Stability enhancement of PSS-UPFC installed power system by support vector regression

Abstract

Ensuring the transient stability of the power system networks is one of the prime challenges in the highly interconnected power systems. Though the low-frequency oscillations are not very harmful initially, failure to damping out may lead the system to go out of synchronism. The employment of flexible AC transmission systems (FACTS) devices may suppress these oscillations effectively in addition to the enhancement of power transfer capability. Among many FACTS devices, unified power flow controller (UPFC) is one of the most sophisticated ones. Tuning the parameters of power system stabilizer (PSS) coordinated with UPFC for a stable system is a multi-objective optimization problem. This paper aims to optimize the parameters of power system stabilizer (PSS) of power network incorporating UPFC using support vector regression (SVR) in real time to damp out the small signal oscillations hence to enhance the transient stability. System eigenvalues obtained from SVR tuned UPFC coordinated PSS and the fixed gain conventional PSS are compared to investigate the efficacy of the proposed technique for different loading conditions. Besides, time domain simulation comparison proves the superiority of the proposed technique over the conventional one. Furthermore, the statistical performance measures for training and testing datasets provide confidence on the developed SVR model.

1 Introduction

Nowadays most of the power system networks are operating almost at their stability limit as the expansion of the systems required to fulfill the increasing demands are not met for most of the cases due to the constraint of limited resources and environmental factors. The power systems which are interconnected through weak tie lines give rise to poorly damped low-frequency oscillations those ranges from 0.1 to 3 Hz [1]. If adequate damping is not provided immediately after their commencement these low-frequency oscillations keep increasing, eventually causing the system to reach out of synchronism. To solve this problem, the control of generator excitation employing automatic voltage regulator (AVR) is one of the solutions to damp out these low-frequency oscillations. But it has the drawback of creating low-frequency oscillations by decreasing rotor damping torque [2]. Consequently, power system stabilizers (PSS) are being used extensively in order to enhance power system stability against low-frequency oscillations [3] but it may fail to suppress the severe disturbances like three phase fault and may affect the voltage profile as well [4].

On the contrary, the application of flexible alternating current (AC) transmission systems (FACTS) has become prevalent with the advancement of power electronics. The capability of very fast power electronic-based control action has made the FACTS devices a strong candidate for improvement in power system damping in addition to improving power transfer capability [3, 5,6,7]. FACTS devices enhance the system stability by controlling the dynamic states of system parameters which include voltage, current, series and shunt impedance, phase angle, and damping of low-frequency oscillations [8]. Applications of FACTS devices have been reported for various control objectives including optimal power flow (OPF) [9], voltage stability [10], damping inter-area low-frequency oscillations [11]. Prominent series structured controllers include thyristor controlled series compensator (TCSC), static synchronous series compensator (SSSC), the thyristor controlled series reactor (TCSR) etc. Among the shunt controllers static VAR compensator (SVC), static synchronous compensators (STATCOM) and the thyristor controlled reactor (TCR) are widely used. The unified power flow controller (UPFC) is the combined unit which takes advantages of independent series and shunt controllers [12].

UPFC is the most popular and promising second-generation FACTS device and capable of improving transient stability, providing voltage support, reducing power loss, controlling the power transfer among transmission lines and improving damping of low-frequency oscillations [13]. Using a power flow controller, supplementary damping controller and a DC voltage regulator, UPFC does its works. In order to guarantee system stability during disturbances, it is required to maintain proper coordination between UPFC and PSS. A great number of researches have been reported for the coordination among PSS and FACTS devices [14,15,16]. Different artificial intelligence (AI) techniques were being employed in power system industries in order to solve many complex problems as well as optimizing the parameters of PSS coordinated with FACTS employing backtracking search algorithm (BSA) [16], genetic algorithm(GA) [17], differential evolution (DE) [18], ant colony optimization (ACO) [19], particle swarm optimization (PSO) [20] to improve the power system stability by damping out the small signal oscillations. However, most of these techniques work in the offline mode as these required a very long time to get the optimized parameters for different operating conditions. To estimate the parameters of PSS coordinated with FACTS devices different machine learning techniques, for instance, artificial neural networks, support vector regressions can be employed effectively.

Support vector machines (SVM) can efficiently and effectively solve the multidimensional, nonlinear and complex function estimation problems. Vapnik [21] was the first to introduce it. SVM’s are now widely used for both classification and regression problems and the machines dedicated to regression problems are renamed as support vector regression (SVR). Power system industry is also employing these machine to solve many complex problems including stability forecasting and prediction [22], fault location [23], assessment of voltage instability [24]. Although artificial neural network can be used for solving the multidimensional optimization problem, sometimes it suffers from rule obscureness and local convergence [25].

As mentioned earlier, the PSS parameters coordinated with UPFC need be optimized in real-time fashion as the loading conditions of electric networks change continually. The trained SVR model can optimize PSS settings within very short period of time (within two cycles of power system) for any loading condition. The faster computational speed and higher accuracy make the proposed SVR approach suitable for online parameter estimation of PSS coordinated with UPFC. Therefore, this paper contributes to the development of an SVR-based technique for the tuning of optimal parameter setting of PSS connected with UPFC in a single machine infinite bus (SMIB) electric network. In this paper, the performance of the SVR algorithm is evaluated to estimate the parameters of PSS coordinated with UPFC in real time depending on varying operating conditions to enhance the stability of power system.

