Saad Dilshad

Received: 18 October 2019 Revised: 20 February 2020 Accepted: 21 March 2020 DOI: 10.1002/- RESEARCH ARTICLE Adaptive Type-2 NeuroFuzzy wavelet-based supplementary damping controls for STATCOM Rabiah Badar | Saad Dilshad Department of Electrical and Computer Engineering, COMSATS University, Islamabad, Pakistan Correspondence Rabiah Badar, Department of Electrical and Computer Engineering, COMSATS University, Islamabad, Pakistan. Email:- Summary Static Synchronous Compensator (STATCOM) is a shunt-connected Flexible AC Transmission System (FACTS) controller with the primary goal of power flow control on a transmission line. However, it can effectively be used for improving power system stability by incorporating a Supplementary Damping Control (SDC) for bus voltage regulation. This paper presents an SDC strategy based on the integration of Type-2 fuzzy logic and Wavelet Neural Network Funding information Higher Education Commision, Pakistan, Grant/Award Number: 2EG2-034 (WNN) in NeuroFuzzy structure. The fuzziness in Type-2 membership function increases the controller capability to handle uncertainties. The performance improvement has been investigated by introducing uncertainty in mean initially and then in both the mean and SD of the Gaussian Type-2 membership function. A multi-machine power system installed with SDC-based STATCOM is used to test the controller performance using different fault scenarios. The comparative evaluation of proposed controllers is done with Adaptive NeuroFuzzy TSK Control (ANFTSKC) and without SDC scenarios. In results, significant performance improvement has been observed in both steady-state and transient regions for recommended Type-2 NeuroFuzzy Wavelet Control (NFWC) strategies. KEYWORDS fuzzy Type-2, gradient descent, multi-machine power system, NeuroFuzzy, STATCOM, wavelet neural network 1 | INTRODUCTION The growing population and rising energy demand have caused significant problems for power systems and the global environment. Non-conventional renewable energy sources are assisting in meeting the rapid growth in supply and demand for electricity; however, their integration with conventional grid brings several new challenges of power Abbreviations: ANFTSKC, adaptive NeuroFuzzy TSK control; ANN, artificial NN; AVRs, automatic voltage regulators; FACTS, flexible AC transmission system; DE, differential evolution; FIS, fuzzy inference system; IAE, integral absolute error; ICA, imperialist competitive algorithm; ISE, integral square error; ITAE, integral time absolute error; ITSE, integral time square error; LFOs, low frequency oscillations; NFWC, NeuroFuzzy wavelet control; NNs, neural networks; PIs, performance indices; PSO, particle swarm optimization; PSSs, power system stabilizers; PWM, Pulse Width modulation; SDC, supplementary damping control; SOA, Seeker Optimization algorithm; STATCOM, static synchronous compensator; STD, standard deviation; SVD, Singular Value decomposition; TSK, Takagi-Sugeno-Kang; VSC, voltage source converter; WNN, wavelet neural network. Int Trans Electr Energ Syst. 2020;e12429. https://doi.org/10.1002/- wileyonlinelibrary.com/journal/etep © 2020 John Wiley & Sons Ltd 1 of 25 2 of 25 BADAR AND DILSHAD quality, reliability, power system stability, harmonics, etc.1 Reliable and secure operation of a power system depends on its ability to withstand various contingencies, faults, and different environmental factors. Low-Frequency Oscillations (LFOs) are critical for the uninterrupted operation of power system.2 LFOs, once started in a power system, may grow with time and lead to partial or complete system blackout. Interruptions in power system operation and cascaded failures due to LFOs have been reported in the literature.3 Automatic Voltage Regulators (AVRs) along with Power System Stabilizers (PSSs) are being used as a common damping approach for several years. It has been found that AVRs regulate the voltage effectively but show poor damping performance for LFOs. On the other hand, PSSs may fail to stabilize the power system due to performance vulnerability for significant variations in operating conditions.4 This led to the emergence of other alternatives to damp LFOs such as Flexible AC Transmission System (FACTS) controllers. FACTS are useful in power flow control with the additional capability of damping LFOs using suitable Supplementary Damping Control (SDC). FACTS controllers and grid-connected energy storage devices are the key solutions for the continuously varying load demand and the centralized nature of electricity.5 The use of FACTS controller has resulted in reliable transmission system operation with the least amount of infrastructure expenditure and minimum environmental influence. Thus, making the whole power system cost-effective and environment friendly. Many FACTS controllers are being used for power system stability enhancement. However, Static Synchronous Compensator (STATCOM) has found vast applications due to its capability of bus voltage magnitude regulation and LFOs damping utilizing the efficiently designed SDC. Its output current is fully controllable to