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Stability of Passivity-Based Control for Power Systems and Power Electronics

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Stability of Passivity-Based Control for Power Systems and Power Electronics Stability of Passivity-Based Control for Power Systems and Power Electronics Kevin D. Bachovchin Department of Electrical & Computer Engineering Carnegie Mellon University Pittsburgh, PA 15213 USA [email protected] Marija D. Ilić Department of Electrical & Computer Engineering Carnegie Mellon University Pittsburgh, PA 15213 USA [email protected] 1. Introduction Passivity-based control is a nonlinear control method, which exploits the intrinsic physical structure and energy properties of the system dynamics when designing control for stabilization or regulation [1]. For this reason, enhanced robustness and simplified controller implementation are achieved with passivity- based control compared to feedback linearization, due to the avoidance of exact cancellation of nonlinearities [2]. Passivity-based control has been applied and demonstrated for robot arms [1], dc/dc converters [1,3,4], one-phase ac/dc converters [5], three-phase ad/dc converters [2], three-phase ac/dc/ac converters [6], torque regulation of induction motors [1,7], and speed regulation of Boost-converter driven dc-motors [8]. For underactuated systems (systems with less controllable inputs than state variables), one challenge with passivity-based control is that the dynamics of the non-directly controlled desired state variables can go unstable [1,2]. Therefore to ensure internal stability with passivity-based control, it is necessary to check the stability of the zero dynamics for the non-directly controlled desired state variables [2]. Often, it is seen that the passivity-based controller can be unstable when one state variable is chosen to be directly controlled but stable when another state variable is chosen to be directly controlled. Also sometimes the passivity-based controller is only stable under certain conditions and hence depends on the parameters in the system. 2. Description of the Problem Consider a general non-linear dynamic system with state space model xf (,)xu (1) A systematic automated approach for deriving the passivity-based control law for electrical systems is introduced in [6]. Using this algorithm, the control designer specifies the original state space model, the set point equations, the closed-loop energy functions (the closed-loop magnetic co-energy and the closed- loop electric energy), and the closed-loop dissipation function, and the automated method symbolically derives the control law. It should be noted that the number of set point equations should match the number of controllable inputs in the system. The closed-loop energy and dissipation functions are expressed in terms of the error state variables x where x x xD and xD denotes the desired state variables. For an underactuated system, these desired state variables cannot be all arbitrarily selected, but rather will be determined from the set point equations and the error dynamics [1]. Given the closed-loop energy and dissipation functions, the error dynamics are computed by evaluating the Lagrange equations. The Lyapunov function for the closed-loop system is the sum of the closed-loop magnetic co-energy and the closed-loop electric energy. If the closed-loop energy and dissipation functions are chosen so that the Lyapunov function is positive definite and the derivative of the Lyapunov function is negative definite, then the error dynamics will be asymptotically stable and the state variables will converge to their desired values [9]. As explained in [6], given the error dynamics and the set point equations, the following control law can be derived Dn ug 1 (,,)x x r (2) Dn Dn xg 2 (,,)x x r (3) where r denotes the external set points and xDn represents the non-directly controlled desired state variables. Since with an underactuated system all state variables cannot be regulated, the non-directly controlled desired state variables have dynamics. While the positive definite Lyapunov function and the negative definite time derivative of the Lyapunov function guarantee that the state variables will converge to the desired state variables, it is possible that the non-directly controlled desired state variables will go unstable. Therefore it is necessary to assess the zero dynamics of the desired state variables in order to design control with provable performance. (In the zero dynamics, it is assumed that the state variables have all converged to the desired state variables and only the dynamics of the desired state variables are considered.) 