Elsevier

Automatica

Volume 82, August 2017, Pages 29-34
Automatica

Brief paper
Stability of decentralized model predictive control of graph-based power flow systems via passivity

https://doi.org/10.1016/j.automatica.2017.04.026Get rights and content

Abstract

This work presents a passivity-based stability guarantee for the decentralized control of nonlinear power flow systems. This class of systems is characterized using a graph-based modeling approach, where vertices represent capacitive elements that store energy and edges represent power flow between these capacitive elements. Due to their complexity and size, these power flow systems are often decomposed into dynamically coupled subsystems, where this coupling stems from the exchange of power between subsystems. Each subsystem has a corresponding model predictive controller that can be part of a decentralized, distributed, or larger hierarchical control structure. By exploiting the structure of the coupling between subsystems, stability of the closed-loop system is guaranteed by augmenting each model predictive controller with a local passivity constraint.

Introduction

Decentralized control of large systems comprised of dynamically coupled subsystems spans many application areas including thermal systems (Chandan, 2013, Jain et al., 2014, Morosan et al., 2010), water distribution networks (Cantoni et al., 2007, Negenborn et al., 2009, Ocampo-Martinez et al., 2012), chemical process networks (Christofides et al., 2013, Tippett and Bao, 2012), microgrids (Guerrero et al., 2013, Riverso et al., 2013, Zamora and Srivastava, 2010), and flow networks (Bauso et al., 2013, Blanchini et al., 2016). These applications, characterized by the flow and conservation of a resource, can be considered cases of a larger class of power flow systems. Power flow systems are governed by the transportation, conversion, and storage of energy across domains. Graph-based system representation is a widely adopted modeling technique that readily captures the structure of the governing mass and energy conservation laws for these systems (Blanchini et al., 2016, Heo et al., 2011, Moore et al., 2011, Preisig, 2009). Vertices, or nodes, represent capacitive elements that store energy, and edges represent power flow paths between these capacitive elements. While local parameters and functional relationships for power flow depend on the energy domain, system structure, analysis, and control are energy domain agnostic. This makes a graph-based approach a powerful tool for the modeling and control of a complex system-of-systems, comprised of multiple systems with various energy domains.

Additionally, the governing energy conservation laws suggest another unifying inherent feature of these systems: passivity. The notion of passivity in system modeling and control originated from the physical principles of energy conservation and dissipation in electrical and mechanical systems (Hill & Moylan, 1976) and has become a widely used and highly general methodology in nonlinear system analysis and control (Khalil, 2002, Sepulchre et al., 1997, van der Schaft, 1996). Thus, passivity-based control has been applied to a variety of power flow systems in centralized (Mukherjee et al., 2012, Ortega et al., 1998, Ulbig, 2007) and decentralized control architectures (Bao & Lee, 2007).

Model Predictive Control (MPC) is well suited for controlling power flow systems. The ability to account for actuator and state constraints and utilize communication and disturbance preview information allows MPC to maximize the performance and efficiency of these systems. Centralized, passivity-based, MPC has been implemented in Falugi (2014), Løvaas, Seron, and Goodwin (2007), Raff, Ebenbauer, and Allgöwer (2007), Sredojev and Eaton (2014) and Yu, Zhu, Xia, and Antsaklis (2013). Decentralized passivity-based MPC extends this approach to systems with a large number of states and actuators (Tippett and Bao, 2012, Varutti et al., 2012). In these approaches, along with those developed in Arcak and Sontag (2008) and Yu and Antsaklis (2010), stability is assessed with a global, system-wide matrix condition that accounts for the subsystem interconnection topology and the gain of the coupling between subsystems.

The aim of this paper is to present a purely decentralized and easily implementable method for augmenting existing decentralized control frameworks that guarantees stability of the overall closed-loop system. The relative simplicity of the approach is enabled by focusing on the control of power flow systems represented as graphs. The proposed approach identifies a set of inputs and outputs that render each subsystem passive. Neighboring subsystems form a negative feedback connection, establishing passivity of the overall system. While the approach relies on a graph-based representation of the system, a nonlinear, affine in control, power flow representation provides applicability to a wide class of systems. Actuator input and state constraints are considered, with slack variables on the state constraints to avoid infeasibility issues. Through the addition of a nonlinear constraint to each controller, the proposed approach provides simple implementation and reduced conservatism compared to standard passivity-based approaches.

The remainder of this paper is organized as follows. Section  2 introduces the graph-based modeling framework for the class of power flow systems. Section  3 presents the main results of the paper including establishing passivity of individual subsystems, analyzing the passivity-preserving interconnections between subsystems, developing a passivity constraint for each MPC controller, and proving the stability of the closed-loop system. Concluding remarks are provided in Section  4.

The symbol R denotes the set of real numbers. For the scalar function f(x),N(f(x))={x|f(x)=0} denotes the zero set of f(x). A vector v with elements vi is defined as v=[vi]. Similarly, a matrix M with elements mjk in the jth row and kth column is defined as M=[mjk]. The eigenvalues of matrix ARn×n are λk(A),k[1,n] and their real parts are denoted Reλk(A),k[1,n].

