Stability assessment using adaptive interval type-2 fuzzy sliding mode controlled power system stabilizer

The low-frequency electromechanical oscillations (LFEOs) in electric power system are because of weaker interties, uncertainties, various faults and disturbances. These LFEOs (0.2–3 Hz) are less in magnitude and are responsible for lower power transfer, increased losses and also threaten the stability of power system. An adaptive interval type-2 fuzzy sliding mode controlled power system stabilizer (AIT2FSMC-PSS) is presented to neutralize the LFEOs and enhance stability under uncertainties and external disturbances. The AIT2FSMC is a hybridization of type-2 fuzzy logic system (T2FLS) with conventional SMC to lower the chattering effect, enhance the robustness of reaching phase and improve system’s performance. Here, T2FLS is used for estimating the unknown functions of SMC. A robust sliding surface is presented to keep the system in the desired plane and remain stable under disturbance conditions. A modified control law is proposed for selecting the control parameters and Lyapunov synthesis is used to make the error asymptotically converging to zero. The effectiveness of the AIT2FSMC-PSS is accessed in single and multimachine power systems subjected to various uncertainties and disturbances. Again, comparison of performance indices (PIs), eigenvalues, damping ratios, oscillating frequencies, integral time absolute error (ITAE), figure of demerit (FD) and frequency domain plots like Bode, root locus and Nyquist plots are also analysed to access the efficacy of the proposed stabilizer. The simulated responses, comparative study and frequency plots confirm the supremacy of the proposed AIT2FSMC-PSS in suppressing the LFEOs to lesser settling characteristics, offer stable performance and assure transient stability of power system as compared to other stabilizers.


