Design and small signal stability enhancement of power system using interval type-2 fuzzy PSS

Abstract

In this paper the performance analysis of an interval type-2 fuzzy logic controller (IT2 FLC) as a power system stabilizer (PSS) is carried out. The input signals to the controller are considered as speed deviation and acceleration. The application of IT2 FLC as PSS (IT2 FPSS) is considered to improve small signal stability of power system. It is considered for three different benchmark systems such as single-machine infinite-bus (SMIB), two-area four-machine and IEEE ten-machine thirty nine-bus power system. The performance with IT2 FPSS is compared against the performance of the power system without PSS, with conventional PSS (CPSS) and with fuzzy PSS (FPSS). The application of IT2 FPSS is extended to two-area four-machine ten-bus and IEEE New England ten-machine thirty nine-bus power system and speed response comparison is carried out with FPSS. The superiority of performance with IT2 FPSS over type-1 FPSS is validated in terms of ITAE, IAE and ISE.

Keywords

Power system stabilizer fuzzy logic based power system stabilizer (FPSS)interval type-2 fuzzy power system stabilizer (IT2 FPSS)conventional PSS (CPSS)single-machine infinite-bus (SMIB)two-area four-machine power system IEEE New England ten-machine thirty nine-bus power system

1 Introduction

Electrical power systems (EPSs) are large, interconnected and complex. It generally consists of generating units, transmission lines and load centers. These are prone to disturbances of temporary or permanent in nature and may result to occur small signal oscillations. These small signal oscillations (SSOs) are of low in magnitude and frequency within the range of 0.2–3.0 Hz. Most of SSOs may be damped out but some of them may persist for a while, grow gradually and result to system separation. The EPS stability is to control excitation of the generator by mean of an automatic voltage regulator (AVR). However, the EPS is dynamically complicated, which often confront with the changes of operating conditions and disturbances.

To enhance dynamic performance, i.e. to provide fast damping to power system, a supplementary control signal in the excitation system to a generating unit can be used. As cheap, simple, easy to install; a power system stabilizer (PSS) has been widely used to suppress the SSOs and to improve dynamic stability of EPS. The most popular structure of PSS is a lead-lag type (phase compensator), whose gain and pole-zeros are determined using linear control theory with a linearized dynamic model of EPS. The PSS parameter adjustment and damping controller design are reported as root locus and sensitivity analysis method [1], pole placement [2] and robust control [3, 4]. The design of PSS with these techniques depends on linearized model and certain nominal operating conditions. Moreover, such designed PSS shows degraded performance of other operating conditions in case of large disturbances. Intelligent optimization based methods have been adopted to design PSS parameters such as genetic algorithm (GA) [5], tabu search (TS) [6], simulated annealing (SA) [7], differential evolution (DE) [8], particle swarm optimization (PSO) [9], evolutionary programming (EP) [10], gravitational search algorithm (GSA) [11], artificial bee colony algorithm (ABC) [12], bat algorithm [13], harmony search algorithm [14 , 16], have recently gained acceptance due to their effectiveness and the ability to investigate the near-global optimal results in problem space. Artificial neural network (ANN) based PSS design methods used the gradient algorithm for learning its parameters either with input/output data [17] or on-line at different operating points of the power system. In the category of online system parametric detection is presented using spectral analysis of online measured signals and detection of damping and oscillation frequencies using Phasor measurement unit (PMU) in power systems [18]. It presents installation of one PMU on each generator bus to determine network and generator parameters online. The estimated model is being used to observe the small signal stability of multimachine power system [18]. The identification of the inter-area oscillations in power system have been presented in [19], by employing continuous wavelet transform (WT) with complex Morlet function. The frequency and damping of power system can be identified using the instantaneous amplitudes and phase angles of the complex-valued signals determined using WT. The input signal to PSSs is determined and transmitted by PMUs and model analysis for simulations studies to enhance small signal stability is presented in [20]. However, it is neither economical nor possible to place all the buses of the system with PMUs because of their high cost and communication facilities [21].

