Control and Energy Management of Hybrid Renewable DC Microgrid by Using Flatness Method with Predictive Neural Network and Fuzzy PI Regulation

The global energy transition to renewable energy resources has risen significantly in the last few years, propelled by the pressing need to reduce greenhouse gas emissions and reliance on fossil fuels, which have limited reserves and cause pollution. Photovoltaic (PV), wind turbine (WT) and water turbine systems are the most deployed renewable technologies, due to their ecological sustainability, economic effectiveness, and modularity. However, their intermittent nature and dependence on meteorological conditions present significant challenges to ensure stability, reliability, and efficient energy utilization in power systems. To address these issues, hybrid renewable energy systems (HRES) are an attractive solution, as they integrate several power sources managed using power electronics converters [1]. A combination of PV, WT, and battery energy storage systems (BESS) have garnered increasing attention as a good solution for the two operational modes of microgrid: isolated mode and grid-connected. In these systems, the battery is essential for balancing generation and demand, mitigating renewable variability and regulating the DC-link voltage [2, 3].

Despite their advantages, the operation and management of hybrid microgrids remain challenging. Advanced control strategies are essential due to the nonlinear dynamics of power converters, the fast fluctuations in renewable energy sources, and the uncertain changes in load. Traditionally, proportional-integral (PI) controllers have been frequently used in the regulation of microgrid converters because of their simplicity and ease of installation [4]. Nevertheless, their limitations have been emphasized in numerous studies when operating in dynamic conditions. For example, Ali et al. [5] found that PI controllers have longer settling periods and undershoot compared to Model Predictive Control (MPC). In general, PI regulators have a fixed-gain structure that renders them susceptible to parameter variations and external disturbances, which can restrict their robustness in microgrids with significant renewable integration [6]. To overcome these shortcomings, several studies have investigated intelligent control strategies. Among them, fuzzy logic controllers (FLCs), which were initially proposed in 1975 [7], are appealing because of their capacity and ability to manage uncertainties while avoiding the need for precise system modeling. In particular, fuzzy-PI schemes combine a traditional PI regulator with the flexibility of fuzzy inference, enabling real-time adjustment of gains [8]. Recent research demonstrates that fuzzy-PI regulators enhance the robustness and convergence of renewable energy systems [9, 10].

In addition to local regulation, predictive control approaches have become more significant for the supervisory control of nonlinear systems. In particular, MPC is extensively utilized in microgrids, providing optimization and improved performance over a finite horizon compared to traditional strategies [11]. However, the computational complexity of MPC frequently restricts its application in real-time scenarios [12]. Another option is predictive neural networks (PNNs), which use the approximation abilities of a multilayer perceptron to predict future trajectories and learn system dynamics. PNN functions as an anticipatory indicator of DC-link energy, offering real-time adjustments for deviations, thereby improving the robustness of supervisory control strategies [13, 14].

In modern control theory, differential flatness, first introduced by Fliess et al. [15], is another efficient and robust framework that provides explicit reference generation for nonlinear systems. It has been successfully applied across various engineering disciplines, including electric vehicle, robotic control, power electronic systems, etc. [16, 17]. Differential flatness is used in hybrid microgrids to generate clear reference trajectories and facilitate coordinated operation, especially for energy balancing and hierarchical control [18]. In practical applications, inner converter loops are frequently regulated using PI controllers, while the integration of predictive and intelligent controllers within the flatness framework can further enhance robustness and dynamic performance under uncertainties.

In this study, an efficient energy management system is proposed based on flatness non-linear control theory hybridized with a PNN control for a microgrid comprising photovoltaic, WT, and battery systems. To enhance robustness and dynamic adaptability, fuzzy-PI controllers are used in the inner loops. This combination of strategies provides a resilient architecture that improves transient response, reduces overshoot, and enhances power sharing between the load, renewable energy sources, and energy storage systems.

The remainder of the paper is structured as follows: Section 2 describes the structure of the hybrid system considered in this study. Section 3 highlights the suggested efficient energy management strategy. Section 4 provides the simulation results and their discussion. Finally, Section 5 presents the conclusion of the paper.

