Fractional-Order Fuzzy Control Approach for Photovoltaic/Battery Systems under Unknown Dynamics, Variable Irradiation and Temperature

For this paper, the problem of energy/voltage management in photovoltaic (PV)/battery systems was studied, and a new fractional-order control system on basis of type-3 (T3) fuzzy logic systems (FLSs) was developed. New fractional-order learning rules are derived for tuning of T3-FLSs such that the stability is ensured. In addition, using fractional-order calculus, the robustness was studied versus dynamic uncertainties, perturbation of irradiation, and temperature and abruptly faults in output loads, and, subsequently, new compensators were proposed. In several examinations under difficult operation conditions, such as random temperature, variable irradiation, and abrupt changes in output load, the capability of the schemed controller was verified. In addition, in comparison with other methods, such as proportional-derivative-integral (PID), sliding mode controller (SMC), passivity-based control systems (PBC), and linear quadratic regulator (LQR), the superiority of the suggested method was demonstrated.


Introduction
Today, photovoltaic (PV) panels are extensively used for energy production, due to their renewability, availability, and clarity [1,2]. However, the main problem in the use of PV panels, is their natural dependance to the weather conditions. Then, to get a stable output voltage, it is necessary that PV panels to be combined with some energy storage systems, such as batteries. A powerful management system is required to make a balance between energy generation, saving, and consumption. The accurate management of this class of hybrid systems is so challenging because of high dependence of renewable energy generators to the level of irradiation and temperature and the existence of high level of uncertainties, such as time-varying output load.
its proficiency is studied. In Reference [28], the effectiveness of a fuzzy controller is examined under partial shading conditions and variable temperature. In Reference [29], a fuzzy controller is designed for power management, and it is shown that the power quality is improved by the use of FLS-based control system.
In the most of the above studies, a simple type-1 FLS was used to cope with unknown mathematical models. However, high-order FLSs have more capability in practical nonlinear systems [30][31][32][33][34]. In addition, the tuning process in the most of above mentioned controllers is done as a non-adaptive and off-line approach. Furthermore, the robustness against fluctuation of irradiation and abrupt variation of load demand are not investigated. By the above motivations, in the current study, a new fractional-order control scenario was introduced on the basis of type-3 (T3)-FLSs such that T3-FLSs are online-optimized through the robustness investigation against perturbations. The main advantages are: • A new type-3 fuzzy fractional-order control scenario is proposed.

•
The dynamics of PV, converters, and battery are assumed to be unknown and are perturbed by variable irradiation, random temperature, and sudden changes in output load.

•
The new fractional-order adaptation rules are derived for T3-FLS such that the stability to be ensured.

•
New combustors are proposed such that the robustness to be guaranteed.

General View
The general block diagram is shown in Figure 1, and the detailed control block diagram is given in Figure 2. It is seen that the currents of the PV/battery I p /I b and output voltage V c are measured by current and voltage sensors. In Figure 1, T3-FLSs (F 1 ,F 2 ) are used for online dynamic estimation, and the controllers (µ p , µ b ) are designed on the basis of the online T3-FLS model. The parameters of T3-FLSs (θ 1 , θ 1 ) and control gains (ĝ 1 ,ĝ 1 ) are online-tuned. The compensators (µ cp , µ cb ) are carried out through the robustness study on basis of fractional-order calculus, tracking errors (e 1 , e 2 ), and upper bounds of approximation errors (Ē 1 andĒ 1 ) such that a good robustness to be achieved against dynamic perturbation and common irradiation and temperature disturbances. The detailed descriptions are given in Theorem 1.  The output load energy is supplied by PV or battery at each sample time. If the required energy cannot be supplied by the PV panel, then the battery systems get to work. In addition, whenever the generated energy by PV panel is more than needed, the extra energy is stored in battery. The converters are used to construct a switching mechanism between PV and battery. The schematic of the converter is depicted in Figure 3. By considering the states of switchers SW 1 , SW 2 , and SW 3 , four modes, as shown in Figure 4, can be derived [35]. If SW 1 and SW 3 are open and SW 2 is closed, then the switching mode is obtained as shown in Figure 4a. If SW 1 and SW 2 are closed and SW 3 is open, the switching mode is obtained as shown in Figure 4b. If SW 1 and SW 2 are open and SW 3 is closed, the switching mode is obtained as shown in Figure 4c. Finally, if SW 1 and SW 3 are closed and SW 2 is open, then the switching mode is obtained as shown in Figure 4d. For each switching mode, one state space formulation can be written. Then, by taking the average of four state space representation, the dynamics of plant are given as:ς where ς 1 , ς 2 , and ς 3 are the current of PV, current of battery, and load voltage, respectively. ι p /ι b , R, and C are values of inductances, resistor, and capacitor in the converter. ν p and ν v are the voltages of PV and battery, respectively, and µ p and µ b are control signals.