operate it in inductive or capacitive mode by absorbing or generating the reactive power, respectively.4,6 Researchers have proposed different linear, non-linear, and artificial intelligence-based SDCs like Particle Swarm Optimization (PSO),7 Seeker Optimization Algorithm (SOA),8 Imperialist Competitive Algorithm (ICA),9 Singular Value Decomposition (SVD),10 Differential Evolution (DE),11 PI,12 etc., to damp power system oscillations. However, linear controllers fail to retain the performance over a wide range of operating conditions, while non-linear control designs are intricate and generally involve the tedious task of the complete system model. Therefore, soft computing-based control approaches like fuzzy systems and Neural Networks (NNs) appear as the sophisticated choice for SDC design.13,14 Fuzzy Inference System (FIS) is a computing system based on fuzzy set theory, fuzzy if-then rules, and fuzzy reasoning. Artificial NN (ANN) can learn by adjusting the interconnection between the hidden neurons. NeuroFuzzy is the synergy of two paradigms: FIS and ANN. The NeuroFuzzy systems have gained the widespread interest of researchers working in the field of engineering and applied sciences. NeuroFuzzy-based controllers have good approximation and control ability. Another fast-growing research trend in the area of NeuroFuzzy is to embed wavelets in their structure to improve their learning performance and rectify the vulnerabilities of learning algorithms. Wavelet-based NeuroFuzzy systems combine wavelet theory with FIS and ANN which embeds the optimal approximation property of WNN in NeuroFuzzy stucture.15 Nowadays, instead of using conventional lead-lag control, researchers have proposed intelligent control techniques like Fuzzy, NeuroFuzzy, Fuzzy Sliding-Mode Control (SMC)-based control for STATCOM. A MultipleInput Single-Output (MISO) and Multi-Input Multi-Output (MIMO)-based NeuroFuzzy wavelet control for STATCOM and Static Synchronous Series Compensator (SSSC) has been proposed in Reference,16 taking generator speed and power deviations and their derivatives as a control input for fuzzy controller. The Modified PSO (MPSO)-based NeuroFuzzy damping controllers for STATCOM with Battery Energy Storage System (BESS) has been proposed in Reference.17. In Reference,18 the NeuroFuzzy SMC-based controllers have been introduced for STATCOM for power stability enhancement. A Legendre wavelet-based NeuroFuzzy damping control for multiple FACTS has been proposed in Reference,19 and the test system includes both SSSC and STATCOM in fourmachine system. These control strategies revealed promising results of integrating wavelets in NeuroFuzzy structures. However, these NeuroFuzzy control strategies have Takagi-Sugeno-Kang (TSK)-based structure having membership functions with fixed mean and SD (STD) in the antecedent part. On the other hand, Type-2 based fuzzy control with uncertain mean and SD provides better management of complex systems with uncertainties that make them suitable for highly dynamic plants.20 The optimization of interval Type-2 fuzzy systems employing genetic algorithms and PSO are presented in Reference.21,22 Furthermore, Type-2 NeuroFuzzy controllers have been proposed for a variety of applications like control of; mobile robot navigation,23 induction motors,24 flexible joint manipulators,25 solid-state transformers,26 electro-hydraulic servo system,27 belt grain dryer,28 power system,16,29,30 and time-series predictions,31,32. BADAR AND DILSHAD 3 of 25 Getting motivated by these interesting results of improving the antecedent and consequent part of NeuroFuzzy controls, the authors have proposed the synergistic integration of Type-2 systems and WNNs to damp LFOs using STATCOM.33 However, the proposed NeuroFuzzy wavelet control considered the Gaussian membership function with the uncertainty introduced in the mean parameter only. In this paper, the work has further been extended by introducing uncertainty in both mean and STD parameters. Also, the detailed mathematical description of both schemes has been given in this work. The parameters of the proposed Type-2 NFWC schemes are adapted online, without using any offline training data, by applying a gradient descent-based backpropagation algorithm. This paper is structured as follows: the detailed mathematical modeling of power system installed with STATCOM is presented in 2. Algorithmic description of both Type-2 NFWCs is presented in 3. The simulation results are discussed in Section 4 and a brief conclusion with a future perspective of this research is presented in the last section. 