3. History of the Problem With passivity-based control, the stability of the desired state variables often depends on which state variables are chosen to be directly controlled. In [1], passivity-based control is designed for Boost converters. It is shown that when directly regulating the voltage across the output capacitor, the dynamics of the desired variables are unstable. However when directly regulating the current across the inductor, the controller is stable. Therefore indirect control of the capacitor voltage must be used, where the inductor current set point is determined by the desired steady-state capacitor voltage [1]. Another example is in [2], where the passivity-based control is designed for a 3-phase AC/DC voltage source converter analyzed. When regulating the capacitor voltage and the quadrature component of the inductor current, the desired direct component of the inductor current has dynamics which are unstable. However, when regulating both the direct and quadrature components of the inductor current, the desired capacitor voltage has dynamics which are stable. With passivity-based control, sometimes the desired state variable dynamics are only stable under certain conditions which depend on the set points and the parameters of the system. For example, in the 3-phase AC/DC/AC converter analyzed in [6], the direct and quadrature components of the load and source currents are regulated, and the desired capacitor charge has dynamics. It is shown that the desired capacitor charge dynamics are only stable when the set points and parameters are such that the power input to the AC/DC/AC converter is greater than the power dissipated. Another example is in [10], where passivity-based control is designed for variable speed drives for flywheel energy storage systems. In the torque controller, the rotor current and the quadrature component of the stator current are regulated, and the desired direct component of the stator current has dynamics. It is shown that the desired dynamics are unstable unless the set point for the rotor current is chosen to be the rotor voltage divided by the rotor resistance. This example shows that the physical equilibrium conditions of the system cannot be violated. 4. Motivation In power systems literature, the analysis and control of microgrids often begin with the droop characteristic between frequency and active power [11,12]. In order for the droop characteristic analysis to be valid, it is necessary for the fast dynamics and control to be stable. In industry today, most controllers in power systems and power electronics use linearized models of nonlinear dynamics and linear control logic. While these linearized models are accurate for small disturbances, large disturbances can perturb the system far away from the equilibrium where linearized models are no longer accurate. Hence linear control logic cannot be used with provable performance when there are large disturbances. Since passivity-based control is a nonlinear control method, it can be used in response to large disturbances. In order to design passivity-based control with provable performance, it is necessary to first derive the non-linear dynamic equations for the interconnected system, including for the power electronics, as in [1,2,6]. It is then necessary to analyze the stability of the desired state variables and find conditions where they remain stable. Once control of the fast dynamics has been designed with provable performance, then the droop characteristic analysis can be used. Bibliography [1] R. Ortega, A. Loria, P. Nicklasson, and H. Sira-Ramirez, Passivity-based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications. New York: Springer Verlag, 1998. [2] T.S. Lee, "Lagrangian modeling and passivity-based control of three-phase AC/DC voltage-source converters," IEEE Transactions on Industrial Electronics, vol. 51, no. 4, pp. 892-902, Aug. 2004. [3] . ira- a r a . r a a i i -based controllers for the stabilization of DC-to-DC power converters," in Proceedings of the 34th IEEE Conference on Decision and Control, 1995, pp. 3471-3476. [4] D. Jeltsema, "Modeling and control of nonlinear networks: a power-based perspective," Delft University of Technology, Ph.D. dissertation 2005. [5] D. del Puerto-Flores et al., "Passivity-Based Control by Series/Parallel Damping of Single-Phase PWM Voltage Source Converter," IEEE Transactions on Control Systems Technology, vol. 22, no. 4, pp. 1310-1322, July 2014. [6] K.D. Bachovchin and M.D. Ilic, "Automated Passivity-Based Control Law Derivation for Electrical Euler- Lagrange Systems and Demonstration on Three-Phase AC/DC/AC Converter," Carnegie Mellon University, EESG Working Paper No. R-WP-5-2014 August 2014. [7] R. Ortega and G. Espinosa, "Torque regulation of induction motors," Automatica, vol. 29, no. 3, pp. 621-633, 1993. [8] . i ar - l r . r a . ira- a r ad Torque Estimation and Passivity-Based Control of a Boost-Converter/DC-Motor Combination," IEEE Transactions on Control Systems Technology, vol. 18, no. 6, pp. 1398-1405, Nov. 2010. [9] J. Slotine and W. Li, Applied Nonlinear Control.: Prentice-Hall, 1991. [10] K. D. Bachovchin and M. D. Ilic, "Passivity-Based Control Using Three Time-Scale Separations of Variable Speed Drives for Flywheel Energy Storage Systems," Carnegie Mellon University, EESG Working Paper No. R-WP-6-2014 October 2014. [11] R.H Lasseter and P. Paigi, "Microgrid: a conceptual solution," in 2004 IEEE 35th Annual Power Electronics Specialists Conference, June 2004, pp. 4285-4290. [12] J.A. Pecas Lopes, C.L. Moreira, and A.G. Madureira, "Defining control strategies for MicroGrids islanded operation," IEEE Transactions on Power Systems, vol.
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