Fuzzy logic based power system stabilizer (FPSS) have been developed to improve dynamic stability and adapt to changes when the operating point drifts as a result of continuous load changes or unpredictable major disturbances such as three-phase fault [22]. The performance of FPSS outperforms CPSS for improving the stability of the power system. The FPSS uses the human expert knowledge to design its prominent rule base, which usually consists of uncertainties to a certain degree [23, 24]. Zadeh in 1975 [25] have generalized the concept of ordinary fuzzy systems as reported [26], and termed as Type-2 (T2) fuzzy systems to handle uncertainties involved in type-1 fuzzy logic systems. Mendel et al. in [27], have extended this concept and explored many aspects of type-2 fuzzy sets and systems. The reported four possible sources of uncertainty present in type-1 fuzzy logic systems are (1) the antecedents and consequent of rules can be uncertain because linguistic terms mean different things to different people, (2) consequences may have a set of values associated with them, especially when knowledge is extracted from a group of experts, who do not agree at all, (3) measurements that activate a type-1 fuzzy logic system may be noisy and, therefore, uncertain and (4) the data that are used to tune the parameters of a type-1 fuzzy logic system may also be noisy. Recent research had reported the limitations of traditional type-1 fuzzy logic theory in treating large uncertainty factors due to unexpected severe faults on the power system [28, 29]. Application of type-2 fuzzy logic in various fields, represents compensation to the limitations that of with type-1 fuzzy logic [30, 31]. Castillo et al., have proposed GA and PSO optimized type-2 fuzzy logic controllers and demonstrated better performance as compared to type-1 controllers [32, 33]. Type-2 fuzzy logic is computationally intensive because its inference and type reduction are very intensive. Type-2 fuzzy set can be converted to interval type-2 by considering secondary memberships equal to zero or one [34]. The type-2 fuzzy logic controller based on the T2-fuzzy sets includes a third dimension and footprint of uncertainty which results to deal with both the linguistic and the numerical uncertainty. The type-2 fuzzy logic controller can obviously outperform its type-1 counterpart under the situation of high uncertainty [35]. Because of such better performances and applicability, interval type-2 fuzzy logic is considered for PSS design.

The later developed fuzzy, i.e. IT2 FS has a few applications in power system-related problems. In [36], fault currents of a power distribution system are calculated based on type-2 fuzzy numbers. The uncertainties associated with the fault current calculations of electrical distribution systems are carried out. A type-2 FLC and a type-2 FSMC (fuzzy sliding-mode controller) is proposed in [37] for control of a buck DC–DC converter. The IT2 FS based TCSC (thyristor controlled series capacitor) has been proposed in [38] to improve stability of the power system. Robandi and Kharisma [39], have presented the design of IT2 FPSS for two-area four-machine power system in decentralized manner. The effectiveness of IT2 FPSS is reported as compared to type-1 FPSS, IEEE PSS4B and IEEE PSS2B with settling time approximately 5.36 seconds. Panda et al. 2012 [40], have presented the application of IT2 FPSS to study small signal stability on SMIB and four-machine power systems. It considered a linearized model and has not added contingency such as faults to system for more non-linearity’s in the system. The application of IT2 FPSS is compared to optimally tuned CPSSs for both systems in [41]. The system response settles to steady-state value in between 4-5 seconds and around 10.5 seconds for SMIB and four-machine power systems, respectively. Meziane et al., 2015 [42], have proposed small signal stability enhancement for three-machine nine-bus power system. It presents hybrid application of optimal H∞ tracking control along with interval type-2 fuzzy control. The effectiveness is proved in terms of reduced overshoot and settling time. The overshoot and settling time are the graphical observations and may not give accurate information in non-linear system analysis. Therefore, the performance indices based analysis of system response is considered in this paper. A more complex IEEE benchmark system (10-machine 39-bus power system) is also considered to prove effectiveness of the proposed method. In [40], an Interval Type-2 fuzzy logic controller with Gaussian type-2 MF is proposed for designing an automatic voltage regulator (AVR) system. The choice of Gaussian type-2 MF is not validated.