Research background
Modern power systems are complex and extremely nonlinear in nature. They operate at increased stress and sometimes near their stability limits to ensure continuity of power supply. Disturbances such as faults, load perturbations, natural disasters, etc., are responsible for the generation of LFEOs in the range of (0.2-3) Hz (Kundur 1994). LFEOs are small in magnitude but sustain for long durations and result in degraded power transfer, loss of synchronism, subsequent blackouts and power outages (Anderson and Fouad 2003). LFEOs are classified into local and interarea mode of oscillations. Oscillations in local generators located in one geographical area are local oscillations (0.8-3) Hz., whereas oscillations in generators located in different areas are called interarea mode oscillations (0.1-0.7) Hz. Among which interarea oscillations should be handled more cautiously than local oscillations because it can lead to generation failure (Paital et al. 2018;Obaid et al. 2017). Stability of power system depends on the operating conditions. Out of different classes of stability, transient stability is at higher priority because it is related to maintaining synchronism between the generators under severe disturbance conditions. Conventionally, oscillation damping is achieved through fast acting high gain excitation systems. But, it produces negative damping torques in the power system (deMello and Concordia 1969). In past decades, lead-lag-based PSS are adopted to provide stabilizing signals to the exciter for damping the LFEOs. It has predefined parameters and provides good damping characteristics in a linearized model of power system. But, due to fixed parameters of PSS, its performance degrades under change in operating point which results in oscillatory instability (Hannan et al. 2018). Hence, to overcome the limitations of CPSS, several compositions of PSS like H-infinity (Chen and Malik 1995), pole placement (Kashki et al. 2010), LMI method (Werner et al. 2003), proportional integral derivative (PID) (Ray et al. 2016), and fractional order PID (FOPID) (Chaib et al. 2017) have been proposed in the literature. For improved performance, various nature inspired optimizations mentioned in the literature (Ray et al. 2018a;Shayeghi et al. 2010;Chitara et al. 2018;Ali 2014;Abd-Elazim and Ali 2016;Singh et al. 2019) are used for optimizing the parameters of PSS.
In addition to the existing control approaches, fuzzy logic control (FLC) has been extensively used for the design of PSS as reported in the literature (Hariri and Malik 1996;Paital et al. 2017). Being a nonlinear control technique, FLC-based PSS (FPSS) provide good oscillation damping performance to that of CPSS. It is applicable to problems where mathematical model is not available. But, its performance gets degraded while handling nonlinearities and uncertainties. Hence, to overcome this limitation, adaptive control strategy-based FLC is introduced (Lie and Sharaf 1996; Hussein et al. 2010). But, it cannot offer the required oscillation damping characteristics under severe disturbances. Nowadays, a robust control technique called sliding mode control (SMC) was gaining popularity because of its simplest structure, superior disturbance elimination and insensitive to parameter variations (Paital et al. 2021a). But, it has its own limitations such as (i) it requires proper knowledge of system dynamics (ii) chattering phenomenon due to discontinuous control law. SMC performs efficiently under normal operating conditions. But its performance gets degraded under large and continuously varying disturbance conditions. To overcome the above said limitations, SMC is hybridized with FLC to ameliorate the system performance (Paital et al. 2021b). However, oscillations are still present under transient disturbance conditions. Hence, different higher-order SMC is adopted in the literature (Nechadi et al. 2012;Saoudi and Harmas 2014;Ray et al. 2018b;Farahani and Ganjefar 2017). In this regard, an AFSMC-based PSS without reaching phase is presented in Nechadi et al. (2012) for damping out LFEOs under disturbance conditions. An indirect AFSMC-PSS is suggested by Saoudi et al. for minimizing oscillations in multimachine power system (Saoudi and Harmas 2014). Again, an AFSMC-PSS is suggested in Ray et al. 2018b for improving stability in SMIB and MMPS and also validated their approach in a real-time simulator. An AFSMC with PI surface is proposed in Farahani and Ganjefar (2017) to damp out oscillations and enhancing stability. Although, the above said methods are based on type-1 FLC (T1FLC) which has limitations in dealing with large uncertainties and unexpected disturbances in the power system. Hence, an extension of T1FLC called type-2 FLC (T2FLC) has been introduced. The T2FLC is a nonlinear control approach which can deal with various nonlinearities and uncertainties of the system in a better way. T2FLC has been extensively used in system modelling, real-time applications, industrial control applications, etc. T2FLC can easily handle the external disturbances and model uncertainties more effectively because of its particular structure of its membership functions (MFs) and additional degrees of freedom (Castillo and Melin 2012).
Many applications of T2FLC for improvement of power system stability can be found in the literature (Shokouhandeh and Jazaeri 2018; Sun et al. 2014;Ray et al. 2019;Adjeroud et al. 2015). Shokouhandeh and Jazaeri have applied a robust T2FLCbased PSS for handling uncertainties due to loading and line parameters (Shokouhandeh and Jazaeri 2018). Sambariya et al. have implemented IT2FPSS enhancing stability in single and MMPS subjected to disturbances . Sun et al. presented a differential evolution (DE) tuned T2FLC-based PSS for stability improvement of power system (Sun et al. 2014). A hybrid firefly swarm algorithm tuned IT2FOFPID-PSS is suggested in Ray et al. (2019) for enhancing stability in both single and MMPS subjected to disturbances and uncertainties. Again, application of T2FLC-based SMC can be found in Akbarzadeh et al. (2017); Nechadi and Harmas 2015). Nechadi et al. proposed a T2AFSMC-based PSS without reaching phase for damping oscillations of power system under disturbances (Nechadi and Harmas 2015). In (AbouOmar et al. 2022), a hybrid neural network-differential evolution technique is applied for optimizing an observer-based IT2FPID (OB-IT2FPID) is presented for a PEMFC air feeding system. A single-input interval type-2 fractional order fuzzy (SIT2FOF) controller is proposed in Aliasghary and Mohammadikia (2022), for improving the control performance of AVR in a power system. Furthermore, a real-time stability analysis using an IT2FLC-based damping controller for a wind-based power system is presented in Anil Naik (2022). The author used OPAL-RT simulator to demonstrate the effectiveness of the proposed damping controller. A T2FLC is presented in Mohammadi Moghadam et al. (2022) for regulating active/ reactive power and energy storage management in an autonomous microgrid system under different disturbance scenarios. Xing et al. presented a networked IT2FS for addressing dynamic event triggered control under multiple cyber-attack (Xing et al. 2023). A variant shark smell optimization (VSSO)-based T2FLC is presented in Cuevas et al. 2022a for testing and navigation of an autonomous mobile robot under varying environmental conditions. An application of T2FLS is presented in Cuevas et al. 2022b for parameter adjustment of marine predator algorithm (MPA) under various disturbances. Again, an improved T2FLC called type-3 FLC (T3FLC) is proposed in Taghieh et al. (2022) for addressing the problem of current sharing and voltage balancing in a distributed generation (DG)based autonomous microgrid.