2 | MODELING OF POWER SYSTEM WITH STATCOM The dynamic model of ith synchronous machine with q axis leading d axis can be found in Reference30,34 and it is sufficient to study the STATCOM dynamic model for damping control of power system oscillations.30 Figure 1 shows a STATCOM installed on bus l between nodes 1 and 2. STATCOM has been modeled as a controllable voltage source in series with resistance and inductance on the AC side, while as capacitance connected in parallel with current source on DC side. The output of Voltage Source Converter (VSC) in phasor form, is given as: STC = εvDC ðcosγ + jsinγ Þ = εvDC ∠γ V ð1Þ where vDC is the capacitor voltage; ε = ck where c is modulation index for Pulse Width Modulation (PWM) inverter and k is AC and DC voltage ratio of the converter. γ = θ + ϑ controls the exchange of reactive power with θ and ϑ being synchronization and firing angle of the PWM inverter, respectively. The DC-side voltage of the STATCOM is given as: C dvDC = ε I d cosγ + I q sinγ : dt ð2Þ The complete AC- and DC-side models of STATCOM are given as:35 d RSTC 2 1 ia = − ia ðt Þ− vDC ðt Þsinðωt + ϑÞ + va LSTC s dt s πLSTC LSTC S ð3Þ d RSTC 2 1 ib s = − i b s ðt Þ − vDC ðt Þsin ωt −1200 + ϑ + vb LSTC dt πLSTC LSTC S ð4Þ d RSTC 2 1 ics = − ics ðt Þ− vDC ðt Þsin ωt + 1200 + ϑ + vc LSTC dt πLSTC LSTC S ð5Þ 1 d 2 vDC = iaS sinðωt + ϑÞ + ibS sin ωt −1200 + ϑ + icS sin ωt + 1200 + ϑ − vDC , dt πC RC ð6Þ where RSTC is the resistance representing total losses of transformer windings and converter; LSTC is the leakage inductance of the coupling transformer; ϑ is the firing angle; and R and C are the resistance and capacitance of the DC-side of the STATCOM, respectively. The fundamental frequency model given in Equations (3)-(6) can be generalized in the following state-space form: 4 of 25 FIGURE 1 BADAR AND DILSHAD Closed-loop system x_ ðt Þ = Ex ðt Þ + Fvðt Þ where, x ðt Þ = ½iaS ibS icS vDC T and vðt Þ = ½vaS vbS vcS 0T ð7Þ BADAR AND DILSHAD 5 of 25 2 6 6 6 6 6 E=6 6 6 6 6 4 − RSTC LSTC 0 −k1 sinðωt + ϑÞ 0 3 7 7 7 RSTC 0 −k1 sinðωt + ϑ− 1200 Þ 7 0 − 7 LSTC 7 7 RSTC 0 7 0 0 − −k1 sinðωt + ϑ + 120 Þ 7 LSTC 7 5 1 0 0 − k2 sinðωt + ϑÞ k2 sinðωt + ϑ −120 Þ k2 sinðωt + ϑ + 120 Þ RC 2 1 0 0 6 LSTC- F =6 LSTC 6 1 6 0 0 4 LSTC 0 0 0 ð8Þ 3 0 7 7 7 07 7, 7 7 07 5 ð9Þ 1 where k1 = 48=2πL and k2 = 48=2πC are for 48-pulse converter. Park's transformation is used to convert from the 3-phase to dqo coordinate system. After applying Park's transformation dqo, reference frame model of the STATCOM is obtained as: 2 2 3 6 6 id 6 7 6 d6 6 iq 7 6 6 7=6 dt 4 io 5 6 6 6 6 vDC 4 − RSTC LSTC −ω − 0 ω 0 RSTC LSTC 0 − 0 3k2 −3k2 sinϑ cosϑ 2 2 3 3k1 2 1 sinϑ 0 0 72 2 7 i 3 6 LSTC 6 7 d 3k1 6 1 cosϑ 7 76 iq- LSTC- io- vDC L 5 STC 1 0 0 0 − RC − RSTC LSTC 0 3 0 2 3 7 e 7 d 76 7 eq 7 07 76 7: 76 4 7 eo 5 07 5 0 ð10Þ 0 STATCOM installed in m-machine power system is presented in Figure 1. VSTC and ISTC are the output voltage and current of the converter, respectively. Before the installation of STATCOM, only m generator nodes, node 1 and node 2, are present and the network equations in matrix form are expressed as: 2 Y 11 6 4 Y 21 g1 Y Y 12 Y 22 g2 Y 32 3 2 3 1g 1 0 Y V 7 6 7 6 7 Y2g 54 V 2 5 = 4 0 5, Ig gg g Y V ð11Þ T g = V g1 V gm T : The following network equation is obtained using KCL with g2 , …, V where I g = I g1I g2 , …, I gm and V STATCOM installed on node l:34 " Y 011 0 #" 1 V # " I 1l # " 1g Y # + + −VG = 0 2g 2 I 2l Y Y 022 V " # " # I 1l 1 V gg −VG = Ig : + +Y Yg1 Yg2 2 I 2l V 0 The terms Y 011 and Y 022 are obtained from Y 11 and Y 22 by eliminating x12 = x1l + x2l. From Figure 1: ð12Þ 6 of 25 BADAR AND DILSHAD l = jx STCI lSTC + V STC = jx STC ðI 1l + I 2l Þ + V STC V ð13Þ i = jx ilI il + V l i = 1, 2: V ð14Þ Using Equations (13) and (14): V1 V STC I 1l −1 =U − I 2l 2 STC V V 2 ð15Þ 3 jðx 2l + x STC Þ − jx STC 6 7 xT xT 7, x T = x 2 − ðx 1l + x STC Þðx 2l + x STC Þ. where U − 1 = 6 STC 4 −jx jðx 1l + x STC Þ 5 STC xT xT Substituting Equation (15) in Equation (12): " # 1g − 1 Y gg Y g1 Y g2 Ig = Y Y 2g E12 Y ! 2 3 jx 2l − 1 6 xT 7 7 g + Y g1 Y E12 6 g2 Y V 4 jx 1l 5V STC , xT ð16Þ where, 2 jðx 2l + x STC Þ 0 6 Y 11 + xT E12 = 6 Y 4 −jx STC xT r =Y gg Here, K " # 1g − 1 Y g1 Y g2 Y Y 2g E12 Y 3 − jx STC 7 xT 7 5 j ð x + x Þ 1l STC Y 022 + xT rV 0V g + K STC Ig = K 2 3 jx 2l − 1 6 xT 7 7 0 = Y g1 Y 6 g2 Y and K E12 4 jx 5. 1l xT ð17Þ The generator terminal voltage for an m-machine power system in common coordinates are given as:36 g = E0 q −jx 0 Ig − j x q −x 0 Iq : V d d ð18Þ Using Equations (17) and (18): h g E0 e Ig = D q jð90o − δi Þ i 0n V STC , + x qn −x 0dn I qn e − jδi + D ð19Þ o =K − 1K 0 . The generator currents in dq-coordinates are obtained by applying the g = −1 1 0 and D where D r + jx K r d transformation. IG = h X Gn E 0q e D jð90o − δi Þ e jδk + x qn −x 0dn I qn e − jδi e jδk STC e 0n V +D jδk i : n Separating the real and imaginary parts in Equation (20), the generator currents dq-coordinates are; ð20Þ BADAR AND DILSHAD 7 of 25 I dk = i X h Dgn −E 0qn sinδng + x qn −x 0dn I qn cosδng + D0n εvDC cosδ0n ð21Þ i X h Dgn −E 0qn cosδng + x qn −x 0dn I qn sinδng + D0n εvDC sinδ0n , ð22Þ n I qk = n where δng = δk − δi + βgin and δ0n = γ + δk + βgin + βoi. 3 | CONTROL S YSTEM DESIGN The closed-loop system showing the layered NeuroFuzzy structure of proposed Type-2 NFWC strategies and a plant consisting of a two-machine power system are illustrated in Figure 1. The conventional Adaptive NeuroFuzzy TSK Control (ANFTSK) has also been developed for comparative evaluation. The SDC block consists of the conventional and introduced adaptive controls with their adaptation mechanism. The input to the NeuroFuzzy SDC is (yr − y) and its derivative, where y and yr are the actual and reference relative rotor speed deviations, respectively. The output of the SDC block is given as; u∈ utsk uType − 2 − 1 uType− 2 − 2 for ANFTSKC, Type-2 NFWC-1, and Type-2 NFWC-2, respectively. The output u is modulated with the reference voltage and then fed to the internal control of STATCOM. The details of the proposed control strategies are given in the following sub-sections. 