The rest of the paper is organized as follows: Sect. 2 presents the detail mathematical model of the system as well as the PSS-UPFC-based control architecture. Optimization problem formulation and data generation strategy have been presented in Sect. 3. Section 4 describes details of support vector regression technique used for problem-solving. Simulation results have been shown and discussed in Sect. 5 and finally, the conclusion is drawn in Sect. 6.

2 Power system dynamic model

2.1 Power system model

A single machine infinite bus (SMIB) system equipped with UPFC has been considered [14] as shown in Fig. 1 where a generator is connected to an infinite bus through the transmission line. UPFC is connected to the power system via excitation transformer (ET) and boosting transformer (BT).Two voltage source converters (VSCs), namely VSC-B and VSC-E are connected through a DC-link common capacitor.

Fig. 1

SMIB system equipped with UPFC [20]

This arrangement works as an ideal bi-directional AC-to-AC power converter. VSC-B is connected in series with the line through boosting transformer and does the work of injecting voltage of variable magnitude. Thus, the output voltage from VSC-B is seen as synchronous AC voltage source. On the other hand, the main task of VSC-E is to supply or absorb the real power which is demanded by VSC-B at the common dc-link in order to support the real power exchange resulting from the series voltage injection. It can also absorb or generate controllable reactive power, if needed, providing the line an independent shunt reactive compensation.

There are four input control parameters of the UPFC which are given by $m_{E}, m_{B},\delta _{E }$ and $\delta _{B}$. Amplitude modulation ratio plays a key role in all power electronics devices which actually reflects the ratio between modulation amplitude and peak amplitude of the given signal to FACTS devices.

The nonlinear model of SMIB can be represented by the following three differential equations (1)–(3) [26]:

$$\begin{aligned}&\dot{\delta }=\omega _b (\omega -1) \end{aligned}$$

(1)

$$\begin{aligned}&\dot{\omega }=\frac{1}{2H}[P_m -D(\omega -1)-P_e ] \end{aligned}$$

(2)

$$\begin{aligned}&\dot{E}_q^{\prime }=\frac{1}{T_{do}^{\prime } }\left[ E_{fd} -(x_d -x_d^{\prime } )i_d -E_q^{\prime } \right] \end{aligned}$$

(3)

Terminal voltage $({v}_{t})$ and output electrical power $(P_{e})$ can be written in terms of the direct axis and quadrature axis voltage and current as given below:

$$\begin{aligned} P_e= & {} v_d i_d +\hbox { }v_q i_q \end{aligned}$$

(4)

$$\begin{aligned} v_t= & {} \sqrt{\left( {v_d^2 +v_q^2 } \right) } \end{aligned}$$

(5)

$v_{d }$ and $v_{q}$ can be expressed as the following:

$$\begin{aligned} v_d= & {} x_q i_q , \\ v_q= & {} E_q {^{\prime }}-x_d {^{\prime }}i_d , \\ i_q= & {} i_{Eq} +i_{Bq} ,i_d =i_{Ed} +i_{Bd} \end{aligned}$$

Taking the d–q axis, four current quantities $i_{Ed},i_{Eq}, i_{Bd}, i_{Bq}$ could be found.

Fig. 2

Block diagram of the excitation system [16]

2.2 Exciter and power system stabilizer (PSS)

Institute of Electrical and Electronics Engineers (IEEE) permits some of the standard-type excitation systems. The one used for the proposed system model is shown in Fig. 2. An AVR has been used to provide excitation to the system. The Power System Stabilizer (PSS) is the widely used lead–lag controller which contains a lead–lag block along with a limiter. PSS also includes a washout filter to suppress the steady-state quantity of the input signal. The output of the PSS is a supplementary stabilizing control signal $(U_\mathrm{{PSS}})$ which is found when a change in angular speed $(\Delta \omega )$ is given as the input in lead–lag compensator.

This excitation system can be represented by the state equation:

$$\begin{aligned} \Delta \dot{E}_{fd} =K_A ((V_\mathrm{{ref}} -v_{t}+U_\mathrm{{pss}})-E_{fd} )\frac{1}{T_A } \end{aligned}$$

(6)

2.3 The dynamic model of the UPFC

The UPFC model has been obtained after park transformation and neglecting the resistances as well as the transients of the Excitation and Boosting transformers [27].

$$\begin{aligned} \dot{v}_{dc}= & {} \frac{3m_E }{4C_{dc} }(\cos \delta _E i_{Ed} +\sin \delta _E i_{Eq} )\nonumber \\&+\frac{3m_B }{4C_{dc} }(\cos \delta _B i_{Bd} +\sin \delta _B i_{Bq} ) \end{aligned}$$

(7)