The paper basically includes the performance analysis of IT2-FPSS with three power systems as controller for a wide range of operating conditions of the power system. The two inputs to the IT2 FPSS are taken as speed deviation (Δω) and acceleration ( $Δ \dot{ω}$ ) while the output is taken as stabilizing voltage (ΔV_pss) signal which is applied to the automatic voltage regulator. The application of IT2 FPSS is extended to two-area four-machine ten-bus and IEEE New England ten-machine thirty nine-bus power system and speed response comparison is carried out with FPSS. The superiority of performance with IT2 FPSS over type-1 FPSS is validated in terms of ITAE, IAE and ISE.

2 Problem formulation

The general representation of a power system using nonlinear differential equations can be given by $\dot{X} = f (X, U)$ (1)

Where, X and U represents the vector of state variables and the vector of input variables. As in [43 , 45], the power system stabilizers can be designed by use of the linearized incremental models of power system around an operating point. The system representation based on differential equations and used data is given in [15]. The state equations of a power system can be written as $Δ \dot{X} = A Δ X + BU$ (2)

2.1 Single-machine infinite-bus power system

The system under consideration consists of the single machine connected to an infinite bus (SMIB) through a transmission line. The infinite bus can be represented by the Thevenin’s equivalent of a large interconnected power system. The single line diagram of single-machine connected to infinite bus system is shown in Fig. 1.

The generator is being equipped with an external excitation system. This excitation system is equipped with an AVR and PSS to control the low-frequency oscillations produced due to lack of damping of the system’s mechanical mode. In case of inadequate damping torque, the system may lose stability. Since the high gain fast acting AVR leads to generate negative damping, therefore, to provide extra damping by using subsidiary excitation control like, PSS is developed [46 , 48]. The linearized model of SMIB was the result of a first serious investigation by DeMello and Concordia in 1969 [47]. In system representation by Equation 2, A is the system matrix with order as 4×4 and is given by δf/δX, while B is the input matrix with order 4×1 and is given by δf/δU. The order of state vector is 4×1, the order of is 1×1. Here, the well known Heffron-Phillip linearized model and the connection to FPSS with scaling factors is shown in Fig. 2 [43 , 49].

2.2 Two-area four-machine power system

Analysis of four-machine ten-bus power system as in Fig. 3 can be carried out by simultaneous solution of equations consisting of prime movers, synchronous machines with excitation systems, transmission line network, dynamic and static loads, and other devices like static VAR and HVDC converters based compensators. The dynamics of generator rotors, excitation, prime movers, and other related devices are being represented by differential equations. Thus, the complete multi-machine model consists of large numbers of ordinary differential equations (ODE) and algebraic equations [50, 51]. These are linearized about an operating point (nominal) to derive a linear model for the small signal oscillatory behaviour of power systems. The range of variation in operating point can generate a set of linear models corresponding to each operating point/condition.

The state equations of a power system, consisting ‘N’ number of generators and N_pss number of power system stabilizers can be written as in Equation 2. Where, A is the system matrix with order as 4N × 4N (16 × 16) & is given by δf/δX, while B is the input matrix with order 4N × N_pss (16 × 4) and is given by δf/δU. The order of state vector is 4N × 1 (16 × 1), the order of is N_pss × 1 (4 × 1). Here, the well known Heffron-Phillip linearized model is used to represent the large multimachine power system as in Fig. 4 [43 , 49].

2.3 IEEE ten-machine thirty nine-bus power system

The state equations of a power system, consisting N number of generators and N_pss number of power system stabilizers can be written as in Equation 2. Where, A is the system matrix with order as 4N×4N (40×40) and is given by δf/δX, while B is the input matrix with order 4N × N_pss (40 × 10) and is given by δf/δU. The order of state vector is 4N × 1 (40 × 1), the order of is N_pss × 1 (10 × 1). Here, the well known Heffron-Phillip linearized model is used to represent the large multimachine power system as in Fig. 5 [8].

