# (371h) Nonlinear Model Predictive Control of a DC-DC Boost Converter with Constant Power Load: Classical and Explicit Approaches

Authors:
Tecnologico de Monterrey
Tecnologico de Monterrey
Tecnologico de Monterrey
The development of microgrids has become a necessity for integration of renewable energy sources and a crucial step towards the development of smart grids [1]. For example, a photovoltaic system is usually connected to a common bus through a dc-dc boost converter. This converter steps up the input voltage E to a level required (vref) for an adequate operation of the microgrid while supplying a constant power load (CPL). The presence of the CPL introduces negative impedance instabilities [2]. Controllers designed by linearizing the converter model about the equilibrium point are generally insufficient to mitigate CPL effect. Sliding-mode controller is a technique that has been used to overcome this difficulty [3]. Model predictive control (MPC) has been applied to power electronics as it offers the possibility of managing multiple inputs and outputs as well as the incorporation of state constraints [4]. However, MPC for dc-dc boost converters with CPL has not been studied sufficiently, especially by taking into account the nonlinearity of the model. In this work, a nonlinear MPC is proposed in order to regulate the output voltage v(t) of the boost converter to provide a constant level while satisfying the CPL requirement. Additionally, an explicit MPC is developed to reduce the computational cost. The two-dimension dynamic model of the boost converter was discretized with orthogonal collocation (Radau points) on finite elements, resulting in a set of nonlinear algebraic equations with good numerical properties. A nonlinear programming (NLP) problem was formulated such that the difference between the output voltage and the reference is minimized subject to the constraints introduced by the discretized model. Then, a nonlinear MPC controller was developed and then implemented on MATLAB/Simulink® interfaced with GAMS® (optimizer). At each sample time ts, a measurement of the current i(ts) and voltage v(ts) is taken, the controller passes these values to the optimizer which solves the NLP and delivers the optimal control input (switching) u(t). This input is injected to the circuit block and this process is repeated along the simulation time. Then, an explicit Model Predictive Control (eMPC) was developed via offline multi-parametric programming in order to replace the online optimization by a simple evaluation of an affine optimal control law. We used an algorithm for approximate multiparametric convex programming [5], in which several NLPs are solved, to get an affine control expression for each partition of the parametric space. The initial simplicial partition was generated by using Delaunay triangulation, then after applying the algorithm the final partition consisted of 166 simplices. The implementation of this solution leads to the problem known as â€œpoint locationâ€, which was addressed with hash tables to manage the time-storage complexity, hence for a given states value the corresponding control law is found in short time [6]. The explicit solution was tested in the same way as the previous classical approach but substituting the optimizer by the solution obtained offline. Simulation results with a sample time of 0.002 s for classical MPC and 0.00125 s for explicit MPC show that both strategies are able to drive the input voltage E to the desired level v(t) ~ vref and stabilize it there. However, the explicit one reacts much faster as it does not carry out online optimization, which can be advantageous for real applications. We conclude that MPC is a promising technique to control the dc-dc boost converter feeding a CPL. The results of this work can be useful for studies about integration of renewable energy systems into microgrids.

References

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