The currents of excitation and boosting transformers $(i_{Ed},\ i_{Eq},\ i_{Bd},\ i_{Bq})$ in terms of other different parameters can be represented as follows:

$$\begin{aligned} i_{Ed}= & {} \frac{x_{BB} }{x_{d\sum } }E_q {^{\prime }}-\frac{m_E \sin \delta _E v_{dc} x_{Bd} }{2x_{d\sum } } \\&+\frac{x_{dE} }{x_{d\sum } }\left( v_b \cos \delta +\frac{m_B \sin \delta _B v_{dc} }{2}\right) \\ i_{Eq}= & {} \frac{m_E \cos \delta _E v_{dc} x_{Bq} }{2x_{q\sum }} \\&-\frac{x_{qE} }{x_{q\sum } }\left( v_b \sin \delta +\frac{m_B \sin \delta _B v_{dc} }{2}\right) \\ i_{Bd}= & {} \frac{x_E }{x_{d\sum } }E_q {^{\prime }}+\frac{m_E \sin \delta _E v_{dc} x_{dE} }{2x_{d\sum } } \\&-\frac{x_{dt} }{x_{d\sum } }\left( v_b \cos \delta +\frac{m_B \sin \delta _B v_{dc} }{2}\right) \\ i_{Bq}= & {} -\frac{m_E \cos \delta _E v_{dc} x_{qE} }{2x_{q\sum }} \\&+\frac{x_{qt} }{x_{q\sum } }\left( v_b \sin \delta +\frac{m_B \cos \delta _B v_{dc} }{2}\right) \end{aligned}$$

where

$$\begin{aligned} {x}_{{qt}}= & {} {x}_{q} +{x}_{{tE}} +{x}_{E} ,{ x}_{{qE}} ={x}_{q} +{ x}_{{tE}} ,\\ {x}_{{dt}}= & {} {x}^{\prime } _{d} +{x}_{{tE}} +{x}_{E} ,{ x}_{{dE}} \\= & {} {x}^{\prime } _{d} +{x}_{{tE}} ,{x}_{{BB}} ={x}_{B} +{x}_{{BV}} , \\ {x}_{{q}\sum }= & {} {x}_{{qt}.} {x}_{{BB}} +{ x}_{e} .{ x}_{{qE}} \\= & {} ({x}_{q} +{x}_{{tE}} +{x}_{E} )({x}_{B} +{x}_{{BV}} )+{x}_{e} ({x}_{q} +{x}_{{tE}} ), \\ {x}_{{Bq}}= & {} {x}_{{BB}} +{ x}_{{qE}} ={ x}_{B} +{x}_{{BV}} +{x}_{q} +{x}_{{tE},} \\ {x}_{{d}\sum }= & {} {x}_{{dt}} {x}_{{BB}} +{ x}_{E} {x}_{{dE}} \\= & {} ({x}^{\prime } _{d} +{x}_{{tE}} +{x}_{E} )({x}_{B} +{x}_{{BV}} )+{x}_{E} ({x}^{\prime } _{d} +{x}_{{tE}} ), \\ {x}_{{Bd}}= & {} {x}_{{BB}} +{ x}_{{dE}} ={ x}_{B} +{x}_{{BV}} +{x}^{\prime } _{d} +{x}_{{tE}} \end{aligned}$$

The constant values of reactance used for the modeling purpose are provided in appendix section.

2.4 UPFC coordinated PSS (damping controller)

The damping controller produces an electrical torque that is in phase with speed deviation using phase compensation technique. The damping torque is controlled by the four control parameters of UPFC such as $\delta _{B}, m_{B}, m_{E }$ and $\delta _{E}$. The lead–lag damping controllers equipped with UPFC are of the structure as presented in Fig. 3 [28], where the output of the UPFC is the control signal input (u). The value of control signal(u) can be any one of the signals mE, mB or $\delta B$.

The DC voltage is regulated by changing the voltage phase angle $(\delta _{E})$ of the excitation transformer (ET). The modulated value of $\delta _{E}$ is obtained from the controller shown in Fig. 4 where PI controller acts as a voltage regulator.

Fig. 3

Lead–lag controller-based UPFC [28]

Fig. 4

Lead–lag controller and DC voltage regulator with UPFC

2.5 Linearized model

For controller design, the linear model of the system has been considered. The nonlinear dynamic equations presented in (1)–(7) can be linearized by taking a small variation of the parameters around a given operating point and the linearized system can be written in state-space form as [27]:

$$\begin{aligned} \dot{X}= A\Delta X + B\Delta U \end{aligned}$$

(8)

Here, A and B are the constant matrices of the system which is directly dependent upon the operating state of the system. The full forms of the matrices of equations 8 are presented in “Appendix C”.

