Model Predictive Real-Time Control of Electric Power Systems Under Emergency Conditions

Abstract

Model Predictive Control (MPC) is a widely used method in process industry for the control of multi-input and multi-output systems. It possesses features that make it attractive for power system applications. Power systems exhibit complex characteristics such as hybrid nature (mixed continuous and discrete dynamics), nonlinear dynamics, and very large size. The optimization computations involved in MPC further increase the challenge of handling such features in a reasonable time. Therefore, the reduction of the computational burden associated with MPC is a crucial factor for real-time applications. In this chapter, we describe a formulation of MPC for power systems based on trajectory sensitivities. Trajectory sensitivities are time-varying sensitivities derived along the predicted nominal trajectory of the system, which allow an accurate reproduction of the nonlinear system behavior using a considerable reduced computational burden as compared with the full nonlinear integration of the system trajectories. Therefore, their deployment opens application possibilities for MPC in new, previously restricted, areas.

Appendix: Trajectory Sensitivities Analysis

An efficient framework for modeling of nonlinear systems featuring discrete states has been presented in Hiskens and Pai (2000). Its application on power systems modeling has been further shown in Hiskens and Sokolowski (2001) and Hiskens and Gong (2004), where a natural flexible modular structure following power systems components classification has been adopted.

Omitting parameter λ from the original formulation (Hiskens and Pai 2000) and introducing control input u, we can write the system equations in compact form:

$$ \dot{x}=f\text{(}x,y,z,u\text{),} $$

((12.11))

$$ 0={{g}^{0}}\text{(}x,y,z,u\text{),} $$

((12.12))

$$ 0=\left\{ \begin{matrix} {{g}^{i-}}\text{(}x,y,z,u\text{)} & {{y}_{d,i}}<0\\ {{g}^{i+}}\text{(}x,y,z,u\text{)} & {{y}_{d,i}}>0\\\end{matrix}\text{ }i=1,\cdots ,d \right., $$

((12.13))

$$ \begin{matrix} {{z}^{+}}={{h}_{j}}({{x}^{-}}{,}{{y}^{-}},{{z}^{-}},{{u}^{-}} ) \\ \dot{z}=0 \\{{y}_{e{,}j}}=0 \\ {{y}_{e{,}j}}\ne 0 j\in \left\{ 1{,}\cdots {,}e\right\}\end{matrix}$$

((12.14))

$$ \begin{matrix} {{y}_{d}}=D.y\\ {{y}_{e}}=E.y\\\end{matrix},\,\text{and} $$

((12.15))

$$ \begin{matrix} x\in X\subseteq {{\Re }^{n}} & z\in Z\subseteq {{\Re }^{k}}\\ y\in Y\subseteq {{\Re }^{m}} & u\in U\subseteq {{\Re }^{l}}\\\end{matrix}. $$

((12.16))

Dynamic state variables are denoted as x, algebraic state variables y, and discrete state variables z. Switching of the status of discrete variables is governed by the equation (12.14) when the corresponding auxiliary variables of y _e are equal to zero. Auxiliary variables y _d determine the region of validity of the equation (12.13). In the power systems context, this may be explained with an example of a line, which changes its status.

When the line is in service, equations linking the current through the line and voltages at both ends of the line as well as line parameters (line impedance and shunt admittance) are valid. When the line is out of service (i.e., disconnected), current flowing through it is zero. The auxiliary variable is in that case the difference between the time and the instant when the line was tripped.

Matrices D and E have normally a very sparse structure and their nonzero elements are equal to one in the positions aiming at the auxiliary variables.

Flow (i.e., time evolution) of the system from its initial point can be characterized by the time evolution of its variables, e.g., for the algebraic state variables we can write

$$ {{\varphi }_{y}}( {{y}_{0}},t )=y\text{(}t\text{)}. $$

((12.17))

Note that the initial state y ₀ is obtained by solving equations (12.12–12.13) by substituting initial values of x(t), z(t), and u(t) by x ₀ , z ₀ , and u ₀ , respectively.

An impact of small changes of the initial conditions on the system flow can be investigated by trajectory sensitivities . The impact of manipulated inputs (i.e., controls) on algebraic states time evolution can be obtained by a Taylor expansion of equation (12.17). When neglecting higher order terms:

$$ \Delta y\text{(}t\text{)}=\frac{\partial y\text{(}t\text{)}}{\partial {{u}_{0}}}\cdot \Delta {{u}_{0}}. $$

((12.18))

Note that trajectory sensitivities are generally time-varying quantities. Note that a numerical approximation of trajectory sensitivities can be computed by solving (12.11–12.16) for an incremental change of each control input. However, that would represent a large computation effort if many control inputs were considered. The methodology for computation of trajectory sensitivities , described in Hiskens and Pai (2000), involves only minimal additional computations, since it uses parts of Jacobian evaluated when solving (12.11–12.16).