Main contribution
Based on the above motivations, it is clear that the conventional control design of PSS such as PID-PSS, FOPID-PSS, FPID-PSS, and FPID-PSS does not provide a clear idea of improving stability of power system under different uncertainties and nonlinearities of the power system. These are slow in operation, specifically under unexpected disturbances and parametric fluctuations. The conventional PSS damps out the LFEOs but cannot completely neutralize it. The oscillations still remain in the power system which threatens the stability. In the other hand, the application of robust control approaches, like SMC, FSMC, AFSMC, and AFFSMC-based optimal-PSS, has become necessary for neutralizing LFEOs and improving system stability under a wide range of disturbances, uncertainties and nonlinearities in the power system. Among the various robust control approaches, AIT2FSMC-based PSS is the most promising technique which can enhance the system stability under external disturbances and uncertainties. The performance of conventional SMC is degraded due to chattering effect due to its discontinuous control law. Also, the estimation of unknown parameters are complex. However, the proposed stabilizer applies the concept of T2FLS to approximate the unknown functions of SMC and robust sliding surface to force the system to be in the sliding surface during disturbances. The AIT2FSMC is a hybridization of T2FLC and conventional SMC which has the ability to minimize the effect of chattering phenomenon, improving the performance of reaching phase and has superior disturbance rejection characteristics. The T2FLC easily handles the model uncertainties and external disturbances efficiently due to its specific configuration of its membership functions (MFs) and additional degrees of freedom. The proposed AIT2FSMC is applied here to design a stabilizer to minimize the LFEOs and improving stability is single and multimachine power systems under various disturbance conditions and uncertainties.

Main highlights
Comparing the previously published papers, the following contributions are made and are being highlighted as follows: 1. An AIT2FSMC-based PSS is proposed for the first time for damping out LFEOs in single and multimachine power system under various system uncertainties like noise and external disturbances. 2. The IT2FLC estimates the unknown functions and a modified control law is proposed to avoid chattering effect and Lyapunov stability criteria is used for assuring stability such that the error is asymptotically converging to zero. 3. Speed deviation (Dx) and accelerating power (DP) are considered as input signals for improvements in the effectiveness and efficacy of AIT2FSMC-PSS in single and 2-area, 16-machine, 68-bus power system under various uncertainties and disturbances. 4. Comparison of time domain PIs like settling time, overshoots, eigenvalues, damping ratios, oscillating frequencies, ITAE, FD and frequency plots like Bode, root locus and Nyquist plots are provided to access the stability. 5. The simulated responses, comparative study and frequency plots confirm the supremacy of the proposed AIT2FSMC-PSS in minimizing LFEOs with lesser peak and settling characteristics, disturbance rejection, offer more stable performance, ensure better robustness against parameter uncertainties and assures transient stability to that of AFSMC-PSS, FSMC-PSS, FPSS and CPSS.

Article organization
The paper is arranged as follows: the comprehensive modelling of single and multimachine power system is highlighted in Sect. 2. In Sect. 3, introduction to AIT2FSMC is presented. Section 4 presents the simulations and the comparative analysis. Section 6 gives the concluding remarks and future scope followed by references.

Power system modelling
The stability analysis is performed through modelling of different power system components. The dynamics of various components are framed using algebraic equations. The dynamics of multimachine power system is characterized by an ith synchronous generator, exciter and other components. These equations are formulated by assuming fixed input mechanical power under disturbance conditions. The expressions representing the rotor dynamics of generator-i are given by Devarapalli et al. (2021).
where d i , x i , x s denote rotor angle, synchronous speed and base speed. The change in rotor angle of ith generator in terms of swing equation is given by: where H; T m ; T e are called inertia, mechanical and electrical torques. The electrical torque of i th generator is calculated from the sub transient model. Hence, Eq.
(2) can be rewritten as: where I di ; I qi denote the stator currents in d and q-axis, w 1di ; w 2qi represent d and q-axis transient and sub-transient EMFs, X di ; X 0 di ; X 00 di and X qi ; X 0 qi ; X 00 qi represent synchronous, transient and sub-transient reactances in d and q-axis. X lsi is the leakage reactance of armature.
The expression of transient EMFs are as follows: where T 0 doi ; T 0 qoi denote transient time constants in d and qaxis. E 0 di ; E 0 qi represent transient voltage in d and q-axis, E fdi is the field voltage of d-axis. The expression for transient and sub-transient EMFs is as follows: The expressions representing stator dynamics of generator-i are as follows: where V i ; R si denote terminal voltage and armature resistance of ith synchronous generator. The saturation function of exciter is given by: where A s ; B s are the saturation constants. The algebraic equations representing the dynamics of excitation system are as follows: where V tri ; V ti represent measured voltage state variable and terminal voltage. The expression of ith PSS is given as: where T w ; T 1i ; T 2i ; T 3i ; T 4i denote washout and lead-lag time constants.
The concept of the proposed adaptive interval type-2 fuzzy sliding mode control in stability enhancement of the power system is discussed in the following section.