The following state equations can be derived from the block diagram of Fig. 4 which will be further added to state Eq. (8) at the time of finding out the closed loop state matrix $A_{c}$.

$$\begin{aligned}&\Delta \dot{y}_1 =\left[ K \Delta \dot{\omega } T_w -\Delta y_1 \right] \frac{1}{T_w } \end{aligned}$$

(9)

$$\begin{aligned}&\Delta \dot{y}_2 =\left[ \Delta y_1 +\Delta \dot{y}_1 T_1 -\Delta y_2 \right] \frac{1}{T_2 } \end{aligned}$$

(10)

$$\begin{aligned}&\Delta \dot{c}=\left[ -\Delta c+\Delta \dot{\delta }_E T_3 +\Delta y_2 \right] \frac{1}{T_4 } \end{aligned}$$

(11)

$$\begin{aligned}&\Delta \dot{L}_2 =K_p \Delta \dot{L}_1 +K_i \Delta L_1 \end{aligned}$$

(12)

$$\begin{aligned}&\Delta \dot{\delta }_E =\left[ K_3 \Delta L_3 -\Delta \delta _E \right] \frac{1}{T_s } \end{aligned}$$

(13)

The Eigenvalues of the state matrix $A_{c }$ will represent the modes of the system after being subjected to any small disturbance. The negative real part will ensure the stable condition of the system. For the case of an unstable situation with positive real part of any of the Eigenvalues, the system can be brought into the stable state again by using the output or the state feedback controller. In that case, the state equation will be modified in the following form containing closed loop state matrix $A_{c}$:

$$\begin{aligned} \dot{X}= A_c X \end{aligned}$$

(14)

Eventually, the dimension of the matrix $\mathbf A _{c}$ will be 9 by 9 which will provide nine eigenvalues and the details about this matrix can be found in [28, 29]. It is worth mentioning that in this paper the PI controller block of Fig. 4 is kept deactivated throughout the simulation period and the considered state variables of equation (14) are $\Delta \delta ,\Delta \omega , \Delta E^{\prime }_{q},\Delta E_{fd} , \Delta V_{dc},\Delta C,\Delta y_{1},\Delta y_{2},$ and $\Delta \delta _{E}$. However, the $\mathbf A _{c}$ matrix generated eigenvalues can be shifted to the negative part of the complex plane ensuring the desirable performance of the system through the proper selection of UPFC coordinated PSS parameters.

3 Optimization problem formulation

In the optimization problem, two objective functions are used which are basically based on Eigenvalues of the states. First one focuses on improving the damping factor of the system and the other one will ensure the satisfactory value of damping ratio. The combination of these two objective functions, $J_{1}$ and $J_{2}$ will form the multi-objective function J which can be represented as follows [30]:

$$\begin{aligned} J= & {} J_1+ \alpha J_2 \end{aligned}$$

(15)

$$\begin{aligned} J= & {} \sum _{\sigma _{mn} \ge \sigma _0 } {(\sigma _0 -\sigma _{mn} )^{2}} +\alpha \sum _{\zeta _m \ge \zeta _0 } {(\zeta _0 -\zeta _m )} ^{2} \end{aligned}$$

(16)

where m represents the Eigenvalue index of the system and n is the index of system operating conditions. $\sigma _{mn}$ is the Eigenvalue’s real part upon which relative stability of the system mostly depends. The damping ratio is represented by $\zeta$. Two objective functions are combined by a weighting factor $\alpha$ and the value has been chosen 10 in this work.

Fig. 5

Eigenvalue locations for corresponding objective function [30]

The graphical representation of each of the objective functions is presented in Fig. 5. Figure 5a reflects that the closed loop Eigenvalues will be forced toward the negative half plane and will be placed in the indicated dashed region if the objective function $J_{1}$ has been taken only. In the same way, Fig. 5b shows that the maximum overshoot of the Eigenvalues is controlled by choosing only $J_{2}$ as the objective function and will limit it within the particular dashed region. When the objective function J is optimized, because of the effect of $J_{1}$ and $J_{2}$, the generated Eigenvalues of the system will be confined inside the D-shaped shaded region as presented in Fig. 5c.

This will ensure the stability of the system by meeting the requirements of the required damping factor and ratio for a stable system. Real parts of the Eigenvalues will be forced to be placed in the negative half plane within a specific overshoot range. So, the optimization problem can be stated as:

$$\begin{aligned} \mathrm{{Minimize}}\qquad J \end{aligned}$$

Subject to

$$\begin{aligned} K^{\min }\le K \le K^{\max } \\ T_{1}^{\min }\le T_{1}\le T_{1}^{\max } \\ T_{2}^{\min }\le T_{2}\le T_{2}^{\max } \\ T_{3}^{\min }\le T_{3}\le T_{3}^{\max } \\ T_{4}^{\min }\le T_{4}\le T_{4}^{\max } \end{aligned}$$

To ensure the stability of the system during transients in the power system the optimum value of the controller parameters must be determined aiming to optimize objective function J. In this work objective function optimization translates into maximizing the minimum value of the damping ratio subject to some inequality constraints. The constraints for this optimization problem are lower and upper limit of time constants $(T_{1}-T_{4})$ and controller gain (K).

Subject to the same five constraints stated above to keep the controller gain and four-time constants of lead–lag compensators within the desired range. The optimum parameters of the controller are predicted online employing trained and tested SVR.