2.4 Conventional power system stabilizer

In early 1960s, the fast-acting high-gain automatic voltage regulators were used to improve the transient stability but simultaneously reduced damping of the synchronous machines [49]. To improve the damping of the system, a supplementary damping torque proportional to rotor speed deviations is introduced to excitation control loop & was termed as power system stabilizer. The most widely accepted type of it is known as lead-lag type PSS or conventional PSS. It consists mainly, the gain block, the washout block & the lead-lag block & can be represented by the following equation. $Δ V_{pss} (s) = K_{pss} \frac{s T_{ω}}{1 + s T_{ω}} \frac{1 + s T_{1}}{1 + s T_{2}} Δ ω (s)$ (3)

The ΔV_pss, is the PSS output and added to the generator excitation as input to modulate the signal. The Δω, is the generator rotor speed deviation being sensed and applied as input to the PSS. In this paper, T_ω and T₂ are kept pre-specified as 10 seconds (range 2 to 15 secs.) and 0.027 seconds (range is 0 to 0.1 secs) [52].

3 Review on interval type-2 fuzzy logic system

The type-2 fuzzy sets are an extension of the concept of the type-1 which were introduced by Prof. Zadeh in 1965 [25]. An FLS that uses one type-1 fuzzy set is called a type-1 FLS while using one type-2 fuzzy set is called type-2 FLS.

The MFs of an IT2FLS are in fuzzy and also represents a footprint of uncertainty (FOU) as in Fig. 6, which is the representation of the area covered in between the upper MFs and lower MFs. FOU represents the capability to handle the uncertainty in the system. Thus, the IT2 FLS can handle and model both numerical and linguistic uncertainties associated with the inputs and outputs of the FLC. Accordingly, the IT2 FLS can give better performance than its counterpart T1 FLS [31 , 54].

3.1 Representation of type-2 FS and IT2 fuzzy sets

Let a type-2 fuzzy set in X is represented by $\tilde{A}$ . The associated membership grade of $\tilde{A}$ is represented by $μ_{\tilde{A}} (x, u)$ , where x ∈ X and u ∈ J_x ⊆ [0, 1], which represents a type-1 fuzzy set (T1 FS) in [0,1]. The domain elements of $μ_{\tilde{A}} (x, u)$ are called primary MFs of x in $\tilde{A}$ [55].

3.1.1 Type-2 FS

The T2 FS denoted by $\tilde{A}$ , is characterized by a type-2 MF $μ_{\tilde{A}} (x, u)$ , in which x ∈ X and u ∈ J_x ⊆ [0, 1], i.e

$\begin{matrix} \tilde{A} & = & [((x, u), μ_{\tilde{A}} (x, u)) | \forall x \in X, \\ \forall u \in J_{x} \subseteq [0, 1]] \end{matrix}$ (4)or $\tilde{A} = \int_{x \in X} \int_{u \in J_{X}} \frac{μ_{\tilde{A}} (x, u)}{(x, u)} | \forall J_{x} \subseteq [0, 1]$ (5)Here the range of $\tilde{A}$ can be represented as $0 \leq μ_{\tilde{A}} (x, u) \leq 1$ . At each value of x, say at x = x′, the vertical slice of $μ_{\tilde{A}} (x, u)$ can be represented by axes u and $μ_{\tilde{A}} (x^{'}, u)$ in 2-dimensional plane. Thus, $μ_{\tilde{A}} (x = x^{'}, u)$ for x ∈ X and ∀u ∈ J_x ⊆ [0, 1], can represented by $μ_{\tilde{A}} (x^{'}, u) = μ_{\tilde{A}} (x^{'}) \int_{u \in J_{X}} f_{x^{'}} (u) / u | J_{x^{'}} \subseteq [0, 1]$ (6)where in Equation 6 the f_x′ (u) is in between [0,1]. A T2 FS ( $\tilde{A}$ ), can be represented in terms of embedded fuzzy sets ( ${\tilde{A}}_{e}^{j}$ ). The ${\tilde{A}}_{e}^{j}$ denotes the j^th type-2 embedded fuzzy set of $\tilde{A}$ . $\tilde{A} = \sum_{j = 1}^{n} {\tilde{A}}_{e}^{j}$ (7)where $n \equiv \prod_{i = 1}^{N} M_{i}$ . Further ${\tilde{A}}_{e}^{j}$ can be represented by ${\tilde{A}}_{e}^{j} \equiv {[u_{i}^{j}, f_{x_{i}} (u_{i}^{j})], i = 1, 2, . . ., N}$ (8)where in Equation 8, $u_{i}^{j} \in (u ik), k = 1, 2, . . ., M_{i}$ . If the secondary MF f_x′ (u) =1 ∀ u ∈ J_X, then the T2 FS are called as interval type-2 fuzzy sets [29, 51].