3.1 | Proposed Type-2 NeuroFuzzy wavelet controls (NFWCs) The Gaussian membership function in conventional ANFTSKC contains certain mean and STD with a linear polynomial or a crisp value in the consequent part. However, Type-2 NFWC involves fuzziness in the boundaries of membership function having an uncertain mean or STD. Initially, the controller named as Type-2 NFWC-1 involves uncertainty in mean only, later the performance of the proposed controller has further been investigated by the introduction of uncertainty in both mean and the STD, named as Type-2 NFWC-2. The mathematical expression for the Gaussian membership is: 2 ! 1 x i −dij , G dij , ςij , x i = exp − ς2ij 2 ð23Þ where ς and dare the STD and mean parameters of the membership function. The ς and dcan be uncertain in intervals ςij ∈ (ς1ij, ς2ij) and dij ∈ (d1ij, d2ij). h i or C~ ji ,μC~ h In Type-2i NFWC-1, the upper and lower Gaussian Type-2 membership functions denoted by μ ji μ j ðx i Þ, μ j ðx i Þ for uncertain mean and certain STD are given as: μ j ðx i Þ = G d1ij , ςij , x i ð24Þ μ j ðx i Þ = G d2ij , ςij , x i ð25Þ Type-2 NFWC-2 having uncertainties in both mean and STD is given below:37 μ j ðx i Þ = G dlij ,ςlij , x i ð26Þ μ j ðx i Þ = G dlij ,ςlij , x i ð27Þ 8 of 25 BADAR AND DILSHAD 2 ! 1 x i −dlij : G dlij , ςlij ,x i = exp − 2 ςl2ij ð28Þ Here, ςl and dl are the STD and mean values, respectively, where l is 1 for lower and 2 for upper membership function. In Figure 1, both Type-2 upper and lower membership functions are collectively written as μ ~ji for simplicity. Type-2 NeuroFuzzy network with h inputs and g outputs is illustrated in Figure 1 based on following fuzzy IFTHEN rule: ~ R j : IF x 1 is C ~ ~ j1 and x 2 is C j2 and x h is C jh THEN ψ j is w j h X ϕ j ðx i Þ: i=1 ~ ji are Type-2 membership functions for Here, x1, x2, …, xh are inputs and ψ 1, ψ 2, …, ψ g are the output variable and C jth rule of ith input with i = 1, 2, 3, …, h and j = 1, 2, 3, …, g. The NFWC network of Type-2 consists of five layers with first layer taking and routing the inputs to the network. The Gaussian membership grade for the upper and lower part is computed in Layer 2. The Gaussian membership function in Type-2 NFWC-1 with an uncertain mean (d) and fixed STD (ς)is given as; 2 ! 1 x i −dij where dij ∈ d1ij , d2ij : μC~ j ðx i Þ = G dij , ςij , x i = exp − 2 ςij 2 ð29Þ In Type-2 NFWC-2, the uncertainty is associated with both mean and STD. 2 ! dij ∈ d1ij ,d2ij 1 x i −dij : where, μC~ j ðx i Þ = G dij , ςij ,x i = exp − ς2ij 2 ςij ∈ ς1ij , ς2ij ð30Þ In layer 3, number of rules are described by the number of nodes. T-norm product operator computes the firing strength of each rule. The output is expressed mathematically as; f j = μC~ ðx 1 Þ*μC~ ðx 2 Þ* *μC~ ðx h Þ ð31Þ f = μ C~ j1 ðx 1 Þ* μC~ j2 ðx 2 Þ* * μC~ j3 ðx h Þ: j ð32Þ j1 j2 jh Layer 4 belongs to the consequent part and each node of this layer contains a WNN. The output of this layer is governed by the Morlet wavelet function, given as: ψ j =wj h X ϕ j ðx i Þ ð33Þ i=1 ϕ j ðx i Þ = cos 5νij e " − 0:5ν2ij # x i − τij : where νij = ξij ð34Þ Here, ξij and τij are the dilation and translation parameters of the Morlet wavelet functionϕj, respectively. Finally, the type reduction and defuzzification are carried out in layer 5. The inference engine adopted to calculate the output of the Type-2 NFWC is expressed as;38 BADAR AND DILSHAD TABLE 1 9 of 25 Comparison between adaptive controllers based on number of update parameters Algorithm parameter ANFTSKC Type-2 NeuroFuzzy Type-2 NFWC-1 Type-2 NFWC-2 Antecedent part (Mean and STD) 8 12 12 16 Consequent part (Translation, dilation and weights) 6 8 10 10 Design factor 0 1 1 1 Total 14 21 23 27 ρ uType − 2 = g P j=1 g P j=1 f jψ j ð1 −ρÞ + f j g P f ψ j j j=1 g P f j ð35Þ j=1 Here, uType − 2 ∈ uType− 2 − 1 uType− 2 − 2 , ρ is the design factor. It controls the influence of the lower and upper membership functions in the controller output. Table 1 shows the comparison between conventional and proposed controllers in terms of the number of parameters to be updated. 