Section snippets

Class of systems

Consider a power flow system composed of N interconnected subsystems Si,i[1,N]. Each subsystem is represented by an oriented graph Gi=(Vi,Ei) with the set of vertices Vi and set of edges Ei. Each oriented edge ei,jEi represents power flow in Si, where positive power Pi,j flows from the tail vertex vi,jtail to the head vertex vi,jhead. Each vertex vi,kVi has an associated state xi,k that represents the energy stored in that vertex. Thus, the dynamic of each vertex vi,k satisfies the energy

Passivity of subsystems

Definition 2

Khalil, 2002

The system H with ẋ=f(x,u),y=h(x,u) where f:Rn×RpRn is locally Lipschitz, h:Rn×RpRp is continuous, f(0,0)=0, and h(0,0)=0 is passive if there exists a continuously differentiable positive semidefinite function V(x) such that uTyV̇=Vxf(x,u),(x,u)Rn×Rp.

If uTyV̇ for only a neighborhood of the origin, H is locally passive.

Fig. 2 shows the interconnection of subsystem Si with “upstream” and “downstream” subsystems Si1 and Si+1. The set of Nis power flows into Si from Si1 is denoted PiinRNi

Conclusions

This paper presented a purely decentralized procedure for augmenting existing model predictive control formulations with a passivity-based constraint to guarantee closed-loop stability of a power flow system. A graph-based modeling procedure captured the energy storage and transport that govern these systems. By establishing passivity of individual subsystems and analyzing the structure of the interactions between subsystems, a stability guarantee for the overall closed-loop system was achieved

Justin P. Koeln received his B.S. degree in 2011 from Utah State University in Mechanical and Aerospace Engineering. He received M.S. and Ph.D. degrees in 2013 and 2016, respectively, from the University of Illinois at Urbana–Champaign in Mechanical Science and Engineering. He is currently a Postdoctoral Research Associate at UIUC working on distributed control approaches for complex energy systems. He is a member of the NSF Engineering Research Center for Power Optimization of Electro-Thermal

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    Justin P. Koeln received his B.S. degree in 2011 from Utah State University in Mechanical and Aerospace Engineering. He received M.S. and Ph.D. degrees in 2013 and 2016, respectively, from the University of Illinois at Urbana–Champaign in Mechanical Science and Engineering. He is currently a Postdoctoral Research Associate at UIUC working on distributed control approaches for complex energy systems. He is a member of the NSF Engineering Research Center for Power Optimization of Electro-Thermal Systems (POETS) with the long term goal of increasing power density of mobile electrified systems by 10–100 times greater than the current state-of-the-art. His research interests include modeling and control of thermal management systems, model predictive control, and hierarchical and distributed control for electro-thermal systems.

    Andrew G. Alleyne received his B.S. in Engineering Degree from Princeton University in 1989 in Mechanical and Aerospace Engineering. He received his M.S. and Ph.D. degrees in Mechanical Engineering in 1992 and 1994, respectively, from The University of California at Berkeley. He joined the Mechanical Science and Engineering Department at the University of Illinois, Urbana–Champaign in 1994 and is also appointed in Electrical and Computer Engineering and the Coordinated Science Laboratory of UIUC. He currently holds the Ralph M. and Catherine V. Fisher Professorship in the College of Engineering and is the Director for the NSF Engineering Research Center on Power Optimization for Electro-Thermal Systems (POETS). He is the recipient of a CAREER award by the National Science Foundation, has been a Distinguished Lecturer of the Institute for Electronic and Electrical Engineers (IEEE), and a National Research Council Associate. From the ASME, he has received the Gustus Larson Award, the Charles Stark Draper Award for Innovative Practice, and the Henry Paynter Outstanding Investigator Award. He was a Fulbright Fellow to the Netherlands and has held visiting Professorships at TU Delft, University of Colorado, ETHZ, and Johannes Kepler University. He has held several editorial positions for ASME, IEEE, and the International Federation of Automatic Control. He recently chaired the ASME Dynamic Systems and Controls Division and has been active in several external advisory boards for universities, industry and government including the Scientific Advisory Board for the US Air Force. His record of campus service includes the Associate Dean for Research in the College of Engineering and the Associate Head for Undergraduate Programs in Mechanical Science and Engineering. In addition to research and service, he has a keen interest in education and has earned the College of Engineering’s Teaching Excellence Award, the UIUC Campus Award for Excellence in Undergraduate Education and the UIUC Campus Award for Excellence in Graduate Student Mentoring. Further information may be found at http://arg.mechse.illinois.edu or http://mechanical.illinois.edu/directory/faculty/alleyne.

    Research supported by the National Science Foundation Graduate Research Fellowship Program, the Air Force Research Laboratory (AFRL) under grant number FA8650-14-C-2517, and the National Science Foundation Engineering Research Center for Power Optimization of Electro Thermal Systems (POETS) with cooperative agreement EEC-1449548. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Giancarlo Ferrari-Trecate under the direction of Editor Ian R. Petersen.

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