5.1. Fuzzy Logic Energy Management System
The artificial neural network MPPT controller provides a reference voltage VC1ref that the lower level controller may track. A Li-ion battery and a SC are employed in the proposed storage system. The control’s goal is to charge and discharge the storage component while maintaining a steady grid voltage during transients. The current going from/to the storage to/from the DC-link capacitance represents the power necessary to complete the power flow. Due to the buck-boost converter’s non-minimum phase behavior, the DC bus voltage VDC cannot be set directly to VDCref. To keep the bus voltage at the reference voltage VDCref, the PI corrector calculates the DC bus reference current Idcref.
The FLS (
Figure 6) delivers the battery and SC reference currents (
IL2ref and
IL3ref, respectively). These reference currents will allow the DC bus voltage to stay constant regardless of the load behavior or variations in the PV generator power drawn.
To divert unexpected power variations into the SC, a low-pass filter is applied to the Idc current. The Idcref current is transmitted through this low filter to create the battery’s current desired Idcref. The difference between the Idcref and IL2ref′ determines the SC current reference IL3ref′.
The SoC of the battery and the SC must be considered while developing the reference currents. To choose the exact reference current, three switches are utilized, which are controlled by the FLS in function of IL2ref*, IL2ref**, and IL3ref*.
Switch 1: Lets you choose between Idcref and IL2ref′.
If Idcref is Negative and SoCSC is 95%, IL2ref″ equals Idcref; else, IL2ref″ equals IL2ref′.
If Idcref is Positive and SoCSC is 25%, IL2ref″ equals Idcref; else, IL2ref″ equals IL2ref′.
Switch 2: Lets you to choose between IL2ref″ and 0.
If Idcref is Negative and SoCba is 95%, IL2ref equals zero; else, IL2ref equals IL2ref″.
If IL2ref″ is Positive and SoCba is 25%, IL2ref equals zero; else, IL2ref equals IL2ref″.
Switch 3: Lets you choose between IL3ref′ and 0.
If IL3ref′ is Negative and SoCSC is 95%, IL3ref equals zero; else, IL3ref equals IL3ref′.
If IL3ref′ is Positive and SoCSC is 25%, IL3ref equals zero; else, IL3ref equals IL3ref′.
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If the storge reference currentIL2ref is Negative, then the PV generator will produce greater power than the load and the SoC of SC is greater that 95%, meaning the IL3ref must be null.
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If the storge reference current IL2ref is positive, then the PV generator will fail to deliver sufficient power and the SoC of SC is 25%, meaning the IL3ref must be null.
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If the Supercapacitor’s SoC exceeds 25%, the SC will start to discharge.
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When the PV panels supply the power required by the load, the IDCref is zero.
The aim of utilizing a FLS in this study is to regulate the total system power flow while keeping the
SoCba and the
SoCSC at their allowable intervals of their
SoC. As illustrated in
Figure 7 and
Figure 8, the FLS employed in this work comprises four inputs and three outputs.
The fuzzy logic system’s inputs are the storage reference current Idcref, the SoCba, the SoCSC, and IL3ref.
The outputs are IL2ref* for the control of switch 1, IL2ref**for the control of switch 2, and IL3ref* for the control of switch 3.
The FLS calculates the control of the three switches utilized for DC bus regulation using the data from these four inputs, as shown in the flowchart.
Membership functions:
The described methodology’s objective is to develop membership functions for the FLS’s input and output variables. The input membership functions are used to switch between the various operational modes.
Figure 7 and
Figure 8 depict them. To accommodate the needs of the proposed strategy, the membership functions of the storage levels (
Figure 7a,d) are based on two levels; N stands for negative, and P stands for positive, where they represent the sign of the reference current
Idcref and
IL3ref’,with N and P representing the charge and discharge of the SC and battery, respectively. Between 100 and −100 A, the reference currents
IL3ref′ and
Idcref are considered.
The described methodology’s objective is to develop membership functions for the inputs and outputs variables of the FLS. The membership functions are employed to switch between the various operational modes.
Figure 5 and
Figure 6 depict them. To satisfy the demands of the suggested approach, the membership functions are based on two levels, N and P (
Figure 7a–d).
The membership of the
SoCba and the
SoCSC is depicted in
Figure 7b,c. Similarly, three levels are defined as follows:
The SoC is represented by a low level of 25% (between 0 and 25%).
The SoC is represented by a middling level of 25 to 95%.
The SoC is represented at a high level of >95%.
The membership functions of the outputs are illustrated in
Figure 8a–c, with switch1 controlling
IL2ref*, switch2 controlling
IL2ref**, and switch 3 controlling
IL3ref*. These functions are divided into two tiers, N and P, which indicate the sign of the switch commands.