3.1.1 | Adaptation mechanism The parameters of the Type-2 controllers are updated by minimizing the following cost function; 1 ℏ K = ðyr − yÞ2 + u2 : 2 2 ð36Þ Here, yr and y are the reference and actual output of the plant, respectively. u∈ uType − 2 − 1 uType− 2 − 2 represents the output of Type-2 NFWC-1 and Type-2 NFWC-2. The gradient descent-based update rule, used to update the parameters, is given as; Bðt + 1Þ = Bðt Þ + ΔBðt Þ: ð37Þ Here, ΔBðt Þ = −ζ ∂K ∂B and B = [ξij, τij, ςij, d1ij, d2ij, ς1ij, ς2ij, wj, ρ] is the vector containing all the update parameters of the network, and ζ represents the learning rate. The consequent part is the same for both the proposed controllers and design variations lie in the antecedent part. Type-2 NFWC-1: The partial derivatives for antecedent part of Type-2 NFWC-1 are " # μj ∂K X ∂u ∂ f j ∂μ j ∂u ∂f j ∂ , = Λ + μ j ∂B1 ∂B1 ∂ f j ∂μ j ∂B1 ∂f j ∂ j ð38Þ where B1h = [ςij, d1iji, d2ij] is the vector containing the update parameter of the antecedent part of the Type-2 NFWC-1 ∂y and Λ = ∂K ∂y ∂u + ℏu : Each partial derivative in Equation (38) is calculated below and final update law is obtained by putting them in the Equation (37). ∂u = ρℑ ∂f j ð39Þ 10 of 25 BADAR AND DILSHAD ∂u = ð1 −ρÞℑ ∂f j ∂f j ∂μij 0 ψ B where, ℑ = B @ g P j j=1 g P f − j g P j=1 1 f ψ j j=1 2 f j j 0 ψ C B B C,ℑ A =@ j = Πg1 k = 1,k6¼i μkj and g P g P j=1 2 f j− j=1 g P f f jψ ð40Þ ∂f j = Πg1 kj , k = 1,k6¼i μ ∂ μij ð41Þ 1 j C C,i = 1, …, h1, k = 1, …, g1, and j = 1, …, g2. A j j=1 8 x i − d2ij 2 > d1ij + d2ij > > , xi ≤ 3 < G d2ij ,ςij , x i ∂μ j ðx i Þ > ςij 2 = 2 > ∂ςij x i − d1ij > d1ij + d2ij > > G d1ij ,ςij , x i , xi > : 3 ςij 2 ð42Þ 8 > x i −d1ij 2 > > > G d1ij , ςij ,x i , x i < d1ij > > ς3ij > ∂ μ j ðx i Þ < d1ij ≤ x i ≤ d2ij = 0 > ∂ςij 2 > > > x i −d2ij > > > , x i > d2ij : G d2ij , ςij ,x i ς3 ð43Þ ij 8 > > > <0 d1ij + d2ij ∂μ j ðx i Þ 2 ! = x −d1 d1 + d2ij > ∂d1ij i ij ij > > , xi > : G d1ij ,ςij , x i 2 ςij 2 ð44Þ 8 ! > x i −d1ij > > , x i < d1ij > G d1ij , ςij , x i ∂ μ j ðx i Þ < ς2ij = > ∂d1ij 0 d1ij ≤ x i ≤ d2ij > > > : 0 x i > d2ij ð45Þ ! 8 x i − d2ij > d1ij + d2ij > , xi ≤ < G d2ij , ςij , x i ∂μ j ðx i Þ > 2 ςij 2 = > ∂d2ij > d1ij + d2ij > :0 xi > 2 ð46Þ 8 0 > > > > <0 xi ≤ x i < d1ij d1ij ≤ x i ≤ d2ij ∂ μ j ðx i Þ ! = > x i −d2ij ∂d2ij > > , x i > d2ij > : G d2ij , ςij ,x i ς2ij : ð47Þ Combining the partial derivatives from Equations (38)-(47) into Equation (37), the complete updated laws for the antecedent part of Type-2 NFWC-1 are obtained. Final update laws for 'ςij, d1ij, d2ij' of the antecedent part of Type-2 NFWC-1 controller, are given below; BADAR AND DILSHAD 11 of 25 8 3 > x i −d2ij 2 d1ij + d2ij > > > 6 7 G d2ij , ςij ,x i , xi ≤ > < 7 ς3ij 2 X 6 6 7 g1 ςij ðt + 1Þ = ςij ðtÞ −ζ Λ6ρℑ:Πk = 1,k6¼i μkj 7 2 > 6 7 > x i −d1ij j d1ij + d2ij 5 > 4 > > , ς ,x , x > G d1 ij i i ij : ς3ij 2 8 2 3 2 > x −d1 > i ij > G d1ij , ς , x i 6 7 > , x i < d1ij ij > > 6 7 ς3ij > > 6 7 < X 6 7 g1 6 d1ij ≤ x i ≤ d2ij 7 −ζ Λ6ð1 −ρÞℑ:Πk = 1,k6¼i μ kj 0 7 > > 6 7 j 2 > > 6 7 > x i −d2ij > 4 5 > > G d2ij , ςij , x i , x i > d2ij : 3 ςij 2 2 ð48Þ 8 > > 0 > > < 3 d1ij + d2ij xi ≤ 7 2 X 6 6 7 g1 ! 6 7 d1ij ðt + 1Þ = d1ij ðt Þ −ζ Λ6ρℑ: Πk = 1,k6¼i μkj 7 > x − d1 d1 + d2 i ij ij ij > 4 5 j > , xi > > 2 : G d1ij , ςij , x i ςij 2 8 2 3 ! > x i − d1ij > > G d1ij , ςij , x i , x i < d1ij > 6 7 > < 7 ς2ij X 6 6 7 g1 Π μ kj − ζ Λ6ð1 − ρÞℑ: 7 k = 1,k6 ¼ i > 6 0 d1ij ≤ x i ≤ d2ij 7 > j > 5 4 > > : 0 x i > d2ij ð49Þ 8 ! > x i −d2ij d1ij + d2ij > > > G d2ij , ςij , x i , xi ≤ < 2 X 6 6 ς 2 g1 ij d2ij ðt + 1Þ = d2ij ðt Þ −ζ Λ6 6ρℑ: Πk = 1,k6¼i μkj > > 4 j > d1ij + d2ij > :0 xi > 2 8 2 3 0 x < d1 i ij > > > > 6 7 > <0 d1ij ≤ x i ≤ d2ij 7 X 6 6 7 g1 kj −ζ Λ6ð1 −ρÞℑ: Πk = 1,k6¼i μ 7: ! > 6 7 > x i −d2ij j > 4 5 > , x G d2 , ς ,x > d2 > ij ij i i ij : ς2ij ð50Þ 2 3 7 7 7 7 5 Type-2 NFWC-2 The antecedent part of Type-2 NFWC-2 has the fuzziness in the both mean and STD parameters of the Gaussian membership function. The partial derivative for antecedent part of Type-2 NFWC-2 are given as " # μj ∂K X ∂u ∂ f j ∂μ j ∂u ∂f j ∂ , = Λ + μ j ∂B2 ∂B2 ∂ f j ∂μ j ∂B2 ∂f j ∂ j ð51Þ where B2 = [ς1ij, ς2ij, d1ij, d2ij] is the adaptation parameter vector for the antecedent part of the Type-2 NFWC-2. The lower and upper Gaussian Type-II membership functions for Type-2 NFWC-2 are given by Reference37; 8 d1ij + d2ij > < G d2ij , ς1ij , x i , x i ≤ 2 μ j ðx i Þ = > d1 + : G d1 , ς1 , x , x > ij d2ij ij ij i i 2 ð52Þ 12 of 25 BADAR AND DILSHAD 8 > < G d1ij ,ς2ij ,x i , x i < d1ij μ j ðx i Þ = 1 d1ij ≤ x i ≤ d2ij , > : G d2ij ,ς2ij ,x i , x i > d2ij ð53Þ 2 ! 1 x i −dlij : G dlij , ςlij ,x i = exp − 2 ςl2ij ð54Þ where, The differentials in Equation (51) are simplified as: 8 x i −d2ij 2 > d1ij + d2ij > > G d2ij , ς1ij , x i , xi ≤ > 3 < ∂μ j ðx i Þ 2 ς1ij = 2 > ∂ς1ij x i −d1ij > d1ij + d2ij > > , xi > : G d1ij , ς1ij , x i 2 ς13ij ð55Þ ∂μ j ðx i Þ ∂ μ j ðx i Þ = 0, =0 ∂ς2ij ∂ς1ij ð56Þ 8 > x i − d1ij 2 > > > G d1ij , ς2ij , x i , x i < d1ij > > ς23ij > ∂ μ j ðx i Þ < d1ij ≤ x i ≤ d2ij = 0 > ∂ς2ij 2 > > > x i − d2ij > > > , x i > d2ij : G d2ij , ς2ij , x i ς23 ð57Þ ij 8 > > > <0 d1ij + d2ij 2 = x −d1 d1 + d2ij i ij > ij > , xi > > : G d1ij ,ς1ij ,x i 2 2 ς1ij ð58Þ 8 x i −d1ij > > , x i < d1ij > G d1ij , ς2ij , x i ς22ij ∂ μ j ðx i Þ < = >0 ∂d1ij d1ij ≤ x i ≤ d2ij > > : 0 x i > d2ij ð59Þ 8 x i −d2ij > d1ij + d2ij > > , xi ≤ ∂μ j ðx i Þ < G d2ij ,ς1ij ,x i 2 2 ς1ij = > ∂d2ij > d1ij + d2ij > :0 xi > 2 ð60Þ ∂μ j ðx i Þ ∂d1ij xi ≤ BADAR AND DILSHAD 13 of 25 8 0 > > > > <0 x i < d1ij d1ij ≤ x i ≤ d2ij ∂ μ j ðx i Þ ! = > x i −d2ij ∂d2ij > > , x i > d2ij > : G d2ij ,ς2ij ,x i ς22ij ð61Þ Complete updated laws for the antecedent part of Type-2 NFWC-2 are obtained by combining the partial