Type-2 fuzzy logic systems (T2FLS)
T2FLS are the extension of T1FS. It was presented by Lotfi Zedeh in 1975 and then developed by Karnik andMendel in 1999 (Mendel 2007). T2FLS is a nonlinear control technique which can handle the system nonlinearities efficiently and produces better performance with high level of accuracy. It applies the concept of fuzzy sets for handling uncertainties, measurement noise, nonlinearities and unexpected disturbances. The T2FLS provides more robustness against parametric uncertainties of the system due to its additional degrees of freedom. The T1FLS has a two-dimensional MF representing crisp values [0, 1], whereas T2FLS has a 3D membership function. Although the interval MFs in T2FLS is regarded as the key difference between T2FS and T1FS. A T2FS denoted byÃ is expressed as 0 lÃ x; u ð Þ 1,8x 2 X, 8u 2 J x & 0; 1 ½ , which is consisting of upper MF (UMF) lÃ x ð Þ and lower MF (LMF) lÃ x ð Þ separated by a footprint of uncertainty (FOU) as shown in Fig. 1a.
When lÃ x; u ð Þ ¼ 1, 8u 2 J x & 0; 1 ½ , then it is called as an interval type-2 MF and also called IT2FSs. The control block diagram of T2FLC is shown in Fig. 1b. The functioning of each blocks is described as follows (Ray et al. 2019;Paital et al. 2022).

Rule base
The rule base of IT2FLS is similar to T1FLS and is formulated based on the user's knowledge. Basically, the r th rules of IT2FLS are expressed as follows: whereF j i ,G j are the input states, e 1 , e 2 are the inputs whereas y is the output. P denotes the rules.

Inference
The inference engine in IT2FLS combines the rules to generate the output. Here, a product t-norm-based inference engine is adopted and is given by: Stability assessment using adaptive interval type-2 fuzzy sliding mode… 7719

Type reducer
The function of type reducer is to reduce the type of T2FS to T1FS. Here, a centre of sets (COS) type reducer is adopted and is given by: :::: The expression of two end points (yl) and (yr) is computed as follows:

Defuzzification
In defuzzification process, the crisp inputs are extracted from type reduced T1FS. This process computed by taking average of the end points The concept of sliding mode control-based power system stabilizer (SMC-PSS) is explained in the next subsection.

