By Milad Farsi and Jun Liu
Over the last few decades, discussions regarding the maximization of solar energy accumulation have attracted much interest in the field of renewable energy. Scientists in this field conduct research in a few different directions. The main branch of the work encompasses the development and improvement of the photovoltaic (PV) cell and associated hardware. This was known as the most rewarding way to boost the energy gain, which successfully introduced solar energy as a practical source of renewable energy by increasing nominal efficiency and lowering production cost.
Another category of research—designing an effective control scheme as a software to control a given solar PV array—plays an important role in boosting the generated power to achieve nominal gain. Various algorithms, known as maximum power point tracking (MPPT) methods, exist in the literature of power electronics to address this objective. Although most of these techniques can already guarantee achievement of a high efficiency rate, a considerable return is still expected by further promoting this rate, even by a very small amount. In other words, any improvement in the efficiency of MPPT techniques can be multiplied by thousands of solar panels in a solar farm or many grid-connected solar PV systems. This motivated us to formulate and solve a control problem in a way that allowed for consistent tracking of a performance objective, which potentially leads to an improved efficiency rate.
To preserve a uniform quality in the system’s performance, one must define and guarantee a performance measure. A standard approach in the control community is to formulate such problems in the optimal control framework. While various approaches in the literature attempt to improve the performance of control in PV systems, no clear connection has been made between the configuration of the implemented controller and the obtained performance. This is due to the presence of nonlinearity in the model and the control objective’s complexity. The formulation of the MPPT problem in an optimal control framework thus remains a challenging task.
To obtain the mathematical model defining the system, we used an equivalent electrical circuit to model an array of PV modules — assuming that all PV modules in the array have identical electrical characteristics. One can utilize a DC–DC boost converter to couple the PV array with the load and control the operating point of the system; a similar approach can be employed to different configurations of converters. As a result, combining the PV array model with the converter model results in an affine switched system model.
Equivalent electrical model of the solar array.
In our recent research paper [1], we employed nonlinear optimal control (NOC) to track the maximum power point (MPP) of the solar PV system. We first provide conditions necessary for optimality and system stability with respect to a set of equilibrium points and a given performance measure. By designing a nonlinear cost functional—aimed at the stationary point of the solar array power-current curve—we then confirm that the stability and optimality conditions we provided earlier indeed hold for the obtained control law.
Solar photovoltaic (PV) array illustrating the partial shading condition considered in the simulation results.
We exploit this idea to establish two different controllers for tracking the MPP and a given reference voltage that are separately activated via an algorithm to handle the partial shading condition. The presented approach—together with two well-known sliding-mode and second-order sliding-mode controllers—is simulated on a realistic model of the system, including non-ideal components. Results indicate that this NOC illustrates a suitable convergence response with minimal oscillations around the MPP in both uniform and non-uniform insolations.
Sketch of the simulated solar photovoltaic (PV) system together with the proposed control approach in Matlab Simulink.
Constructing the obtained feedback law requires first- and second-order partial derivatives of the voltage with respect to the current. The problem formulation hence relies on our knowledge of the solar PV model parameters. As an extension, we propose a model-free-based control scheme algorithm that uses output voltage and current measurements of the array to approximate partial derivatives.
References
[1] Farsi, M., & Liu, J. (2019). Nonlinear optimal feedback control and stability analysis of solar photovoltaic systems. IEEE Trans. Cont. Syst. Tech.
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Milad Farsi is currently pursuing a Ph.D. in applied mathematics at the University of Waterloo. His research interests include hybrid dynamical systems, optimal control, and learning-based control with applications in power electronics and robotics. |
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Jun Liu is an associate professor of applied mathematics at the University of Waterloo, where he also directs the Hybrid Systems Laboratory. His present research interests encompass the theory and applications of hybrid systems and control, including rigorous computational methods for control design with applications in cyber-physical systems and robotics. |