The theorem of fuzzy-set (FS) was first presented to the academic society by the sole contribution of Zadeh in the mid-sixties. FS is simply an approach to interpret crisp values into changing degrees of belonging (or changing degrees of truth) using lingual variables. In the traditional (nonfuzzy, conventional, classic, hard, ordinary, or crisp) set, any separate element of the universe-of-discourse is either belonging to the set or is not belonging to the set. Thus, the degree of membership associated with any separate element is crisp value, i.e., it is either yes (in the set and takes the number 1) or no (not in the set and takes the number 0). In other words, FS is a generic form of the traditional set. FS is able to process the concepts and notions that humans use in their day-to-day life such as, “very high,” “high,” “medium,” “low,” and “very low” without the necessity to know the definite ranges associated with each concept [28].

Each FS is distinguished by a unique function that determines the degrees of belonging of crisp the values usually named the membership function (MF) which has a nonlinear nature and determines the degrees of belonging of crisp values associated with certain lingual variable utilizing a crisp value lying in the range from 1 to 0. MF has many types, such as sigmoidal or s-shaped, triangular, generalized bell (GB), trapezoidal, and Gaussian. Therefore, the fuzzy logic, which is based upon the FS theory, resembles human thinking and reasoning depending on its reliance on changing degrees of belonging accompanied by the applying of lingual variables. Unlike its Boolean counterpart, which implements two levels of logic (1 or 0), the fuzzy-logic implements logical levels of unlimited values lying between 1 and 0 to find solutions for issues that have uncertainties or ambiguous situations [28].

Despite its origin in the US during the mid-sixties along with its theoretic validation in the UK in during the mid-seventies, fuzzy-logic acquired a broad reputation among the peer community accompanied by practical real-life applications in Japan in the eighties. Fuzzy control is different from the traditional control in that it does not require a mathematical model of the system to be considered [28]. It regulates the inputs to obtain suitable outputs by simply scanning the present output of the system to be considered by depending on humanlike expert’s decision procedures in regulating the plant inputs [28, 29]. FLCs have been proposed for a variety of multidisciplinary network issues with the containment of a significant number of uncertainties since the early nineties [29]. It has been long identified as a very effective candidate for dealing with many issues such as, among others, generator’s excitation voltage control, load frequency control, load forecasting, power flow analyzing, electromechanical oscillation damping, transient stability improving, blade-pitch angle controlling in wind turbines, DC link controlling in energy conversion systems, transformer fault diagnosing, and economic dispatching [29].

Fuzzy Rules, or “fuzzy linguistic conditional statement,” “fuzzy if-then rule,” “linguistic implication,” “linguistic fuzzy information,” “fuzzy implication” take the form “**IF** the rule antecedent or the premise, **THEN** the rule consequence or the conclusion,” [30]. The linguistic fuzzy rules implementation are based upon the procedures performed by skillful humanly operators that do not require to study the mathematical model of the plant under their operation. The skilled humanly operators, with operating experience, modify the system inputs to acquire the desired output levels by just only observing the current system outputs without knowing the dynamics system and parameter variations [31]. The conception of fuzzy lingual (or linguistic) variable was first emerged to the peer community by Zadeh in 1971 as an alternate approach for simulating human intellectual thinking [30]. A fuzzy-rule may have various variables in the premise and/or in the conclusion parts. It can be multi-input–multioutput (MIMO), multi-input–single-output (MISO) single-input–single-output (SISO), or single-input–multi-output (SIMO) [31]. Fuzzy rules perform an essential role in exemplifying the practiced controlling/modeling knowledge and expertise and in interfacing the inputs of the fuzzy-logic systems to their output variables [32]. Fuzzy reasoning is inference proceedings utilizing the fuzzy-logic to conclude the final output from the utilized fuzzy rules and well-known certainties. Fuzzy reasoning builds mainly on two prime frameworks, namely fuzzy relations, and fuzzy extension principle. Fuzzy-rule is then interpreted into fuzzy relation using extension principle where the crisp domain is extended into the fuzzy domain using certain mapping techniques such as one-to-one mapping, one-to-many mapping, or many-to-one mapping [30].

There are two main types of fuzzy relations, namely unary fuzzy relations where the FS possesses a one-dimensional membership function, and binary fuzzy relations where the FS possesses a two-dimensional MF. Zadeh came up with a composition operation, called Maximum–Minimum (Max–Min) composition or Max–Min product, to obtain the fuzzy relations [30]. FLC is a rule-based controller of the nonlinear type that relies on the manipulation of the expert knowledge. FLCs introduce superior performance levels by exploiting the expert’s knowledge conception in treating a broad variety of multidisciplinary control issues. The elementary configuration of generic FLC is typically comprised of the four primary stages: fuzzification, knowledge base, fuzzy-inference-system (FIS), and defuzzification which is graphically supported by Fig. 5 [31].