3.1 Principle of flatness control approach

The differential flatness can be used to control nonlinear dynamic systems. A system is defined to be flat if there exists a flat output that can be employed to express all the system states and control variables as functions of this output and its derivatives [25, 26]. Mathematically, for a nonlinear system described by Eq. (1):

$\dot{x}=f(x, u)$              (1)

The flatness theory reduces the order model and can be expressed as follows:

$y=\phi\left(x, u, \dot{u}, \ldots, u^{(\alpha)}\right)$                (2)

$\left\{\begin{array}{c}x=\varphi\left(y, \dot{y}, \ldots, y^{(\beta)}\right) \\ u=\psi\left(y, \dot{y}, \ldots, y^{(\beta+1)}\right)\end{array}\right.$                (3)

where, x denotes the state variables, y is the flat output, and u is the control input variable. With the functions of the smooth mappings $\phi, \varphi$ and $\psi$. The term $\mathrm{y}^{(\beta+1)}$ represents the derivative of the output $(\beta+{1})^{\text {th }}, \alpha$ is a finite number of the derivative, and rank $(\phi)=\mathrm{m}, \operatorname{rank}(\varphi)=\mathrm{n}$, and $\operatorname{rank}(\psi)=\mathrm{m}$ [27]. The main advantage of this method is that the system trajectories can be determined and estimated by the flat output and its derivatives. This characteristic makes the flatness approach attractive for nonlinear and hybrid systems [28, 29].

3.2 Mathematical modeling of the hybrid system

For the hybrid power system under study, the PV, WT, and battery currents are assumed to follow their respective reference signals with negligible error. Accordingly, the reference currents are given by the following Eqs. (4), (5), and (6):

$I_{P V}=I_{P V r e f}=\frac{P_{P V}}{V_{P V}}=\frac{P_{P V r e f}}{V_{P V}}$              (4)

$I_{W T}=I_{W T r e f}=\frac{P_{W T}}{V_{W T}}=\frac{P_{W T r e f}}{V_{W T}}$             (5)

$I_{\text {Batt }}=I_{\text {Battref }}=\frac{P_{\text {Batt }}}{V_{\text {Batt }}}=\frac{P_{\text {Battref }}}{V_{\text {Batt }}}$               (6)

where, P_PV, and P_WT represent the PV and wind generated power, respectively, while P_Batt denotes the battery power. V_PV, V_WT, V_Batt and I_PV, I_WT, I_Batt are the voltage and current of the hybrid system respectively. In this study, only static losses are considered in converter models, where r_pv, r_wt, and r_batt correspond to the static loss parameters of the PV, WT, and battery converters, respectively.

The capacitive energy for both the DC bus and the battery bank can be expressed, according to the study by Thounthong et al. [30], by:

$Y_{B u s}=\frac{1}{2} C_{B u s} \times V_{B u s}^2$              (7)

$Y_{\text {Batt }}=\frac{1}{2} C_{\text {Batt }} \times V_{\text {Batt }}^2$                 (8)

where, C_Bus is the capacitance of the DC bus, C_Batt the battery capacitance, V_Bus is the DC bus voltage and V_Batt is the instantaneous voltage of the battery.

The total electrostatic energy Y_Tot stored in the DC-link capacitor C_Bus and in the battery C_Batt can also be written as below:

The capacitive energy for both the DC bus and the battery bank can be expressed by:

$Y_{\text {Tot }}=\frac{1}{2} C_{\text {Bus }} \times V_{\text {Bus }}^2+\frac{1}{2} C_{\text {Batt }} \times V_{\text {Batt }}^2$              (9)

As portrayed in Figure 1, the derivative of DC-bus capacitive energy is defined using P_PVo, P_WTo, P_Batto, and P_load by the following differential Eq. (10):

$\dot{\mathrm{Y}}_{\text {Bus }}=P_{\text {Pvo }}+P_{\text {WTo }}+P_{\text {Batto }}-P_{\text {load }}$            (10)

where,

$P_{P v o}=P_{P V}-r_{p v} \times\left(\frac{P_{P V}}{V_{P V}}\right)^2$               (11)

$P_{W t o}=P_{W T}-r_{w t} \times\left(\frac{P_{W T}}{V_{W T}}\right)^2$               (12)

$P_{\text {Batto }}=P_{\text {Batt }}-r_{\text {batt }} \times\left(\frac{P_{\text {Batt }}}{V_{\text {Batt }}}\right)^2$               (13)

The power demanded by the load is defined as:

$P_{\text {load }}=V_{\text {Bus }} \times I_{\text {load }}=\sqrt{\frac{2 Y_{\text {Bus }}}{C_{\text {Bus }}}} . I_{\text {load }}$            (14)

3.3 Flatness of the power system model

To verify the flatness property [31], the system introduced above is analyzed. According to the flatness theory, the flat outputs Y, the control input variables u, and the state variables x are expressed as follows:

$Y=\left[\begin{array}{l}Y_{\text {Bus }} \\ Y_{\text {Tot }}\end{array}\right]=\left[\begin{array}{l}Y_1 \\ Y_2\end{array}\right]$              (15)

$u=\left[\begin{array}{c}P_{\text {Batt }} \\ P_{\text {PVdemd }}\end{array}\right]=\left[\begin{array}{l}u_1 \\ u_2\end{array}\right]$               (16)

$x=\left[\begin{array}{c}V_{\text {Bus }} \\ V_{\text {Batt }}\end{array}\right]=\left[\begin{array}{l}x_1 \\ x_2\end{array}\right]$              (17)

where, we assume that the photovoltaic (PV) is the principal and the first primary source for the hybrid proposed PV-WT-Battery power system, and P_PVdemd is the required power of the differential flatness control algorithm.