Sliding mode control-based power system stabilizer (SMC-PSS)
SMC is a most promising robust control approach that offer superior disturbance rejection characteristics under parametric variations and uncertainties. The execution of SMC involves two phases such as reaching and sliding phase. Out of these phases, the system is exposed to disturbances in reaching phase of SMC. Hence, exclusion of reaching phase enhances system stability (Paital et al. 2021a, b;Nechadi et al. 2012;Saoudi and Harmas 2014;Ray et al. 2018b). The purpose of SMC is to keep the error on the sliding surface by neglecting the consequences of reaching phase and chattering. The nonlinear model of power system under disturbance is expressed as (Paital et al. 2021a): where x t ð Þ 2 R n and u t ð Þ 2 R m are state and control vectors, f x ð Þ is a nonlinear function. Here, state vector, where DP ¼ P m À P e ; u 2 R is the input signal. The tracking error (e) is the difference among trajectory state x ð Þ and command x d ð Þ is given by: The sliding surface of SMC is expressed as follows: where, b is a constant, x 1 0 ð Þ; x 2 0 ð Þ are the states. Taking time derivative of Eq. (24) as Using Eq. (25), the condition in Eq. (26) can be satisfied using the following theorems.
Theorem 1 For the nonlinear system expressed in Eq. (20), a control law is selected neglecting the effects of reaching phase and chattering.
Proof The Lyapunov function (V) is given by: where S, S T denote sliding surface and its transpose. h Substituting Eqs. (24), (25) in Eq. (28) and taking time derivative we get: For stability, following criterion must be satisfied: Remark 1 In the Control law expressed in Eq. (27), calculation of q is difficult and tedious process and cannot be computed directly. Again, calculation of f x; t ð Þ and g x; t ð Þ are also difficult. Therefore, a modified control law is presented in the next subsection to overcome the above limitations.
3.8 Adaptive interval type-2 fuzzy sliding mode power system stabilizer AIT2FSMC is a modification of AFSMC by type-2 fuzzy systems. The details of AIT2FSMC-PSS can be found in Nechadi and Harmas (2015). In this paper, the concept of AIT2FSMC is used for designing a stabilizer for damping out LFEOs under disturbance conditions. The proposed AIT2FSMC-PSS can easily handle nonlinearities and uncertainties of power system and has the unique ability to estimate the nonlinear functions (Nechadi and Harmas 2015).
The adaptive laws are given below: Then, the stability is guaranteed.
Proof Selecting the Lyapunov function (V) as: Applying time derivative to Eq. (43), we get: Substituting the adaptation laws mentioned in Eqs. (41)-(44), we get The inequality conditions are given below: Àh Now, the expression of _ V is given by: Assuming, a ¼ min 2 q 2 ; c 1 ; c 2 and l ¼ 1 Multiplying both sides of Eq. (49) by e at , we get: Integrating Eq. (50) in the range between 0 and t: Stability assessment using adaptive interval type-2 fuzzy sliding mode… 7721 where Applying Barbalat's Lemma, it is found that, sliding surface S ð Þ and derivative of sliding surface _ S À Á is bounded and also small approximation error e ð Þ,h f andh g are also bounded. The block diagram of the proposed AIT2FSMC-PSS is shown in Fig. 2.
Hence, the proof is completed. h Remark 2 It is clear that, the proposed control scheme satisfies the stability criterion and the error asymptotically converges to zero.

Simulation results and discussions
In this section, the viability of the proposed AIT2FSMC-PSS is analysed under uncertainties and disturbances. The proposed scheme is implemented using MATLAB/ Simulink environment and verified in both single and multimachine power systems. Different small and transient disturbance scenarios are considered to analyse the performance of the proposed AIT2FSMC-PSS and the results are compared with AFSMC-PSS, SMC-PSS, FPSS and CPSS based on their PIs, eigenvalues, damping ratios, oscillating frequencies, ITAE and FD. The inputs to the proposed stabilizer are speed deviation Dx ð Þ and accelerating power DP ð Þ. Type-2 fuzzy triangular MFs shown in Fig. 3a, b are considered for the stability analysis. 25 fuzzy rules with 05 linguistic variables like negative big (NEB), negative medium (NEM), zero (ZER), positive medium (POM) and positive big (POB) given in Table 1 are considered.
Small signal stability analysis is executed on a linearized model of power system. The state equations are framed from the state space model of the linearized system and the corresponding eigenvalues, damping ratios and oscillating frequencies are obtained. From the location of eigenvalues in the s-plane, the stabilities of the power system can be analysed.
The procedure for calculating eigenvalues, damping ratios and oscillating frequencies is as follows: From the characteristics equation given in Eq. (55), the eigenvalues (k) can be calculated as follows: The expression for calculating damping ratio (n) and oscillation frequency (r) is given by Paital et al. 2021a: The expression for calculating ITAE and FD is given as follows (Hashemi et al. 2017): where N SD is the number of signals, MP, US denote the under and overshoots. T s is the settling time.

Single machine infinite bus (SMIB) power system
This sub-section, the assessment of stability is performed in SMIB system as given in Fig. 4. The SMIB system comprises of a 200 MVA synchronous generator feeding power to the infinite bus (rest power system) through 100 km line. As, the power system is an elementary one, it can give an idea of small signal stability under various uncertainties and disturbance conditions. To testify the efficacy and robustness of the proposed stabilizer and to determine its supremacy over aforesaid stabilizers, four different disturbance scenarios are considered. The disturbances are selected that, they cover the uncertainties and disturbances of the power system.
The disturbance scenarios in SMIB power system are as follows: • Scenario-1: Normal loading with 5% increase in line parameters. • Scenario-2: 10% increase in loading and 10% increase in line parameters. • Scenario-3: 20% increase in loading and 15% increase in line parameters.