derivatives from Equations (39)-(41) and Equations (51)-(61) into Equation (37). Final relations for update laws are given below; 8 3 > x i − d2ij 2 d1ij + d2ij > > > G d2ij , ς1ij , x i , xi ≤ 6 7 > < 7 2 ς13ij X 6 6 7 g1 ς1ij ðt + 1Þ = ς1ij ðt Þ− ζ Λ6ρℑ:Πk = 1,k6¼i μkj 7 2 > 6 7 > j x − d1 d1 + d2 > 4 5 i ij ij ij > > G d1ij , ς1ij , x i , x > i : 3 2 ς1ij 2 8 3 > x i −d1ij 2 > > 6 7 > G d1ij , ς2ij , x i , x i < d1ij > > 6 7 ς23ij > < 6 7 X 6 g1 d1ij ≤ x i ≤ d2ij 7 ς2ij ðt + 1Þ = ς2ij ðt Þ −ζ Λ6ð1 −ρÞℑ:Πk = 1,k6¼i μ kj 0 7 > 6 7 2 > j > 6 7 > x i −d2ij > 4 5 > > G d2 , ς2 , x , x > d2 ij ij i i ij : 3 ς2 ð62Þ 2 ð63Þ ij 2 8 > > > <0 3 d1ij + d2ij xi ≤ 7 X 6 2 6 7 g1 d1ij ðt + 1Þ = d1ij ðt Þ −ζ Λ6ρℑ: Πk = 1,k6¼i μkj 7 x −d1 > 4 d1 + d2 i ij ij ij 5 > j > , x > i : G d1ij , ς1ij , x i 2 ς12ij 8 2 3 x i −d1ij > > G d1ij , ς2ij , x i , x i < d1ij > > 6 7 < ς22ij X 6 7 g1 Π 7 − ζ Λ6 ð 1 − ρ Þ ℑ: μ k = 1,k6¼i kj > 6 0 d1ij ≤ x i ≤ d2ij 7 > 4 5 j > > : 0 x i > d2ij ð64Þ 8 3 x i −d2ij d1ij + d2ij > > > , xi ≤ < G d2ij , ς1ij , x i 7 X 6 2 ς12ij 6 7 d2ij ðt + 1Þ = d2ij ðt Þ −ζ Λ6ρℑ: Πg1 μ 7 k = 1,k6¼i kj > 4 5 > j d1 + d2 > ij ij :0 xi > 8 2 32 0 x < d1 i ij > > > > 6 7 > < 6 0 d1ij ≤ x i ≤ d2ij 7 X 6 7 g1 Π − ζ Λ6ð1 − ρÞℑ: kj 7 ! k = 1,k6¼i μ > 6 7 > x i −d2ij j > 4 5 > , x > d2 > i ij : G d2ij , ς2ij , x i ς22ij ð65Þ 2 Consequent part The update laws for dilation, translation, weight, and design factor are computed by applying the following chain rules " # ∂K ∂u ∂ψ j ∂ϕ j ∂νij =Λ ∂ξij ∂ψ j ∂ϕ j ∂νij ∂ξij ð66Þ 14 of 25 BADAR AND DILSHAD " # ∂K ∂u ∂ψ j ∂ϕ j ∂νij =Λ ∂τij ∂ψ j ∂ϕ j ∂νij ∂τij ð67Þ ∂K ∂u ∂ψ j =Λ ∂w j ∂ψ j ∂w j ð68Þ ∂K ∂u =Λ ∂ρ ∂ρ ð69Þ The partial derivatives in Equations (66)-(69) are calculated as follows; ρ g P ∂u j=1 = g P ∂ψ j f f j=1 g P f j ð1 −ρÞ j j=1 + g P f j j ð70Þ j=1 ∂ψ j =wj ∂ϕ j ð71Þ h X ∂ψ = ϕ ðx i Þ ∂w j i = 1 j ð72Þ g P ∂u = ∂ρ f jψ j j=1 g P j=1 f j g P f ψ j j − j=1 g P j=1 f j ð73Þ 2 2 ∂ϕ j cos 5νij e − 0:5νij :ν2ij + 5νij sin 5νij e − 0:5νij = ∂ξij ξij ð74Þ 2 2 ∂ϕ j νij :cos 5νij e − 0:5νij + 5sin 5νij e − 0:5νij = ∂τij ξij ð75Þ Using Equations (39)-(41) and Equations (66)-(75) in Equation (37), the complete update laws for the consequent part are obtained as follows: 0 g 1 g P P f " − 0:5ν2 − 0:5ν2 # ρ f ð 1 −ρ Þ jC j ij ν2 + 5ν sin 5ν ij X B cos 5νij e ij ij e j=1 C ij B j=1 ξij ðt + 1Þ = ξij ðt Þ −ζ ΛB g + : w C ij g P @ P A ξij f j fj j j=1 ð76Þ j=1 0 1 g g P P f " # ρ f ð 1 −ρ Þ 2 2 jC j X B νij :cos 5νij e − 0:5νij + 5sin 5νij e − 0:5νij j=1 C B j=1 τij ðt + 1Þ = τij ðtÞ −ζ ΛB g + Cwij : g P @ P A ξij f j fj j j=1 ð77Þ j=1 Using Equations (70) and (71) and Equation (68) in Equation (37), the complete update law for weights is given as; BADAR AND DILSHAD 15 of 25 F I G U R E 2 Two-machine Case I: A, rotor speed deviation B, load angle deviation C, line power flow D, STATCOM injected voltage 0 1 g P ρ f j ð1 −ρÞ f jC h X B j=1 CX B j=1 w j ðt + 1Þ = w j ðt Þ−ζ ΛB g + ϕ ðx i Þ C g P @ P A i=1 j j fj fj g P j=1 j=1 Using Equations (73) and (69) in Equation (37), the complete update law for design factor is given as: ð78Þ 16 of 25 BADAR AND DILSHAD F I G U R E 3 Two-machine Case II: A, rotor speed deviation B, load angle deviation C, line power flow D, STATCOM injected voltage 2 ρij ðt + 1Þ = ρij ðt Þ −ζ g P f jψ j X 6 6 j=1 Λ6 g 4 P j j=1 f j 3 g P f ψ j j7 j=1 7 − g 7: P 5 fj j=1 ð79Þ BADAR AND DILSHAD 17 of 25 F I G U R E 4 Two-machine Case III: A, rotor speed deviation B, load angle deviation C, line power flow D, STATCOM injected voltage Due to direct adaptive control mechanism, used for controller parameters update, the plant sensitivity measure ∂y=∂u is considered as constant.39 4 | R ES U L T S A N D D I S C U S S I O N A two-machine test system, shown in Figure 1, is simulated as a plant to assess the performance of the proposed control strategy. The simulations have been carried out using Intel Core (TM) i5-5200U CPU@ 2.20 GHz 2.20 GHz processor 18 of 25 FIGURE 5 BADAR AND DILSHAD Design factor for Type-2 NFWC-2: A, Case I C, Case II E, Case III, design factor for Type-2 NFWC-I: B, Case I D, Case II F, Case III with 8 GB RAM and 64-bit operating system with MATLAB/Simulink software environment. The system consists of two generators G1 and G2 with rated capacities of 1400 and 700 MVA, respectively, connected with transmission lines through step-up transformers. PSSs are installed on each generating station. STATCOM is installed at bus B2, slightly off-centered toward the generating unit 1 for better control and enhanced stability of power system.40 The initial loading conditions for generating stations are taken as P1 = 0.8937 p.u. and P2 = 0.4 p.u. The three different simulation scenarios of STATCOM installed with Type-2 NFWC-1, Type-2 NFWC-2, and ANFTSKC can be simulated using switch S1, as shown in Figure 1. Switch S2 is used for switching between STATCOM with or without SDC scenarios. 