The electrostatic energy Y_Bus of the DC bus of the system is assumed constant and defined as a flat output Y₁, and can be written as:

$\dot{\mathrm{Y}}_{\text {Bus }}=0=P_{\text {Pvo }}+P_{W T o}+P_{\text {Batto }}-P_{\text {load }}$             (18)

Also, the electrostatic energy Y_Bus is constant and represented as a flat output Y₂. Therefore:

$\dot{\mathrm{Y}}_2=0=P_{P v o}+P_{W T o}-P_{l o a d}$             (19)

It is stocked in the DC bus capacitor C_Bus and in the battery bank.

The stat variables (x₁, x₂) of DC link voltage and the battery voltage respectively, can be written as:

$x_1=V_{B u s}=\sqrt{\frac{2 \times Y_{B u s}}{C_{B u s}}}=\varphi_1\left(Y_1\right)$              (20)

$x_2=V_{\text {Batt }}=\sqrt{\frac{2\left(Y_{\text {tot }}-Y_{\text {Bus }}\right)}{C_{\text {Batt }}}}=\varphi_2\left(Y_1, Y_2\right)$              (21)

From (10) to (14), the input control variables u can be calculated from the flat outputs Y and their time derivatives:

$u_1 = 2 \cdot P_{\text{BattLim}} \left[ 1 - \sqrt{ 1 - \left( \frac{\dot{Y}_1 + I_{\text{load}} \times \varphi_1(Y_1) - P_{Pvo} - P_{WTo}} {P_{\text{BattLim}}} \right) } \right]$           (22)

$u_1=\psi\left(Y_1, \dot{Y}_1\right)=P_{\text {Battref }}$            （23）

$u_2 = 2 \cdot P_{\text{totLim}} \left[ 1 - \sqrt{ 1 - \left( \frac{\dot{Y}_2 + I_{\text{load}} \times \varphi(Y_1)} {P_{\text{totLim}}} \right) } \right] = \psi_2(Y_1, \dot{Y}_2)$               （24）

where,

$P_{\text {BattLim }}=\frac{V_{\text {Batt }}^2}{4 r_{\text {batt }}}$           (25)

$P_{\text {totLim }}=P_{\text {PVLim }}+P_{\text {WTLim }}$               (26)

P_BattLim is the limited maximum power from the battery buck-boost converter.

Therefore, and according to the preceding design, the proposed reduced order-model can be considered as a flat system [32].

3.4 The control law

The control loop of the battery energy depends on the regulation of the total electromagnetic energy. The control law of the battery used in this work can be written as [33]:

$\left(\dot{Y}_2-\dot{Y}_{2 r e f}\right)+K_{21}\left(Y_2-Y_{2 r e f}\right)=0$               (27)

This gives:

$\dot{Y}_2=\dot{Y}_{2 r e f}+K_{21}\left(Y_{2 r e f}-Y_2\right)$             (28)

where, Y_2ref is the reference of total electrostatic energy (see Eq. (9)), and K₂₁ represents the control parameter.

3.5 PNN law for DC-Bus energy stabilization

The neural predictor is designed to allow the user some freedom in selecting the inputs/states/outputs variables and adopting a fully connected MLP (Multi-Layer Perceptron) with tanh hidden activations and a linear output. The number of hidden neurons is determined iteratively until the validation error ceases to improve.

Training is based on time-aligned sequences generated from the MATLAB/Simulink: the control inputs used for the plant and the associated measured outputs. The network is thus identified as a forward predictor, trained by minimizing the prediction error so that its output closely matches the plant’s measured response over time.

During operation, the trained neural network is used within a predictive control framework to capture the short-term evolution of the DC-link over a finite prediction horizon. Based on the reference trajectory (for example, from the DC-bus energy set-point), the proposed PNN directly generates the control action at each sampling instant to ensure accurate tracking while limiting excessive control variations. In this way, the predictive objective is preserved without solving an explicit receding-horizon optimization problem, since the control law is obtained through a single forward evaluation of the trained MLP.

In this work, the neural network model uses the DC-link energy as input (along with optional past values and additional signals) to predict the next step output, which corresponds to the power supplied by the DC bus. This predicted power is the representation of the plant’s dynamic response. The training dataset contains this input value and the output target, and was generated from MATLAB/Simulink simulations. The network is trained offline using the Levenberg-Marquardt algorithm (LM), and the performance was evaluated using the mean square error MSE.