4 Proposed methodology

4.1 Data generation

The loading conditions comprising of the per unit values of three variables of synchronous machine, namely real power $(P_{e})$, reactive power $(Q_{e})$ and terminal voltage $(V_{t})$, have been used for data generation and the ranges of these variables are:

$$\begin{aligned}&0.40\le P_{e}\le 1.10 \\&-0.30\le Q_{e}\le 0.30 \\&0.90\le V_{t}\le 1.10 \end{aligned}$$

With a view to training and testing the SVR, a set of one thousand optimal parameters $(K \& T_{1})$ for PSS coordinated with UPFC were generated for different loading conditions employing backtracking search algorithm (BSA) by setting some constant values to $T_{2}$, $T_{3,}$ and $T_{4}$ as the parameters $T_{1}-T_{4}$ are related to each other. The main operators of BSA are initialization, selection-I, mutation, crossover, and selection-II. Though BSA shares the similar name for the operators with genetic algorithm (GA), but the mutation and crossover operations of BSA are completely different from GA. Additionally, the boundary control mechanism is different from GA or other evolutionary techniques. The details about BSA can be found in [31,32,33,34]. Real power $(P_{e})$, reactive power $(Q_{e})$ and terminal voltage ($V_{t})$ representing the loading conditions were taken as the inputs (independent variables) whereas BSA tuned optimal PSS parameters $(K \& T_{1})$ coordinated with UPFC were taken as the targets (dependent variables) to the SVR. From the prepared dataset 70% of data have been selected randomly for training and rest of them were selected for testing the SVR model.

4.2 Support vector regression model

The methodology of SVR follows Structural Risk Minimization (SRM) principle which tries to minimize the upper bound on the expected risk. This has been found better than empirical risk minimization process, implied by conventional neural networks to minimize the error of the training data [35]. The training data of support vector regression (SVR) is mapped in a high-dimensional feature space which is also known as Hilbert space and turns the nonlinear regression function into a linear one by proper mapping. Because of the following of SVR, it can perform well even with the fewer number of data samples.

Formation of data set is the first step for training model followed by testing phase. Let us consider $\{(x_1 ,y_1 ),\ldots ,(x_m,y_m )\}$ as a set of training data where each of the $x_i$ represents m-dimensional input space of the sample and $y_i$ is the corresponding outputs (target values) for $i=1,2,\ldots ,m$ and m represents the size of the training data. If the fitting function is denoted by f(x), then the SVR can reach the target y for the corresponding testing sample x through f(x) after the training will be done.

With these training data, SVR forms the mapping function (estimation function) like [36]:

$$\begin{aligned} f(x)=w\cdot \phi (x)+b \end{aligned}$$

(17)

Here $\omega$ is the weight vector, b represents the bias and $\phi (x)$ is the high-dimensional space mapped from the input space $x_i$. In order to get the estimated value of $\omega$ and b, minimizing the following regression risk function is required:

$$\begin{aligned} R_{reg} (f)=C\sum _{i=0}^m {\psi (f(x_i )-y_i )+\frac{1}{2}\left\| w \right\| ^{2}} \end{aligned}$$

(18)

where $\psi (.)$ the cost function and C is a particular constant. Substituting the value of the weight vector

$$\begin{aligned} w=\sum _{i=1}^m {(a_i -a_i^*)\phi (x_i )} \end{aligned}$$

in Eq. (17) will give us the modified generic equation:

$$\begin{aligned} f(x)= & {} \sum _{i=1}^m {(a_i -a_i^*)(\phi (x_i )} \cdot \phi (x))+b\nonumber \\= & {} \sum _{i=1}^m {(a_i -a_i^*)k(x_i ,x)+b} \end{aligned}$$

(19)

The dot product of Eq. (19) can is replaced by $k(x_i ,x)$ which is known as Kernel function that helps to use the dot product in low- dimensional space data input without knowing $\phi (x)$. Besides, $\varepsilon \hbox {-insensitive}$ loss function must have to be satisfied in order to ensure robustness of the regression result as well as to ensure the sparsity of the solution. The function is as follows [37]:

$$\begin{aligned} \psi (f(x)-y)=\left\{ \begin{array}{ll} \left| {f(x)-y}\right| -\varepsilon , &{} \mathrm{{for}}\left| {f(x)-y} \right| \ge \varepsilon \\ 0, &{} \mathrm{{otherwise}} \\ \end{array}\right. \end{aligned}$$

(20)

The problem of regression risk in Eq. 18 and $\varepsilon \hbox {-insensitive}$ loss function of Eq. 20 can be solved using quadratic optimization problem and can be represented as:

$$\begin{aligned}&\frac{1}{2}\sum _{i,j=1}^m {(a_i^*-a_i )(a_j^*-a_j )} k(x_i ,x_j )\\&-\sum _{i=1}^m {a_i^*(y_i -\varepsilon )-a_i^ (y_i +\varepsilon )} \end{aligned}$$

Subject to

$$\begin{aligned} \sum _{i=1}^m {a_i -a_i^*=0, a_i ,a_i^*} \in [0,C] \end{aligned}$$

Here, in the optimization problem formulation presented above, $a_i$ and $a_i^*$ are the Lagrange multipliers which represent the solution of the quadratic function. These help the prediction to reach the targeted values $y_i$. Nonzero values of these Lagrange are known as support vectors and they are actually responsible for forecasting the regression line properly. It happens only when the condition $\left| {f(x)-y} \right| \ge \varepsilon$ is satisfied. Thus, the value of w can be obtained.