PV Modeling
By the single-diode modeling technique of PV panels, dynamics of PV are described as [36]: where all parameter definitions are given in Table 1. As depicted in Figure 5, the power of PV panel is not constant but is changed by its current. As it can be observed, at an optimal current, the maximum energy can be achieved. Table 1. Parameter definition of photovoltaic (PV); see Equation (2).

Parameter Description
Reference temperature ph (A) Photo generated currents

Parameter Unit Description
Voltage of battery open circuit η 1 and η 2 -Rates of charge and discharge E Max (J) Maximum storable energy

Uncertainty Estimation by Type-3 FLS
T3-FLSs are the generalization of T2-FLSs. The main idea for T3-FLS is presented in Reference [37], which was developed in this study for dynamic identification of a practical system. The basic superiority of T3-FLSs with respect to T2-FLSs is that, in type-3 membership functions (MFs), the upper and lower of uncertainties are considered to be fuzzy set, while, in the type-2 MFs, the upper and lower of uncertainties are considered to be crisp values. In this section, the proposed T3-FLS is explained. The structure is given in Figure 6. T3-FLSs are used to deal with dynamic perturbation of PV, battery, and other units. The details are given below. (1) The inputs of T3-FLSs are counterparts. It is seen that the secondary membership is also type-2 MF. For example, consider a type-2 MF for input x in which its primary domain is betweenū andū. In a type-3 MF for input x, the upper and lower bounds of primary domain are not constant, but they are the fuzzy sets. For input I p , one has:πψ 1 πψ1 where χ is the value of secondary membership.πψk ς j |χ and πψk ς j |χ are the upper and lower memberships forψ k ς j . Similarly to (9) and (10), for other inputs I b and V c , one has: (3) The lower rule firing is obtained as Equations (15) and (16): Vc |χ . . .
Similarly, for the upper rule firing, one has: (4) The output ofF 1 andF 2 are:F where θ i and φ i are: where M represents number of rules. φ r andφ R are: where n χ is number of slices.

Main Results
The control signals and tuning rules are summarized in the following theorem: Theorem 1. For the given controllers as Equations (23) and (24), tuning rules as Equations (25)-(28), and compensators as Equations (29) and (30), the asymptotic stability is guaranteed.
where υ, δ, k b , k p ,Ē i i = 1, 2 are constant. µ cp and µ cb represent compensators. ς 1r and ς 2r are reference signals for ς 1 and ς 2 , respectively. e 1 and e 2 are errors that are defined as e i = ς i − ς ir .ς i is defined as Proof. The output dynamics (1) are: To design µ p and µ b , the dynamics of ς 1 and ς 2 are estimated as suggested fractional-order T3-FLS model: where D α t ς i , i = 1, 2 are the fractional derivatives, andς 1 andς 2 are the estimation of ς 1 and ς 2 .ĝ 1 and g 2 are the estimations of 1/L p and 1/C, respectively.F 1 andF 2 are the T3-FLSs. From the universal estimation feature of FLSs, the dynamics of ς 1 and ς 2 in (32) can be written as: whereF * 1 andF * 2 are optimal T3-FLS, andĝ * 1 andĝ * 2 are the optimal values ofĝ 1 andĝ 2 , respectively. E i , i = 1, 2 are the approximation errors. From (32) and (33), the estimation error dynamics (ς i = ς i −ς i ) can be obtained as: Considering definitions:θ From (34), one has: Applying the control signals (23), (24), and (32) on the estimated model (32), one has: To examine the robustness and stability, Lyapunov function is considered as: By taking fractional time-derivative, D α t V is written as: From (40), D α t V becomes: From (41), one has: Considering tuning rules D α Then, one has: Now, applying the compensators results in: Then, it is proved that D α t V ≤ 0. Considering Barbalat's Lemma and from the fact thatV is bounded, the proof is completed.