The model used to predict the output of the control law is given by the following energy balance equation:

$\dot{\mathrm{Y}}_{\text {Bus }}=0=P_{\text {Pvo }}+P_{W T o}+P_{\text {Batto }}-P_{\text {load }}$                (29)

The neural network is trained to minimize a performance criterion derived from classical predictive control, expressed as:

$\begin{aligned} J=\sum_{j=N_1}^{N_2}\left(Y_{\text {Bus }}(t\right. & \left.+j)-\dot{\mathrm{Y}}_m(t+j)\right)^2 +\rho \sum_{j=1}^{N_2} u(t+j-1)-u(t+j -2))^2\end{aligned}$               (30)

where, N₁, N₂, and N_u are, respectively, the horizons over which the tracking error and the control increments are computed, Y_Bus is the desired output, Y_m is the output predicted by the neural model, and u is the tentative control signal. The parameter ρ adjusts the influence of control increments in the performance index and is selected through a parametric sweep to achieve an appropriate trade-off between tracking accuracy and control smoothness. This function represents the performance objective that the NN implicitly acquires during training. Thus, the suggested PNN controller operates in a feedforward predictive manner, immediately producing the control signal from the neural outputs.

In online operation, the controller only evaluates a forward pass of the trained MLP at each sampling instant (i.e., a limited number of multiply-accumulate operations and activation function evaluations). Therefore, no online optimization is required, that make the proposed approach compatible with real-time implementation on standard digital control platforms.

The topology of the neural network used is presented in Figure 2. A two-hidden-layer feedforward network MLP [5 2] neurons was trained. Where w is the weight and b represents the biases of the neural network layers.

2.png

Figure 2. Topology of the proposed neural network model

3.6 Inner current control

In the proposed flatness-based control architecture, the inner control loops of the converters are implemented utilizing a fuzzy PI controller as a replacement for the conventional PI regulator. This hybrid strategy combines the simplicity of the PI controller with the intelligence and adaptability of fuzzy logic to provide the capability of adapting the control effort online (implicit gain scheduling) [34]. Within this architecture, the flatness-based control approach generates the reference trajectories, while fuzzy PI regulator controls the converter output currents and guarantees precise tracking in the inner loops [35]. The objective is to enhance robustness and dynamic efficacy of the internal regulation, especially under load changes, renewable generation fluctuations, and system parameters uncertainties [36].

The fuzzy PI controller uses two inputs: the normalized error e(k) and the change in error Δe(k). These crisp inputs are fuzzified using a triangular membership functions with linguistic variables: Negative Big (NB), Negative Medium (NM), Negative Small (NS), Zero (ZE), Positive Small (PS), Positive Medium (PM), and Positive Big (PB). The fuzzy system outputs the incremental control signal Δu(k) from the inputs, which updates the PI control rule. Hence, the controller operates as an adaptive mechanism that implicitly adjusts the equivalent PI action online (implicit gain scheduling). The schematic of the fuzzy PI controller structure is shown in Figure 3.

3.png

Figure 3. Fuzzy PI controller structure

The fuzzy rule base is developed by combining control heuristics and expert knowledge, as illustrated in Table 1. Large errors generate strong corrective actions, while small errors near the steady state produce mild control efforts to reduce oscillations. Defuzzification is achieved through the application of the centroid method for enabling smooth actuation.

This method enables faster settling, reduced overshoot, and enhanced rejection of disturbances compared to a fixed PI controller, with better dynamic behavior of the inner loops of the converters in the flatness-based energy management technique.

Table 1. Rule base

This paper presents a flatness-based control method for a hybrid photovoltaic/wind turbine/battery direct current microgrid using the PNN control law to improve trajectory tracking and address nonlinearities and model uncertainties. The PNN makes the system more resilient and responsive by comprehending the forward dynamics of the DC bus and ensuring more accurate reference tracking for diverse loads with a lower static error. A fuzzy-PI regulation method was employed at the converter level to replace the conventional PI controllers in the inner current loops. MATLAB/Simulink was used to design and simulate the proposed control strategy. The simulation results indicate that this technique ensures stable DC bus regulation with minimized overshoot (0.05%) and seamless power distribution among sources. The research illustrates the capacity of integrating the differential flatness approach with intelligent control strategies for enhanced energy management in DC microgrids.

For completeness and contextualize the proposed contribution relative to commonly adopted PI and MPC strategies, Table 3 summarizes a qualitative comparison relying on commonly reported characteristics in the literature and the behavior observed in the simulation results reported in this paper.

Several directions are suggested for future research to extend this study: 1) include experimental validation; 2) the incorporation of supplementary optimization layers (such as economic dispatch); and 3) design and control of a hybrid DC/AC microgrid configurations.

Table 3. Qualitative comparison with classical PI and MPC controllers