In order to get the value of constant b, well-known KKT (Karush Kuhn Tucker) condition needs to be applied which says that the multiplication of constraints and the Lagrange multipliers will be zero.

$$\begin{aligned}&a_i (\varepsilon +\xi _i -y_i +(w,x_i )+b)=0 \\&a_i^*(\varepsilon +\xi _i^*+y_i -(w,x_i )-b)=0 \\&(C-a_i )\xi _i =0 \\&(C-a_i^*)\xi _i^*=0 \end{aligned}$$

Here, $\xi _i$ and $\xi _i^*$ are the slack variables which are used to calculate the errors $\varepsilon$ outside the boundary. Bias constant b could be found from the above equations as:

$$\begin{aligned}&a_i =a_i^*=\xi _i^*=0\\&b=y_i -(w,x_i )-\varepsilon for a_i \in (0,C) \\&b=y_i -(w,x_i )+\varepsilon for a_i^*\in (0,C) \end{aligned}$$

Thus the SVR will be ready to search for the target for some provided input dataset without even knowing the transformation. The penalty C, imposed for estimation error needs to be examined so as the value of $\varepsilon$ which will decide the specific data to be ignored while doing regression.

In this paper, the proposed SVR model optimizes two key parameters of the PSS coordinated with UPFC, namely gain (K) and time constant $(T_{1})$.It is worth mentioning that the key issue in designing PSS coordinated with UPFC is to ensure appropriate phase lead which can be achieved by keeping $T_{2}$, $T_{3}$, and $T_{4}$ as constants and optimizing K and $T_{1}$. The complete flow diagram of the work procedure is shown in “Appendix D”.

5 Simulation results and discussions

5.1 Performance of support vector regression

In this paper, three variables of synchronous machine, namely real power $(P_{e})$, reactive power $(Q_{e})$ and terminal voltage $(V_{t})$ were taken as the inputs and corresponding optimized PSS parameters $(K \& T_{1})$ with UPFC were taken as outputs. The number of training and testing data, as well as the required constant parameters for SVR, are presented in Table 1 which has been used in this work.

With a view to examining the performance of the proposed SVR in estimating PSS parameters with UPFC, mean squared error (MSE), root mean squared error (RMSE), mean absolute percentage error (MAPE) and the coefficient of determination $(\textit{R}^{2})$ were determined for each parameter. Mathematical formulas used for the error measures are presented in “Appendix A”. The smaller values of MSE, RMSE and MAPE refer the strength of the proposed method. Additionally, the value ‘1’ for $R^{2}$ indicates a perfect match between actual and predicted values whereas the value ‘0’ indicates no match at all. The statistical measures for the proposed technique are tabulated in Table 2, where smaller values of MSE, RMSE, and MAPE give confidence over developed technique. Furthermore, the table also shows the values of ${R}^{2}$ are very close to 1.00 for all the parameters which indicate a very good match of between the BSA optimized generated data and SVR estimated values.

Table 1 Parameter used for SVR model

Table 2 Recorded statistical measures of SVR model

5.2 Required time to estimate UPFC coordinated PSS parameters

The simulation was carried out in MATLAB environment and in an Intel Xeon (3.06 GHz -2 Processors, 24 GB RAM) workstation computer. As the SVR takes different times for training and testing purposes for each run, the average training and testing times for one hundred runs were 58.0938 and 0.0169 s, respectively. Furthermore, the average time required for calculation of initial values and determination of optimal parameters for one hundred loading conditions were also recorded and times were 0.0025 and 0.0021 s, respectively. So the overall time required for SVR to estimate optimized values of UPFC is 0.0046 s which is less than 2 cycles of a 60 Hz electric system. As the proposed technique can stabilize a disturbed network in less than two cycles of time, the technique can be implemented in the real-time application.

Table 3 Comparison of Eigenvalues for nominal loading

Table 4 Comparison of Eigenvalues for light loading

Table 5 Comparison of Eigenvalues for heavy loading

Table 6 Damping ratio comparison for different loading conditions

5.3 Eigenvalues and minimum damping ratio comparison

The SMIB with UPFC was simulated for three loading conditions, namely nominal, light and heavy loading conditions. System stability can be determined through analysis of eigenvalues. Eigenvalues for three different loading conditions, namely nominal, light and heavy loading conditions for both SVR optimized PSS with UPFC and fixed gain conventional PSS with UPFC are tabulated in Tables 3, 4 and 5. The tables show all the eigenvalues have negative real part means the systems are stable but the fixed gain PSS with UPFC needs more time to settle the disturbance down than the optimized one as its complex eigenvalues are relatively close to the imaginary $(\hbox {j}\omega )$ axis.