• Scenario-4: A 3-u fault in the transmission line
With the increase in loading conditions, stability margin gets reduced and entire system is pushed towards instability. In this regard, a gradual increase in loading by 5%, 10% and 15% is applied at t = 0 s for the first three disturbance scenarios. The impacts of variations in loading and transmission line parameters are observed in deviations in speeds of corresponding synchronous generators. In scenario-1, under normal loading condition and 5% increase in line parameters, the speed deviations depicted in Fig. 5a are maximum in case of CPSS and the oscillations go on reducing with the presented stabilizers. The proposed AIT2FSMCPSS show minimum peak and settling characteristics with maximum of eigenvalues, damping ratios and ensure superior oscillating damping performance. Similarly, in scenario-2 and scenario-3, the loadings are increased by 10% and 20% and the transmission line parameters are increased by 10% and 15%. As observed from the simulated responses depicted in Fig. 5b and c, that the increase in loading and line parameters greatly impacts the stability of the power system. The system is highly oscillatory under these disturbance scenarios. Maximum oscillations are present in case of CPSS and also settling time is quite high than other stabilizers. The eigenvalues and its corresponding damping ratios are very close to the origin of s-plane. The performance of CPSS under these disturbances are close to instability.
Again, there are oscillations observed in case FPSS, FSMC-PSS and AFSMC-PSS. But, the proposed AIT2FSMC-PSS easily manages the disturbances with minimum peak values and settling time than other stabilizers. Again, eigenvalues and electromechanical modes damping ratios are found to be shifted well within the left half of s-plane which assures stability of the proposed stabilizer. In scenario-4, the most severe fault called 3u fault is applied as a disturbance. The simulated response is given in Fig. 5d. While accessing stability of the proposed stabilizer under this severe fault disturbance, it is observed that the system is oscillatory. Especially, CPSS possesses maximum settling time and overshoot time than others. Again, while comparing the eigenvalues, it is clear that the roots are closer to unstable region than other stabilizers.
Whereas, the proposed stabilizer shows superior oscillation damping characteristics with minimum of settling time of 2.34 s and peak overshoot of 4.6 p.u. to that of other stabilizers. The comparison of eigenvalues, damping ratios and oscillating frequencies of SMIB system under the   Stability assessment using adaptive interval type-2 fuzzy sliding mode… 7725 aforementioned disturbance scenarios are presented in Table 2.

Multimachine power system
In this sub-section, the viability of the proposed stabilizer is implemented in a 16-machine, 68-bus power system to illustrate its efficacy in a multimachine power system. Figure 6 shows the studied

Small signal stability analysis
In this sub-section, analysis of small signal stability of the proposed AIT2FSMC-PSS is performed for small disturbance. A 10% step increase in load demand is applied to bus-21 of area-5 at t = 1 s and the corresponding speed deviation in generators G13, G14, G15, G12 and G5 located at deferent areas of MMPS representing local mode of oscillations are given in Fig. 7a-e, respectively. The simulation responses of different generators reveal that the proposed AIT2FSMC-PSS is efficient in damping LFEOs than that of AFSMC-PSS, FSMC-PSS, FPSS and CPSS.
This establishes the effectiveness of the proposed AIT2FSMC-PSS in the improvement small signal stability with lesser oscillations in peak overshoot and settling time.