3.1.2 Interval type-2 FS

If the T2 FS denoted by $\tilde{A}$ , is characterized by an IT2 MF as $μ_{\tilde{A}} (x, u) = 1$ , where x ∈ X and u ∈ J_x ⊆ [0, 1], i.e Equation 4. If the uncertainty of $\tilde{A}$ , is denoted by FOU, which is the union of the primary functions, as $FOU (\tilde{A}) = \cup_{x \in X} J_{X}$ . If the upper bound of membership function (UMF) is represented by ${\bar{μ}}_{\tilde{A}} (x)$ and the lower bound of membership function (LMF) is represented by ${\underline{μ}}_{\tilde{A}} (x)$ of $\tilde{A}$ is equal representation as two ‘type-1 MF’. Graphical representation of LMF and UMF for triangular MF and trapezoidal MF is represented in Fig. 7 [55], while the FOU is represented by the shaded area. The embedded fuzzy set ${\tilde{A}}_{e}^{j}$ if represented by a wavy line in Fig. 6.

3.2 Interval type-2 fuzzy logic systems (IT2FLS)

The representation of the IT2 FLS is similar to the type-1 FLS except the defuzzifier block of a type-1 FLS is replaced with the output processing block in a type-2 FLS as shown in Fig. 8. The output processing block in a T2 FLS consists of a type-reduction block and a defuzzification block.

The type-reduce block maps a T2 FS to a type-1 FS, while the defuzzifier block maps a fuzzy value to a crisp value. The membership function (MF) of an IT2 FS is called FOU which is represented by two MFs of a type-1 fuzzy set and are called as lower membership function(LMF) and upper membership function (UMF) as in Fig. 7 [55].

In this paper Mamdani type fuzzy inference system (FIS) is used, which is normally called as Max-Min method. In Max-Min method, 5-step operation is required to produce an output such as fuzzyfication, membership function (MF) operation (selection of MF), implication function (min-max), aggregation, and defuzzyfication. The defuzzyfication involves a mapping from fuzzy value in fuzzy control area to a crisp value (non-fuzzy) in output control area. The defuzzyfication process on an interval type-2 FLS uses the centroid method which is proposed by Karnik and Mendel [56].

If a lower membership function (LMF) is represented by ${\underline{μ}}_{\tilde{A}} (x)$ and an upper membership function (UMF) is denoted by ${\bar{μ}}_{\tilde{A}} (x)$ then the centroid (proposed by Karnik and Mendel) is represented by the interval between C_L and C_R. To determine C_L and C_R, the iterative method proposed by Karnik and Mendel can be used with the following steps [56].

The θ_i can be calculated by the following expression $θ_{i} = \frac{1}{2} {{\bar{μ}}_{\tilde{A}} (x) + {\underline{μ}}_{\tilde{A}} (x)}$ (9)

The value of c′ can be evaluated using $c^{'} = c (θ_{1}, θ_{2}, . . ., θ_{N}) = \frac{\sum_{i = 1}^{N} x_{i} \times θ_{i}}{\sum_{i = 1}^{N} θ_{i}}$ (10)