Obtained minimum damping ratios for fixed gain conventional PSS with UPFC, SVR optimized PSS with UPFC and BSA optimized PSS with UPFC (offline) are tabulated in Table 6 for different loading conditions. It is clear from the table that optimized parameters have better values than conventional fixed gain one which signals that better damping will be achieved with optimized PSS with UPFC. However, as BSA-based optimization is carried out is offline, it cannot be applied in real-time application.

Both minimum damping ratios and eigenvalues comparisons clearly signal the better performance of SVR optimized PSS with UPFC over conventional fixed gain one in settling down the small frequency oscillations after being subjected to any disturbances in real-time operation of power system network.

5.4 Time domain simulation results with disturbance

The performance of SVR optimized PSS with UPFC for damping small frequency oscillations in case of disturbance has been compared with conventional one having fixed gain. To simulate the disturbance a 10% pulse input of mechanical torque has been applied at 1.0 s for 4 cycles and corresponding system responses have been observed. Both conventional and SVR optimized PSS with UPFC stabilize the power system network after being subjected to mechanical disturbance. However, SVR optimized PSS with UPFC exhibits faster response compared to the conventional one as shown in Figs. 6, 7 and 8 for nominal loading condition.

Fig. 6

Response of $\Delta \delta$ for SVR tuned and fixed gain power system stabilizer coordinated with UPFC

Fig. 7

Response of $\Delta \omega$ for SVR tuned and fixed gain power system stabilizer coordinated with UPFC

Fig. 8

Response of $\Delta V_{dc}$ for SVR tuned and fixed gain power system stabilizer coordinated with UPFC

Figures 6, 7 and 8 reflect the transient behavior of three of the system states $\Delta \delta$, $\Delta \omega$, and $\Delta V_{dc}$, respectively. When the PSS parameters are selected randomly for fixed gain PSS with UPFC, the response of the states take very long time to reach the stable state. In each of the three figures below, it is reflected that the system couldn’t reach the stable stage even within the 10 s time span provided in the simulation. Whereas oscillations in $\Delta \omega$ states become stable in around 2.7 s (Fig. 7) and $\Delta \delta$ in 4 s (Fig. 6) when the control parameters are optimized with proposed SVR optimization.$\Delta V_{dc}$ requires a bit higher time than others but still makes the system oscillation-free within around 5 s (Fig. 8). It signifies the effect of using SVR algorithm in the field of power system stability.

6 Conclusions

Investigating the real-time performance of UPFC coordinated PSS connected power system subjected to low-frequency oscillations is the main objective of this paper. Support vector regression model estimates the parameters of PSS coordinated with UPFC in real-time fashion. The performance of proposed technique is tested for three different loading conditions. In addition, the obtained results are compared with the fixed gain PSS coordinated with UPFC. From the eigenvalue analysis and time domain representation of system parameters, it is seen that proposed controller outperforms the conventional one in all respect. Furthermore, it is found that the time required for the SVR model to tune the controller parameters is less than two cycles of a 60 Hz power system. Thus, in terms of computational speed and accuracy, the proposed model presents its efficiency as a real-time optimizer and ensures robustness in power system stability enhancement. Besides, satisfactory values of the statistical performance measures for training and testing datasets provide confidence on the proposed SVR model. However, the idea presented in this paper can be further extended for tuning the PSS parameters of a multimachine power system network.

Appendices

Appendix A

System parameters used are given below:

Generator and transmission line:

$\hbox {M} = 8 \hbox {MJ/MVA}$, $\hbox {D}=0, \hbox {x}_{\mathrm{d}}= 1.0 \hbox {pu}$, $\hbox {x}_{\mathrm{q}}= 0.6 \hbox {pu}$, $\hbox {T}^{\prime } _{\mathrm{d}0}= 5.044 s$, $\hbox {x}^{\prime }_{\mathrm{d}}= 0.3 \hbox {pu},\omega _{\mathrm{b}} =377 \hbox {rad/s}$ and $\hbox {X}_{\mathrm{L}} = 0.1 \hbox {pu}$

Machine excitation system: $K_{\mathrm{A}}=100$, $\hbox {T}_{\mathrm{A}}=0.01s$

Transformer: $\hbox {X}_{\mathrm{T}}= 0.1 \hbox {pu}$, $\hbox {X}_{\mathrm{E}}= 0.1 pu$, and $\hbox {X}_{\mathrm{B}}= 0.1 \hbox {pu}$

DC-link:

$\hbox {v}_{\mathrm{DC}}= 2 \hbox {pu}$ and $\hbox {C}_{\mathrm{DC}} = 1.2 \hbox {pu}$

parameters (Fixed gain):

UPFC: $\updelta _{\mathrm{E}} =68.113 \hbox {deg}$, $\delta _{\mathrm{B}} =41.12 \hbox {deg}$, $\hbox {m}_{\mathrm{B}} = 0.96$, $\hbox {m}_{\mathrm{E}}= 0.7667$