Defuzzification is simply a mathematical averaging function, and it is the union of all the rule consequences of the embedded fuzzy rules [30, 31]. There are six main techniques of defuzzification, namely centroid-of-area, the center-of-gravity, the bisector-of-area or equal-of-area, mean-of-maximum, smallest-of-maximum, largest-of-maximum. The defuzzification procedures, mainly in FLC with Mamdani FIS, represent heavy computational burdens in real-life fuzzy applications [33]. Takagi–Sugeno (TS) controllers do not require the final defuzzification procedure since the involved output MF is either a linear-function or constant value, i.e., numerically represented, which makes the TS controllers more efficient than Mamdani controllers from the computational perspective [28,29,30]. An example of the TS fuzzy-rule is presented as follows [30]. For zero-order Sugeno fuzzy model, if the first input is *x* and the second input is *y*, then output will be constant denoted by “c”. For first-order Sugeno fuzzy model, if the first input is *x* and the second input is y, then output will be *ax* + *by* + *c*. A higher-order TS fuzzy model could be obtained if the rule consequent is a nonlinear equation, if the first input is *x* and the second input is y, then output will be *ax*^{2} + *by* + *c*. The aggregated output of all TS fuzzy rules is achieved either via "weighted average,” or "weighted sum" operators which makes the time-consuming defuzzification process avoidable.

Many works of literature have proved that acquired damping levels from the employment of the alternator, or synchronous machine, speed deviation as an FLC input is much more enhanced than the levels achieved from employing other speed signals from the various shaft sections. Nonetheless, the alternator mass speed is not very challenging to capture suchlike the various steam turbine different sections speed deviations since the turbine is thermally insulated to decrease the thermal heat losses and also the steam prime mover rotor is rigorously sealed through a highly effective steam-sealing system [34]. So, the speed deviation pf the alternator mass in per-unit (p.u.), \({\mathrm{\Delta \omega }}_{\mathrm{Gen}.}\), is chosen as the FLC input and the output is fixed with either 1 or 0. The FLC takes the appropriate action if the speed deviation signal (\({\mathrm{\Delta \omega }}_{\mathrm{Gen}.})\) is different from the specified non-acting band based upon the equation given in (5):

$${\Delta \omega }_{Gen.}={\omega }_{Gen.}-{\omega }_{0}$$

(5)

where \({\Delta \omega }_{Gen.}\) is the deviation of the alternator rotor angular speed in p.u., \({\omega }_{Gen.}\) is the alternator rotor angular speed in p.u., and \({\omega }_{0}\) is the nominal alternator speed of rotation under normal conditions (it equals to 1 p.u.). The battery bank should inject real power if the \({\Delta \omega }_{Gen.}\) is having negative values and should be in the idle operation mode (i.e., not charging or discharging) elsewhere. While the resistive brake should absorb real power if the \({\Delta \omega }_{Gen.}\) is having positive values and should be out of service elsewhere. Three membership functions of the GB type are utilized to represent the inputs for the propositioned FLC and are portrayed in Fig. 6. The BES under consideration shall be selectively charged when the SoC falls below 85% upon normal system operating conditions to be ready for the varied awaited SSR events.

Three lingual variables, PD (Positive-Deviation), SD (Small-Deviation), and ND (Negative-Deviation), are defining the fuzziness of the FLC input speed signal for both controllers. Equation (6) introduces the expression of GB function utilized to determine the membership degree.

$${\mu }_{A}({\Delta \omega }_{Gen.})=\frac{1}{1+{\left|\frac{\Delta \omega-c}{a}\right|}^{2b}}$$

(6)

where \({\mu }_{A}({\Delta \omega }_{Gen.})\) is the membership degree, the ‘a’ parameter decides the bandwidth of the MF curve, the ‘b’ parameter decides the width of the MF curve crispy top, and the ‘c’ parameter decides the MF curve center.

The propositioned FLC output is an invariant number possessing either 1 or 0. Then, the rules employed for the BES controller will be, If (\({\Delta \omega }_{Gen.}\)) is ND, then FLC output will be 1, If (\({\Delta \omega }_{Gen.}\)) is SD, then FLC output will be 0, and finally If (\({\Delta \omega }_{Gen.}\)) is PD, then FLC output will be 0. In a corresponding manner, the rules employed for the resistive resistor brake FLC will be, If (\({\Delta \omega }_{Gen.}\)) is ND, then FLC output will be 0, If (\({\Delta \omega }_{Gen.}\)) is SD, then FLC output will be 0, and finally If (\({\Delta \omega }_{Gen.}\)) is PD, then FLC output will be 1. The output of the BES controller will be forwarded to the PWM control circuitry to decide either the BES will be in idle mode or discharging mode. Accordingly, the output of resistive brake FLC will be forwarded to the thyristor firing angle circuit to decide whether the resistor brake will be in-service or out of service. The switching strategy under consideration in this investigation is fuzzy-based discontinuous ON–OFF control in which the real power injection and dissipation moments are decided by the implemented FLCs.