The value of k is determined such that x_k ≤ c′ ≤ x_k+1

If c " is used for finding C_L, then it can be determined by the following equation, $c " = \frac{\sum_{i = 1}^{k} x_{i} {\bar{μ}}_{\tilde{A}} (x_{i}) + \sum_{i = k + 1}^{N} x_{i} {\underline{μ}}_{\tilde{A}} (x_{i})}{\sum_{i = 1}^{k} {\bar{μ}}_{\tilde{A}} (x_{i}) + \sum_{i = k + 1}^{N} {\underline{μ}}_{\tilde{A}} (x_{i})}$ (11)If c " is used for finding C_R then is determined by the following equation, $c " = \frac{\sum_{i = 1}^{k} x_{i} {\underline{μ}}_{\tilde{A}} (x_{i}) + \sum_{i = k + 1}^{N} x_{i} {\bar{μ}}_{\tilde{A}} (x_{i})}{\sum_{i = 1}^{k} {\underline{μ}}_{\tilde{A}} (x_{i}) + \sum_{i = k + 1}^{N} {\bar{μ}}_{\tilde{A}} (x_{i})}$ (12)

If the value of c′ = c " then go to step-6 otherwise go (set) to step-3.

Put C_L = c′ or put C_R = c′ and calculate the mean of centroid, y by using $y = \frac{C_{R} + C_{L}}{2}$ (13)

3.3 Interval type-2 FMF generation

The used method to generate fuzzy membership function (FMF) is based on heuristics. This method generates the interval type-2 (IT2) FMF based on heuristic type-1 (T1) FMFs and a scaling factor. In this method pre-selected Type-1 FMF function, such as Gaussian, triangular, trapezoidal, S, or pi functions are used along with specific scaling factor, α. The frequently used heuristic membership functions are asfollowings.

Triangular Function $μ (x; a, b, c) = {\begin{matrix} 0, & if x \leq a \\ \frac{x - a}{x - b} & if a \leq x \leq b \\ \frac{c - x}{c - b} & if b \leq x \leq c \\ 0, & if x \leq c \end{matrix}$ (14)Trapezoidal Function $μ (x; a, b, c) = {\begin{matrix} 0, & if x \leq a \\ \frac{x - a}{x - b} & if a \leq x \leq b \\ 1, & if b \leq x \leq c \\ \frac{c - x}{c - b} & if b \leq x \leq c \\ 0, & if x \geq d \end{matrix}$ (15)The appropriate Type-1 FMF which is suitable for PSS design is selected out of MFs as in Equations 14-15 or the combinatorial function of these. The heuristic Type-1 FMF represented by μ (x) is designed by using the parameters (a,b,c) of the function. The selection of parameters is usually provided by experts. As in the previous discussion, the UMF can be represented by ${\bar{μ}}_{\tilde{A}} (x)$ of the IT2 FMF. The LMF denoted by ${\underline{μ}}_{\tilde{A}} (x)$ is determined using the scaling of the UMF by a factor α in between 0 and 1. The FOU for 0 ≤ α ≤ 1 can be expressed as

$\begin{matrix} \cup_{\forall x \in X} [{\bar{μ}}_{\tilde{A}} (x), {\underline{μ}}_{\tilde{A}} (x)] \\ = \cup_{\forall x \in X} [μ_{\tilde{A}} (x), α \times μ_{\tilde{A}} (x)] \end{matrix}$ (16)

The FOU can be determined by taking intersections of all lower MFs and upper MFs for all features as in Fig. 9. If the min operation as intersection is selected then the FOU can be represented by

$\begin{matrix} \cup_{\forall x \in X} [{\bar{μ}}_{\tilde{A}} (x), {\underline{μ}}_{\tilde{A}} (x)] \\ = \cup_{\forall x \in X} [\min {μ_{\tilde{A}} (x)}, \min {μ_{\tilde{A}} (x)}] \end{matrix}$ (17)where ${\underline{μ}}_{\tilde{A}} (x)$ and ${\bar{μ}}_{\tilde{A}} (x)$ are the minimum LMF and UMF among all LMFs and UMFs for all features, respectively. The lower MF is determined by appropriate scaling of the height of upper MF. The area in between the upper MF and scaled LMF represents the FOU for each class in consideration. The heuristic method can be summarized as follows.

The first step involves judicious selection of the heuristic Type-1 Fuzzy MF which is appropriate to represent a system.