PSS: $K=15.71, T_{1}= 0.3~\hbox {s}, T_{2 }=0.3$ s, $T_{3}= 0.39~\hbox {s}, T_{4}= 0.6623$ s

PSS parameters: $1\le K\le 50, 0.01 \le T_{1}\le 1.0$

$$\begin{aligned}&\mathrm{{Eigenvalue}}:\lambda =\sigma +j\omega \end{aligned}$$

(A.1)

$$\begin{aligned}&\mathrm{{Damping Ratio}}: \zeta =-\frac{\sigma }{\sqrt{\sigma ^{2}+\omega ^{2}}} \end{aligned}$$

(A.2)

Appendix B

The performance of the proposed technique was tested with different well-known error measures including mean squared error (MSE), root mean squared error (RMSE), mean absolute percentage error (MAPE) and the coefficient of determination $({R}^{2})$.

For total n data samples, actual values $y_{a}$ and predicted values $y_{p}$, mathematical formulas for the error measures are presented below:

$$\begin{aligned}&\mathrm{{MSE}}=\sum _{i=1}^n {((y_a )_i -} (y_p )_i )^{2} \end{aligned}$$

(A.3)

$$\begin{aligned}&\mathrm{{RMSE}}=\sqrt{\frac{\sum _{i=1}^n {((y_a )_i -} (y_p )_i )^{2}}{n}} \end{aligned}$$

(A.4)

$$\begin{aligned}&\mathrm{{MAPE}}=\frac{1}{n}\sum _{i=1}^n {\left| {\frac{(y_a )_i -(y_p )_i }{(y_a )_i }} \right| \times 100} \end{aligned}$$

(A.5)

$$\begin{aligned}&R^{2}=1-\sum _{i=1}^n {\frac{((y_a )_i -(y_p )_i )^{2}}{((y_a )_i -\bar{{y}}_a )^{2}}} \end{aligned}$$

(A.6)

where $\bar{{y}}_a$ is the mean of actual value.

Appendix C

The state vector $\Delta X = [\Delta \delta ,\Delta \omega , \Delta E^{\prime }_q,\Delta E_{fd}, \Delta V_{dc}]^{ T}$

Control vector $\Delta U = [\Delta U_\mathrm{{pss}},\Delta m_{E},\Delta \delta _{E},\Delta m_{B},\Delta \delta _{B}]^{ T}$

$$\begin{aligned} A= & {} \left[ {{\begin{array}{ccccc} 0&{}\quad {\omega _b }&{}\quad 0&{}\quad 0&{}\quad 0 \\ {-\frac{K_1 }{M}}&{}\quad {-\frac{D}{M}}&{}\quad {-\frac{K_2 }{M}}&{}\quad 0&{}\quad {-\frac{K_{pd} }{M}} \\ {-\frac{K_4 }{T{^{\prime }}_{d0} }}&{}\quad 0&{}\quad {-\frac{K_3 }{T{^{\prime }}_{d0} }} {\frac{1}{T{^{\prime }}_{d0} }}&{}\quad {-\frac{K_{qd} }{T{^{\prime }}_{d0} }} \\ {-\frac{K_A K_5 }{T_A }}&{}\quad 0&{}\quad {-\frac{K_A K_6 }{T_A }}&{}\quad {-\frac{1}{T_A }}&{}\quad {-\frac{K_A K_{vd} }{T_A }} \\ {K_7 }&{}\quad 0&{}\quad {K_8 }&{}\quad 0&{}\quad {-K_9 } \\ \end{array} }} \right] \\ B= & {} \left[ {{\begin{array}{ccccc} 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad {-\frac{K_{pe} }{M}}&{}\quad {-\frac{K_{pde} }{M}}&{}\quad {-\frac{K_{pb} }{M}}&{}\quad {-\frac{K_{pdb} }{M}} \\ 0&{}\quad {-\frac{K_{qe} }{T{^{\prime }}_{d0} }}&{}\quad {-\frac{K_{qde} }{T{^{\prime }}_{d0} }}&{}\quad {\frac{K_{qb} }{T{^{\prime }}_{d0} }}&{}\quad {-\frac{K_{qdb} }{T{^{\prime }}_{d0} }} \\ {\frac{K_A }{T_A }}&{}\quad {-\frac{K_A K_{ve} }{T_A }}&{}\quad {-\frac{K_A K_{vde} }{T_A }}&{}\quad {-\frac{K_A K_{vb} }{T_A }}&{}\quad {-\frac{K_A K_{vdb} }{T_A }} \\ 0&{}\quad {K_{ce} }&{}\quad {K_{cde} }&{}\quad {K_{cb} }&{}\quad {K_{cdb} } \\ \end{array} }} \right] \end{aligned}$$

Appendix D

1.1 Stability enhancement method:

The remote terminal unit (RTU) at synchronous generator end measures the real power, reactive power, and terminal voltage continually and based on these values the proposed method estimates the initial conditions and then the PSS parameters with UPFC are optimized through SVR. The following flowchart illustrates the whole online optimization process (Fig. 9).

Fig. 9

Flowchart of proposed SVR-based approach