Transient stability analysis
In this sub-section, investigation of transient stability is performed to validate the efficacy of the proposed stabilizer in handling transient disturbances like 3-u faults. Different disturbances considered for the analysis of transient stability are as follows: Scenario-1: 3-u fault of 100 ms duration is applied at bus-36 of area-3.
Scenario-2: A 6-cycle 3-u fault at bus-18 of area-5. Scenario-3: A line outage between bus-30-31 of area-5. The simulations are performed for various speeds deviations of generators located at different geographical areas showing local and interarea oscillations are presented in Figs. 8, 9, 10, respectively. The stability analysis under these aforementioned fault disturbances is discussed below.
In scenario-1, a 3-u fault disturbance of 200 ms duration is created at bus-36 of area-3 at t = 1 s to analyse the stability performance. Under this severe fault disturbance, the system undergoes oscillations which is replicated  Table 3. By verifying the eigenvalues and damping ratios, it clearly seen that the Eigen are values well within the left of the S-plane in case of the proposed AITFSMC-PSS. This conforms the stability of the proposed stabilizer. In scenario-2, a 6-cycle, 3-u fault disturbance is created at bus-18 of area-5 at t = 1 s. The resulting speed deviations among G10-G13, G3-G7 and G1-G14 representing local and interarea mode oscillations under the aforementioned fault disturbance are depicted in Fig. 9a-c, respectively. Under this fault disturbance, the speeds of the synchronous generators oscillate from its rated speed. But, due to the application of proposed stabilizers, these oscillations settle back to their nominal value of 1 p.u. More oscillations are seen in case of CPSS whereas comparatively lesser in case of FPSS, FSMC-PSS, and AFSMC-PSS. The proposed AITFSMCPSS shows minimum oscillations with minimum peak values and settling time to that of other approaches. Again, eigenvalue analysis presented in Table 4 also supports the stability study. It is seen that the roots are more negative in case of AIT2FSMC-PSS than that of other stabilizers. This proves the efficacy and effectiveness of the proposed stabilizer handling transient disturbances and enhancing stability under a 3-u fault disturbance.
In scenario-3, the effectiveness of the proposed AIT2FSMC-PSS is studied for a line outage disturbance. In this scenario, the line connecting bus-30-31 of area-5 is tripped at t = 1 s and reclosed after 3-cycles. The corresponding speed deviations between generators G11-G13,  Fig. 10a-c, respectively. The impact of line outage disturbance is reflected on the simulation results, as it undergoes oscillations in speed deviations. The proposed approaches efficiently damps out the oscillations under this fault disturbance. The response of CPSS is quite oscillatory to that of FPSS, FSMC-PSS and AFSMC-PSS. The proposed AIT2FSMC-PSS exhibits efficient oscillation damping with minimum of peak values and settling characteristics. Again, eigenvalue analysis in Table 5 supports the stability as that the roots are shifted towards left half of s-plane in case of proposed AIT2DSMC-PSS. This proves the robustness of the proposed approach in handling the line outage disturbance.

Stability analysis for continuous load fluctuation
In this section, the performance of the proposed AIT2FSMC-PSS is analysed for a continuous load fluctuation. The pattern of load fluctuation is shown in Fig. 11a. Under this load fluctuation, the corresponding deviations in speed of generators located in similar and different regions are shown in Fig. 11a and b, respectively. The deviation of speed of generator-7 of area-5 and generator-13 of area-1 representing interarea oscillating mode is presented in Fig. 11b, whereas deviation of speed of generator-9 and generator-10 of area-3 representing local oscillating mode is presented in Fig. 11c. Here, the performance of the proposed AIT2FSMC-PSS is compared with FPSS, FSMC-PSS and AFSMC-PSS for this continuous load fluctuation. As observed from the simulated responses of local and interarea oscillations, it is found that the proposed AIT2FSMC-PSS is efficient in handling this continuous load fluctuations effectively with lesser values of overshoot and settling time as compared to other approaches and hence assures the stability.

Robustness analysis
In this subsection, the robustness of the proposed AIT2FSMC-PSS is evaluated for varying parameters of the power system. The parameters like mechanical power (P m ) and reference voltage (V ref ) are carried individually from -50% to ? 50% in steps of 25%. The simulations are performed and the corresponding time domain performance indices such as maximum peak overshoot (M p ) and settling time (T s ) are tabulated in Table 6. The comparative analysis clearly confirms the robustness of the proposed stabilizer as the responses vary nominally with the corresponding variations in the system parameters. Stability assessment using adaptive interval type-2 fuzzy sliding mode… 7729