4.1 | Case I: 3-phase fault A 3-phase, self-clearing fault of eight cycles is applied on L3 at t = 0.1 s. The fault resides on the system for 0.1333 seconds. Before restoring it to the pre-fault condition. Figure 2A shows the rotor speed deviation with poor damping performance of STATCOM without SDC. Significant performance improvement has been observed with Type-2 NFWC-2 as compared to that of Type-2 NFWC-1, ANFTSKC, and without SDC. Figure 2B shows the line power flow on L4. The steady-state power flow after the clearance of the 3-phase fault is almost 550 MW. The STATCOM injected voltage modulated with the output of internal control is shown in Figure 2D. The control output has limits between maximum and minimum threshold values of 1.1 and 0.9 p.u. due to STATCOM ratings. BADAR AND DILSHAD 19 of 25 F I G U R E 6 Case I Type-2 NFWC-2: A, antecedent parameters B, consequent parameters TABLE 2 Performance improvement w.r.t. STATCOM without SDC [%] Performance index Test cases Control algorithms Case I ANFTSKC Case II Case III ITAE ITSE 1.36 14.59 5.05 10.35 Type-2 NFWC-1 27.12 24.88 14.77 11.34 Type-2 NFWC-2 29.89 34.93 19.55 15.95 ANFTSKC 18.97 37.71 26.73 35.25 Type-2 NFWC-1 22.61 53.55 33.35 51.59 Type-2 NFWC-2 31.17 56.86 37.11 52.91 3.88 30.32 11.82 25.02 Type-2 NFWC-1 14.74 39.37 19.72 37.91 Type-2 NFWC-2 18.92 48.01 24.64 42.19 ANFTSKC IAE ISE The computation time in the base case taken by internal control of STATCOM with no SDC installed in the system is 36.107 seconds. Also, the computation time taken by ANFTSK, Type-2 based NFWC-1, and Type-2 based NFWC-2 is 43.304, 57.205, and 59.191 seconds, respectively, showing that the proposed controllers have taken more computation time due to increased computational complexity for parameters adaptation. 20 of 25 BADAR AND DILSHAD F I G U R E 7 Case II Type-2 NFWC-2: A, antecedent parameters B, consequent parameters 4.2 | Case II: series of faults A series of fault events are used to examine the robustness of the proposed controllers by applying a 3-phase fault on L4 at t = 0.1 s. Cleared by the permanent outage of line L4 at t = 0.3 s. Another fault event of a 50% reduction in load 2 is applied at t = 5 s. Figure 3 shows the system parameters results revealing that STATCOM without SDC shows poor damping of oscillations in the transient and steady-state region. Type-2 NFWC-2 performs better than all other controllers in both the transient and steady-state regions. Figure 3C shows the line power flow on healthy line L3. It can be seen that power flow increases from pre-fault power flow of almost 555 to 1055 MW as the fault is cleared by line outage. The power flow decreases from 1055 to 685 MW following a load change applied at load 2 at t = 5 s. STATCOM injected voltages are shown in Figure 3D, it shows that injected voltages are higher with smooth oscillations damping for Type-2 NFWC-2 as compared to ANFTSKC and Type-2 NFWC-1. The computation time, in this test case, is 80.224, 123.208, 178.275, and 194.495 seconds., for STATCOM without SDC, ANFTSKC, Type-2 NFWC-1, and Type-2 based NFWC-2, respectively. 4.3 | Case III: rapid variation of operating conditions In this case, cascaded fault events are applied such that the new fault occurs before the system gets stabilize from the effect of the previous fault. A 3-phase fault at t = 0.1 s of 12 cycles duration is applied at L4 followed by partial line outage at t = 0.315 and re-closure at 0.418 s. BADAR AND DILSHAD 21 of 25 F I G U R E 8 Case III Type-2 NFWC-2: A, antecedent parameters B, consequent parameters Figure 4 shows the comparative results of controllers for different system parameters. Figure 4C shows that the steady-state line power is almost 555 MW on L3. It has been observed that Type-2 NFWC-2 retains its superior performance as compared to other controllers for rapid variation in operating point. The computation time for this case is more than Case I but lower than that of Case II. The computation time for STATCOM without SDC, ANFTSK, Type-2 based NFWC-1, and Type-2 based NFWC-2 is 48.086, 62.010, 69.844, and 83.928 seconds, respectively. Figure 5 shows the results for the design factor used in Type-2 based NFWC-1 and Type-2 based NFWC-2. Significant variations have been observed in the design factor when the system undergoes a fault event. It adjusts its value during the transient period using update law given in Equation (79) to determine the contribution of upper and lower membership functions in the final output. The results shown for all the three fault cases reveal that the design factor is more sensitive to faults in the case of Type-2 NFWC-2 as compared to that of Type-2 based NFWC-1. The results for parameter adaptation of the proposed Type-2 based NFWC-2 have been presented in Figure 6 to show the tuning performance of the controller. The update parameters for the Cases I-III are shown in Figures 6-8, respectively. The update parameter results in case of multiple faults show that once the parameters are updated, they quickly get adjusted with minor changes for operating conditions due to subsequent faults. Performance Indices (PIs) give a quantitative measure of the controller performance and thus further insight for controller behavior in both the steady-state and transient regions. PIs are generally expressed as: ð tsim PI = 0 t k jeðt Þjl dt: ð80Þ 22 of 25 BADAR