Specify the parameters (a,b,c) for the selected MF which is generally supplied by an expert.

The UMF and LMF are designed Equations 16 and 17.

4 Simulation results and discussion

4.1 Performance with SMIB system

As in previous section, due to the superior performance of triangular type-2 fuzzy membership function is used to study the performance of SMIB system over a wide range of operating conditions of it. Eight operating points of the SMIB power system are considered with different sets of active power (P_g0) and transmission line reactions (X_e) as [0.5, 0.5, 0.75, 0.75, 1.0, 1.0, 1.10, 1.20] and [0.2, 0.4, 0.2, 0.4, 0.2, 0.4, 0.4, 0.4], respectively.

Models of power system and IT2 FPSS are simulated in an SIMULINK working environment of MATLAB. The output of PSS is generally restricted to ± 0.25 pu; hence consequent parameter limit values are set to ± 0.25 pu. The plant responses are recorded for system without PSS, with conventional PSS (CPSS) and with IT2 FPSS (itraitype2 mf) [57, 58]. The SMIB power system is simulated for all plant conditions (Plants 1-8) without PSS, with CPSS [52] and with proposed IT2 FPSS. However, the graphical response comparison is shown only with plant-2, plant-4 and plant-7 in Fig. 10, Figs. 11 and 12, respectively. The speed response of the system is compared in terms of settling time and presented in Table 1.

The response of the SMIB power system settles with IT2 FPSS at 5.36 seconds with fault at 0.5 seconds [39]. In [40, 41], IT2FLCPSS is damping the oscillations of rotor speed deviation at approximate 10 seconds as compared to approximate 16 seconds that of with PSOLLCPSS and PSOFLCPSS, respectively. The settling time for a wide range of operating conditions is in the range of 10.5 seconds to 14.0 seconds, including fault application at 5.0 seconds and clearing at 5.1 seconds in Table 1. Therefore, the effective settling time is in the range of 5.4 seconds to 8.9 seconds. The system considered is in this paper includes nonlinearity; while in [40, 41] is linear. It proves its effectiveness and applicability in practical power system applications as compared to [39 , 41].

4.2 Performance with four-machine system

The two-area four-machine ten-bus power system is considered as described in [56] and the creation ofsystem models based on operating conditions. The FPSS [59] and IT2 FPSS (Proposed) are connected to the system and simulation carried out for speed response. In each plant condition as listed in Table 2 is considered with fault location at different bus. The disturbances is considered as self clearing at different buses at 1.0 second and cleared after 0.05 second. The different Plant 1-8 is differentiated with fault location at different bus as bus-3, bus-4, bus-5, bus-6, bus-7, bus-8, bus-9 and bus-10, respectively. The speed response of four-machine system without PSS for plant-2 configuration is shown in Fig. 13 and found that none of generators of plants is showing stable operation, therefore, not needed to compare in the simulation results. As a sample, the speed response of Gen-1 –Gen-4 for plant-1 is compared with FPSS [59], SPEA-CPSS [60] and IT2 FPSS (Proposed) in Figs.14 –17. Speed response of four-machine system with FPSS [59], SPEA-CPSS [60] and IT2 FPSS (Proposed) considering all eight plant conditions is recorded in Table 2. It is not possible to show comparative response to all plant conditions because of space constraints. However, the response of system for all eight plants is shown in Figs. 18 –21 for Gen-1 to Gen-4, respectively to show robust operation of the proposed controller.

As the speed response of four-machine system with SPEA-CPSS [60], FPSS [59] and IT2 FPSS (Proposed) are not shown graphically for plant configurations but the performance indices of speed is recorded during each simulation carried out for 40 seconds and being recorded in Table 2 [13, 14]. It is clear that the performance of system with IT2 FPSS is greatly improved as compared to FPSS [59] and SPEA-CPSS [60].