Uncertainty analysis
This section is presented to confirm the viability of the proposed AIT2FSMC-PSS in handling uncertainties of power system. Uncertainties are due to error in measurement, disturbances, parameter variations, noise, etc. The AI2TFSMC-PSS has shown its effectiveness in tackling various small and transient disturbances. In order to verify the performance of AIT2FSMC-PSS to deal with uncertainties, a random noise is supplemented with the input signal e t ð Þ (Paital et al. 2021a).
e t ð Þ ¼ e t ð Þ þ c Â r and n where c denotes random numbers, represents uncertainty level and its value is taken as 0.05 (Adjeroud et al. 2015).
The simulation results are presented in Fig. 12a-c, respectively. As observed, the proposed stabilizer can easily handles the noise present in the power system and it shows lesser peak and settling characteristics to the response of the stabilizer with noise. A comparison of ITAE values with and without noise is given in Table 7. Both simulation and comparison prove the effectiveness of AIT2FSMC-PSS in handling uncertainties present in the power system.

Comparative analysis
This sub-section presents a quantitative performance analysis of the designed stabilizers. Different PIs like maximum peak overshoot (Mp), settling time (Ts), integral of time absolute error (ITAE) and figure of demerit (FD) are considered for evaluating the performance of the stabilizers. A comparison of PIs of responses of SMIB system under different disturbance scenarios is presented in Table 8. As analysed in the previous subsections, the proposed AIT2FSMC-PSS shows superior oscillation damping with minimum of its PIs under different disturbance scenarios. The results are bolded to show its superiority over others. Again a comparison of PIs of MMPS is presented in Tables 9 and 10 respectively. Small signal stability analysis is given in Table 9, whereas transient stability analysis is given in Table 10. Both small signal and transient stability analysis indicate the robust performance of the proposed AIT2FSMC-PSS with least values of its peak and settling characteristics.
A comparison of ITAE values and FD values of the proposed stabilizers is presented in bar charts as presented in Figs. 13a-c and 14a-c respectively. The least values of these indices indicate better stability performance of the stabilizers. The bar charts indicate that AIT2FSMC-PSS possesses minimum of ITAE and FD values to that of other stabilizers. It proves the improved stability performance of the proposed stabilizer.

Frequency domain analysis
This sub-section presents frequency stability analysis using Bode, root locus and Nyquist plot shown in Fig. 15a-d. From the linearized model, state matrices are derived and the above said plots are plotted. The root locus and its zoomed response are depicted in Fig. 15a and b. The plot clearly shows that the roots are more negative and located farther from the origin than that of other stabilizers. It shows the stable performance of the proposed stabilizer. Again, Bode plot in Fig. 15c shows stable performance as both gain margin (GM) and phase margin (PM) are quite small in case on no control action which makes the system  unstable. Whereas, GM and PM lies within infinity and -1800 in case of the proposed AIT2FSMC-PSS than others.
The large values of GM and PM guarantee performance without loss of stability. Again, Nyquist plot shown in Fig. 15d shows stable performance of the proposed AIT2FSMC-PSS as the critical point (-1 ? j0) is encircled in counter clockwise direction which guarantees the stability.

Conclusion
A robust AIT2FSMC-PSS is presented for damping LFEOs in both SMIB and MMPS subjected to various uncertainties involving noise and external disturbances. A robust sliding surface is implemented to keep the system stable under disturbance conditions. A modified control law is presented to avoid chattering phenomenon while selecting the parameters of IT2FLS. Lyapunov synthesis is adopted for the stability analysis and to assure that the error converges asymptotically to zero even under uncertainties and  Again, frequency domain stability analysis is also performed to ensure the stability performances of the proposed stabilizers. Both simulation and comparative analysis suggest the dominance of the proposed stabilizer in damping LFEOs in the power system. In both time and frequency domain, performance indices offer lesser peak and settling characteristics, disturbance rejection, offer more stable performance, ensure better robustness against parameter uncertainties and assure transient stability in comparison with that of AFSMC-PSS, FSMC-PSS, FPSS and CPSS. M p (9 10 -3 ) T s (s) M p (9 10 -3 ) T s (s) M p (9 10 -3 ) T s (s) M p (9 10 -3 ) T s (s) M p (9 10 -3 ) T s (s)  In future work, the control structure of adaptive interval type-2 fuzzy sliding mode controlled PSS will be improved and the proposed control approach will be studied in multimachine power system integrated with distributed generation systems (DGs) under different penetration levels and disturbances. Further, the study will be extended to wide-area monitoring and control with the design of robust and adaptive damping controller under various disturbances, parametric variations and uncertainties.