AND DILSHAD Here, k and l are constants such that (k, l) ∈ {(0, 1), (0, 2), (1, 1), (1, 2)} for Integral Absolute Error (IAE), Integral Square Error (ISE), Integral Time Absolute Error (ITAE), and Integral Time Square Error (ITSE), respectively. Percentage improvement, obtained from PIs curves, is calculated using the following expression: Percentage Improvemnt for X w:r:t:Y = PI Y −PI x %: PI Y ð81Þ Here, Y represents STATCOM without SDC and X is the set of all other controls. The percentage improvement results are provided in Table 2. The performance margin is significant from ANFTSKC to Type-2 NFWC-1 due to the inclusion of uncertainty in the mean parameter of the membership function and the use of WNN in consequent instead of the linear polynomial. The performance has been further improved using both uncertain mean and STD in Type-2 NFWC-2. Case I reveals that more percentage improvement is achieved for ITSE and ITAE as compared to IAE and ISE. The improvement for Type-2 NFWC-1 and ANFTSKC is marginal for ISE. Type-2 NFWC-2 gives better performance in all the PIs with most improvement seen in ITSE depicting the improved performance of the proposed controller in the transient region. Similarly, in Case II the performance margin is more significant for ISE and ITSE as compared to IAE and ITAE; this is due to the better performance of proposed controllers in the steady-state region. The performance margin is significant in this scenario even the system undergoes series of faults which shows the effectiveness of Type-2 NFWC strategy in more complex scenarios. Percentage performance improvement in Case III indicates that performance margin is more in ITSE and ISE. The performance of both NFWCs is almost the same in the transient region as seen by ITEA and IAE, however, much better than ANFTSKC and without SDC. Overall, Type-2 NFWC-2 gives better performance than Type-2 NFWC-1 in steady-state and transient regions. 5 | C ON C L U S I ON This research presents a Type-2 based NeuroFuzzy wavelet control scheme that integrates Gaussian Type-2 membership function in the antecedent part and WNN in the consequent part for damping LFOs using STATCOM in a multimachine environment. The two variations of uncertain mean only and both uncertain mean and STD have been considered for performance improvement. The graphical simulation results and PIs based analysis validates the notable performance improvement of Type-2 NFWC based controls under different fault scenarios. Furthermore, it can be concluded from quantitative results that the performance improvement margin of Type-2 NFWC-2 is large for more complex fault scenarios and thus shows the robustness of Type-2 NFWC-2. Longer computation time has been observed for Type-2 NFWC-2 as trade-off for performance improvement due to increased computational complexity. However, the computational time difference in moving from NFWC-1 to NFWC-2 is smaller as compared to that of ANFTSKC and NFWC-1 which leads toward an important conclusion that major delay has been introduced due to the inclusion of wavelets rather than the introduction of uncertainty in both mean and STD. The performance enhancement of Type-2 NFWC-2 using indirect adaptive control strategy and optimization to reduce latency can be investigated as potential future research. A C K N O WL E D G E M E N T The financial support by the Higher Education Commission (HEC) Pakistan under 5000 Indigenous scholarship scheme PIN No. 2EG2-744 is highly acknowledged. Symbols used in Modeling of Power System STC Output voltage of VSC V vDC c k θ ϑ γ ias , ibs , ics Capacitor voltage Modulation index Turn ratio PWM inverter Synchronizing angle Firing angle of the PWM inverter Control the exchange of reactive power AC side current (Phase a, b and c) BADAR AND DILSHAD RSTC LSTC R C LSTC k1, k2 Y 23 of 25 Total resistance Leakage inductance of the transformer Resistance of the DC-side of the STATCOM Capacitance of the DC-side of the STATCOM Total inductance of STATCOM Constant 1 and 2 for 48-pulse converter Admittance Symbols used in Control System Design y Actual relative rotor speed deviation yr Reference deviation in relative rotor speed u Output of the SDC block (Controller output) Output of the ANFTSKC utsk uType − 2 − 1 Output of the Type-2 NFWC-1 uType − 2 − 2 Output of the Type-2 NFWC-2 G(dij, ςij, xi) Gaussian membership function ς STD parameter d Mean parameter Upper Gaussian Type-2 membership function μ j ðx i Þ μ j ðx i Þ Lower Gaussian Type-2 membership function ~ ji Type-2 membership functions for jth rule of ith input C f j and f Firing strength of each rule (lower and upper membership functions) ψj Output of Layer 4 (WNN) Morlet wavelet function ϕj(xi) Weight wj Dilation parameter of wavelet ξij τij Translation parameter of wavelet ρ Design factor K Output of cost function ℏ h cut (constant) B Vector containing all update parameters of the network ζ Learning rate ∂y=∂u Plant sensitivity measure G1 and G2 Generator 1 and 2 Speed deviation between generator 1 and 2 Δω21 Line Power PL v*ref STATCOM injected voltage (p.u.) 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