4.3 Performance with ten-machine system

Speed response of New England system with IT2 FPSS (Proposed) is carried out by simulation of nonlinear plant configurations. The plant 1-8 are subjected with different fault location as at bus no. 16, 13, 11, 9, 7, 17, 19 and 21, respectively. The way of creating plants is well illustrated in [16]. To show the hardness of the system condition, plot for speed response without PSS with plant-1 configuration is shown in Fig. 22. It is clear that none of the generator is showing stable response, therefore, not included in the forthcoming comparative responses. The performance of the proposed controller is to be compared that of with FPSS [61], cultural algorithm based CPSS [62], adaptive mutation breeder genetic algorithm based CPSS [63], strength pareto evolutionary algorithm based CPSS [60]; therefore the speed response for plant-1 configuration is recorded and compared in Figs. 23 –26 for Gen-1, Gen-5, Gen-7 and Gen-10, respectively for plant-1 configuration. In each response of Gen-1, 5, 7, 10, the response with IT2 FPSS (Proposed) is superior as compared to response with FPSS [61], CA-CPSS [62], ABGA-CPSS [63], SPEA-CPSS [60]. The simulation results with all comparing controllers for all plant conditions is not possible because of space constraint, therefore, speed response of system with IT2 FPSS for all plant conditions is shown in Figs. 27–30 for Gen-1, Gen-5, Gen-8 and Gen-10, respectively. it is clear that the proposed IT2 FPSS is able to stabilize all plant conditions (covering wide range of operating conditions and different system configurations distinguished by separate fault location in each case).

To evaluate the robustness of the proposed IT2 FPSS, simulation is carried out for all eight plant configurations which represent the wide range of operating conditions and system configurations [16]. The system is simulated with FPSS [61], CA-CPSS [62], ABGA-CPSS [63], SPEA- CPSS [60] for comparison purpose with eight plant conditions. Each time the performance indices (ITAE, IAE and ISE) [13, 14] are recorded and enlisted in Table 3. Since the system possess ten generators, therefore the PI values recorded for a particular type in these tables is the sum of PIs of ten generators. Comparatively lower value of PI refers to better performance. It is clear from this table, that the performance of the system is enhanced by using proposed IT2 FPSS (Proposed) as compared to performance with FPSS [61], CA-CPSS [62], ABGA-CPSS [63], SPEA- CPSS [60].

5 Conclusion

In this paper, an interval type-2 fuzzy logic based power system stabilizer (IT2 FPSS) is designed and the performance is evaluated for three systems such as single-machine infinite-bus power system (SMIB), two-area four-machine ten-bus power system and IEEE New England ten-machine thirty nine-bus powersystems.

The itriatype-2 mf based IT2 FPSS is considered to compare speed response over eight plant conditions of SMIB power system and compared to response of CPSS [52] and system without PSS. It is found that the IT2 FPSS outperforms the considered CPSS [52].

An IT2 FPSS is designed by considering rule base in [57] and itritype-2 mf and connected to four-machine system in decentralized manner. The such designed IT2 FPSS, FPSS [[59]] and SPEA-CPSS [60] are subjected to simulate over eight plant conditions and found that the IT2 FPSS outperforms over FPSS [57] and SPEA-CPSS [58]. The comparative results are validated by considering performance indices of speed response of system with IT2 FPSS and FPSS [59], SPEA-CPSS [60].

Similarly, An IT2 FPSS is designed by considering rule base in [59] and itritype-2 mf; it is connected to ten-machine system in decentralized manner. The such designed IT2 FPSS and FPSS [[59]] are subjected to simulate over eight plant conditions and found that the IT2 FPSS outperforms over FPSS [61], CA-CPSS [62], ABGA-CPSS [63], SPEA- CPSS [60]. The comparative results are validated by considering performance indices of speed response of system with IT2 FPSS and FPSS [61], CA-CPSS [62], ABGA-CPSS [63], SPEA- CPSS [60].

Footnotes

Acknowledgments

This research was supported by All India Council of Technical Education, New Delhi, India. I am grateful to University College of Engineering, Rajasthan Technical University, Kota for sponsoring me under Quality Improvement Programme. I also thank my colleagues for sharing the responsibility at the parent Institute during my stay at Roorkee.

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