Rules of FLS:
The rules of the FLS generated for energy management storage are derived from system behavior analysis. It must be recognized in their formulation that utilizing different control rules based on operational situations might increase the performance of the energy management storage. The rules that link the FLS are shown in the following tables (
Table 2,
Table 3 and
Table 4):
5.2. PV Subsystem Controller
Lets us look at the control u
1 that is used to stabilize
x1 and
x2; it is created via utilizing a sliding-mode approach to create an
x1ref desiredvalue to
x1 that is employed to guide
VC1 to its equilibrium point after imposing a desirable dynamic behavior. From system (2),
Let us first establish the output tracking error as
; the control input’s u
1 goal is to correctly incorporate the PV energy while also obtaining the maximum power possible from the PV generator. This is known as MPPT, and it consists of regulating the voltage
VC1 (
x1) to its reference
VC1ref (
x1ref) delivered by a higher-level controller and deemed constant throughout each time period
T. The desired dynamic for
x1 is introduced as
From Equations (3)and (4), we can then design
To track the state
x2 to its reference value
x2ref, we establish the following error term for the controller’s design:
To develop an SMC, an SF was chosen that allows the system to reach the SF and achieve the required desired value. Since the PV energy subsystem’s state space model has just only one control law input
u1, the SF has been defined as follows:
where
a2 is a positive constant for the SF design parameter. If we derive the time of Equation (7), we obtain the following:
Equation (6)’s time derivative provides the following equation:
Replacing
from Equation (3) into Equation (9) yields
Substituting
from Equation (10) into Equation (8) yields
The control law
u1 was selected in a manner so that it achieves global asymptotic stability, as indicated below:
where
uSC1 denotes the switching control that keeps the track on the SF and
uNC1 denotes the nominal control that brings the trajectory state to the SF. Using
to obtain the value of
uNC1 yields
and
In Equations (13) and (14), the condition (always met in practice)
,
is a constant ranging from 0 to 1 that converges the subsystem (3) to the SF,
is the degree of nonlinearity used to prevent chattering, and
k1 and
k2 are gains (positive) that are used to modify the control law
u1.
Sgn is the Signumfunction, which is in the following form:
Substituting the value of
u1 in (12), (13), and (14) results in the following:
For stability analysis, rearrange (16) as follows:
where
and
.
To evaluate the subsystem’s (2) stability, the following Lyapunov candidate function was used:
The Lyapunov function in (18) has a quadratic form as follows:
where
,
,
, and
.
Its time derivative along the solution of (16) yields the following results:
where
k1 and
k2 are positive and
.
In this case, if the controller gains
k1 and
k2 are positive, then
. The controller
u1 fits the Lyapunov stability condition, as demonstrated by (18), ensuring that the error convergence is zero in the limited time. It also shows that the PV subsystem can provide maximum power throughout the day.
5.3. Design of the Battery’s Current Control
We must create a control law
u2 to direct
IL2 toward its reference
x4ref =
IL2ref. It is developed using a sliding-mode technique. From system (1),
To track the state
x4 to its referencevalue
x4ref, we establish the following error term for the controller’s design:
To develop an SMC, an SF was chosen that allows the system to reach the SF and achieve the required desired value. Since the current battery subsystem’s state space model just has only one control law input
u2, the SF has been defined as follows:
where
a4 is a positive constant for the SF design parameter. We derive the time of Equation (24), obtaining the following:
Equation (23)’s time derivative provides the following equation:
Substituting the value of
from Equation (22) into Equation (26) yields
Substituting
from Equation (27) into Equation (25) yields
The control law
u2 was selected in a manner that achieves global asymptotic stability, as indicated below:
where
uSC2 denotes the switching control that keeps the track on the SF and
uNC2 denotes the nominal control that brings the trajectory state to the SF. Using
to obtain the value of
uNC2 yields
and
In Equations (30) and (31), the condition (always met in practice)
,
is a constant ranging from 0 to 1 which converges the subsystem (22) to the SF,
is the degree of nonlinearity used to prevent chattering, and,
k3 and
k4 are gains (positive) that are used to modify the control law
u2. Substituting the value of
u2 in (29), (30), and (31) results in the following:
For stability analysis, rearrange (32) as follows:
where
and
.
To evaluate the system’s stability, the following Lyapunov candidate function was used:
The Lyapunov function in (34) has a quadratic form
, where
and
Its time derivative along the solution of (34) yields the following results:
where
k3 and
k4 are positive and
.
In this case, if the controller gains k3 and k4 are positive, then .
The controller
u2 fits the Lyapunov stability condition, as demonstrated by (38), ensuring that the error convergence is zero in the limited time. This controller guarantees that the battery is effectively managed and that the DCMG operates consistently under different load situations.
5.4. Control Law Design for Supercapacitor
We must design a control law
u3 to control the dynamics
IL3 to ensure that the supercapacitor charges and discharges as required. It is developed using a sliding-mode technique. From system (1),
To track the state
x6 to its referencevalue
x6ref, we establish the following error term for the controller’s design:
To develop an SMC, an SF was chosen that allows the system to reach the SF and achieve the required desired value. Since the current supercapacitor subsystem’s state space model has only one control law input
u3, the SF has been defined as follows:
where
a3 is a positive constant for the SF design parameter. We derive the time of Equation (41), obtaining
Equation (40)s’ time derivative provides the following equation:
Substituting the value of
from Equation (39) into Equation (43) yields
Substituting
from Equation (44) into Equation (42) yields
The control law
u3 was selected in a manner that achieves global asymptotic stability, as indicated below:
where
uSC3 denotes the switching control that keeps the track on the SF and
uNC1 denotes the nominal control that brings the trajectory state to the SF. Using
to obtain the value of
uNC3 yields
and
In Equations (47) and (48), the condition (always met in practice)
,
is a constant ranging from 0 to 1 that converges the subsystem (39) to the SF,
is the degree of nonlinearity used to prevent chattering, and
k1 and
k2 are gains (positive) that are used to modify the control law
u3. Substituting the value of
u3 in (46), (47), and (48) results in the following:
For stability analysis, rearrange (49) as follows:
where
and
.
To evaluate the system’s stability, the following Lyapunov candidate function was used:
The Lyapunov function in (51) has a quadratic form
,where
and
Its time derivative along the solution of (51) yields the following results:
where
k5 and
k6 are positive and
.
In (54),
P6 is a positive definite matrix with positive
k5 and
k6 values. By carrying out the Lyapunov function stability analysis as described in the preceding paragraph, it is possible to deduce that the control law
u3 renders the subsystem (39) asymptotically stable, as demonstrated by (55). These controllers guarantee that the supercapacitor is effectively managed and that the DCMG operate consistently under different load situations.