COMPLEXITY SCIENCE WORKSHOP 18, 19 June 2015 Systems & Control Research Centre School of Mathematics, Computer Science and Engineering CITY UNIVERSITY LONDON Modeling and control of distributed generation power systems as complex nonlinear Hamiltonian systems Presentation: Antonio T. Alexandridis

The deployment of modern power systems is mainly based on the high penetration of renewable energy sources (RES) a decentralized structure known as distributed generation (DG). In DG the power electronic interfaces used to connect the different parts, play an important role that: exploits the local capabilities of the system on managing energy in the scheme of a microgrid enables a cooperated and fast self-controlled local implementation Note: The microgrid is defined as an integrated energy system consisting of distributed energy sources and multiple electrical loads operating as a single, local grid either in grid connected-mode or in islanded-mode with respect to the existing utility power system. Introduction

The fast response of power electronic devices provide the possibility of new control strategies in a wide area of DG sources and loads As a result, DG should be planned and analyzed as a whole (at least on a microgrid level): All system dynamics should be taken into account To this end, it is needed: a complete dynamic analysis of the DG system a systematic methodology of obtaining the complete DG system nonlinear model It is shown that this model is in Hamiltonian form with certain damping properties can be systematically obtained from the integration of all the DG components, i.e., from their individual Hamiltonian models damping and other structural properties can be effectively used for stable control designs through Lyapunov analysis. Problem formulation

Based on the Hamiltonian modeling, the entire system is described as a large nonlinear system with external uncontrolled inputs Advantages: The Hamiltonian formulation provides an immediate construction of energy based Lyapunov functions (!) Drawbacks: The error dynamic model is not Hamiltonian The entire system is nonautonomous with f=f(x,u(t)) To prove stability is not an easy task (a particular sequence of stages for the analysis and design are needed: This is our current work) The integrated method is very efficient and complete Why Hamiltonian modeling?

Hamiltonian systems In the case where the system is given in the form: where is the system Hamiltonian function, is a semi- positive definite matrix and is skew-symmetric, the system is called generalized Hamiltonian system. In many cases, such as almost all DG system components can be written as Where matrix is symmetric and positive definite, is semi-positive definite, is a skew-symmetric matrix and. represents the external input vector. DG components in Hamiltonian form

Almost all Distributed Generators (PV-arrays, Wind generators, Storage batteries, Power converters: dc/dc or ac/dc) are modeled as Hamiltonian systems The transmission line, local load and capacitor bank models can also be written in the general nonlinear Hamiltonian form Thus, the complete DG network can be integrated and modeled as Hamiltonian system. DG components in Hamiltonian form

Passivity Preliminaries Let the nonlinear system where are smooth. Theorem 1. [Khalil] Assume that there is a continuous function such that for all functions, for all and all. Then the system with input and output is passive. DG systems and components

Passivity analysis of Hamiltonian systems Considering the storage function: The derivative of the storage function is calculated as: Assuming as output the vector and input the vector the above inequality becomes. Integrating from zero to, according to Theorem 1, proves that the nonlinear Hamiltonian system is passive.

Typical configuration of a DG network A DG example

Nonlinear Hamiltonian form of the system: where: with: and with: Hamiltonian model of a wind turbine induction generator

and Hamiltonian model of a wind turbine induction generator

Hamiltonian model of a Photovoltaic Generator Nonlinear Hamiltonian form of the system: where: with: and

Hamiltonian model of a Photovoltaic Generator

Transmission line: Local load: Capacitor bank: Hamiltonian model of a transmission line, a local load and a capacitor bank

The complete DG system can be integrated in the form of: the state vector is: Obtaining the complete Hamiltonian Dynamic Model of a DG System

The external vector is: and Obtaining the complete Hamiltonian Dynamic Model of a DG System

and Obtaining the complete Hamiltonian Dynamic Model of a DG System

Hamiltonian Dynamic Model of a DG System Basic features  Matrix continues to be a symmetric, positive definite matrix  is a semi-positive definite matrix.  is skew-symmetric and depends from the duty-ratio control inputs of the generator-side converters, the grid-side converters and the boost converters of the complete DG system. Since matrix does not appear in the passivity inequality, the complete DG system is passive for any non dynamic control scheme applied at the control inputs.

The Example

Simulations results We consider a squirrel-cage induction generator (SCIG) wind turbine and a Photovoltaic (PV) system. At time. the power of the PV system drops by and at time. the wind power increases almost by. Main task in a DG system is to extract the maximum amount of energy from the generators and achieve unity power factor. Passivity-based control is applied.

Simulations results

Conclusions A new approach that integrates the partial individual unit models of a distributed power system in a common Hamiltonian form has been presented. The Hamiltonian form with certain damping properties can be effectively used for stable control designs.* Passivity and stability analysis can be obtained from the complete nonlinear Hamiltonian model.* The interactions between the individual units are not ignored. Thus, contradictory performances can be avoided.

*Issues in DG control Next steps constructing on the proposed model A new suitable DG control is applied with main goals: The most possible simple decentralized structure Controllers independent from system parameters and operating conditions For example, local cascaded PI controllers are examined The stability analysis includes Current Inner-loop controllers Special stability analysis for external-loop controllers A hard analysis is needed in order to obtain simple and efficient controllers guaranteeing stability

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