Simulation Studies
The efficiency and good output voltage and power regulation performance of suggested technique is shown in this section. Simulation and control parameters are given in Tables 3 and 4.   Example 1. In this example, the normal condition is taken to account such that the irradiation is assumed to be fixed at level 300 w/m 2 . The trajectories of ς i , i = 1, 2 and power of PV P and controllers µ p and µ b are shown in Figures 7-11. From Figures 7-11, one can realize that the suggested scenario results in desired regulation performance, and the control signals have smooth shape. It is seen that the trajectories of the current of PV I p and the output voltage V c are converged to the desired level less than 10 s. Example 2. In this example, the irradiation is changed from 200 w/m 2 into 600 w/m 2 at time t = 55 s and also the temperature is changed as T = 50 − 10 sin(t). The trajectories of ς i , i = 1, 2 and power of PV P and controllers µ p and µ b are shown in Figures 12-16. It is observed that a well power regulation is achieved in spite of time-varying temperature and irradiation. It is seen that the effect of variation of irradiation is well handled, and output voltage is well regulated on its desired level. In addition, the current of PV track its optimal level in less than 10 s. Example 3. In this example, a Gaussian noise with variance 0.05 is added to temperature and the temperature is randomly changed between 30 and 50, irradiation is changed as same Example 2, and output load is suddenly changed from 70 (Ω) into 40 (Ω) at time t = 55 s. The trajectories of ς i , i = 1, 2 and power of PV P and controllers µ p and µ b are shown in Figures 17-21. It can be realized that a well tracking performance is achieved versus abrupt changes in load, time-varying temperature, and variable irradiation. In addition, it should be remembered that the dynamics of all units are unknown for the controller unit. It is seen that the effect of noise is also well eliminated, and the output voltage well tracks the reference signal. It should be noted that the T3-FLSs have better performance is noisy conditions, in contrast to the type-2 counterparts. Example 4. For the last example, a numerical and graphical comparison is provided with conventional PID [38], PBC [39], SMC [40], LQR [41], and our method on the basis of integer-order calculus (IOC). The simulation conditions are the same as Example 3 with the difference being that temperature and irradiation are changed at times t = 40 s and t = 80 s, respectively. The values of mean square errors (MSEs) in Table 5 and the trajectories of I p and V c in Figure 22 clearly show the superiority of the suggested control scenario. It is observed that the suggested method on the basis of fractional-order calculus (FOC) results in more accurate power/voltage regulation proficiency. It is seen that, when the dynamic perturbation occurrs at times 40 s and 80 s, the classic controllers LQR and PID failed to track reference signals. Furthermore, the settling time for the suggested controller is less than the others we compared.

Remark 1.
It should be noted that the perturbations, such as changes of irradiation, output load, and temperature, are not predicted. But, these perturbations are considered as a part of dynamic uncertainties. The dynamic uncertainties are handled by the online optimization of T3-FLSs and compensators.

Remark 2.
From the trajectories of I p and V c in all Examples, a small initial over shout can be seen. This problem is the cost of unknown dynamics assumption. In other words, the dynamics of all units are assumed to be unknown and are estimated by the suggested T3-FLSs. Furthermore, the effects of dynamic perturbations, such as changes of irradiation, temperature, and output load, are considered as a part of approximation error. The effect of approximation errors are eliminated by the online optimization of T3-FLSs and compensators. Because of stability considerations, the adaptation rate of T3-FLSs are chosen to be small. On the other hand, the upper bound of approximation errors in compensators are concentratively considered to be large. Then, it takes few times for regulation error to be reached on the zero level, and a small over shout begins to be seen. However, if the upper bounds of approximation errors are decreased, the initial peaks are diminished, but settling time is increased, and the subsidence of regulation error takes more time.

Remark 3.
The regulation accuracy and speed of the suggested controller depend on the modeling accuracy and speed. In other words, if the estimation error reaches zero level in less time, accordingly, the subsidence of the regulation error is increased. On the other hand, it has been shown in literature that, by the use of fractional-order calculus, the nonlinear dynamics can be modeled with more accuracy. Because of this motivation, the suggested controller was designed on basis of fractional-order calculus, and its effectiveness is shown in Example 4.

Conclusions
In this study, a new control approach considering T3-FLSs and fractional-order calculus was developed for voltage/power management in PV/battery systems. The uncertain time-varying dynamics are online-modeled by the suggested fractional-order T3-FLS. Because of one more degree of freedom in type-3 MFs, the capability of type-3 MFs to represent the high level of uncertainties is more than the type-2 and type-1 counterparts. It should be noted that the secondary membership in type-3 MFs is a type-2 MF. The new factional-order tuning rules are derived to optimize T3-FLS such that the stability to be guaranteed. In addition, the robustness is ensured by the adaptive compensators. In four simulation examples, the superiority and well performance of the suggested control scenario was demonstrated. In the first example, a normal condition was taken to account. In the second example, the impacts of variable irradiation and temperature were examined. In the third example, in addition to the variable irradiation and temperature, an abrupt change in output load was also applied on the system as dynamic perturbation. Finally, in the last example, a comparison with other popular techniques was completed. For future studies, the optimization of the value of fractional-order and the rule database of the suggested fuzzy controller can be considered. Also the uncertainties can be estimated with other intelligent systems such as group method for data handling neural networks (GMDH-NNs) and the fuel cells can be added to PV/Battery system to ensure the maximum charging of battery. Furthermore, the upper bound of approximation error (AE) is assumed to be known and fixed in this paper, for the our future studies, the upper bound of AE is online estimated and then a new compensator in integer-order calculus is designed to deal with the effects of AEs and by the Lyapunov and LaSalle's invariant set theorems the asymptotic stability is investigated.