, there was saturation in some moments of the output values of the PANs and in the multiplier factors of the weight. However, the system was able to carry out the control more efficiently than the other methods.

7. Conclusions

In this paper, we presented a predictive controller, implemented with the fundamentals of Paraconsistent Annotated Logic (PAL) in their extended form, which we denoted Paraconsistent Annotated Logic with annotation of two values—PAL2v. This Paraconsistent Predictive Controller (PPC-PAL2v) was applied in the control of a Synchronous Generator in the case of stability to small signals. For the validation tests of the PPC-PAL2v, a mathematical model of a Single-Machine Infinite Bus (SMIB) was developed, as well as the model of an Automatic Voltage Regulator (AVR) in conjunction with a Power System Stabilizer (PSS) and a classical Model Predictive Control (MPC). After this, the results of the performances of the three control methods were compared. From the computational simulation of the proposed model, it was verified that the PPC-PAL2v effectively presented a superior result to the classical control of the AVR in conjunction with the PSS in the two cases studied. Regarding the MPC, which presents constant weight adjustment, the PPC-PAL2v was also superior due to its characteristic of variable adjustment in a continuous space of values with predetermined limits. The application of the algorithms of PAL2v, forming the Paraconsistent Analysis Network (PANnet) which acts on the stability of the small signals of power systems, is innovative and has shown very promising properties. The results found in this research will serve as a basis for new developments and future applications of the PAL2v in the control and automation of electric power systems.