Passivity based control applied to power converters

Passivity based control applied to power converters Marco Liserre liserre@poliba.it

Passivity based control history • 70’s definition of dissipative systems (Willems) • 1981 application to rigid robots (Arimoto e Takagi) in power electronics . . . • 1991 first theoretical paper (Ortega, Espinoza & others) • 1996 first experimental paper (Cecati, & others, IAS Annual Meeting) • 1998 first book “Passivity-Based Control of Euler-Lagrange Systems” (Ortega and Sira-Ramirez, Springer, ISBN 1852330163) • 1999 application to active filters (Mattavelli and Stankovic, ISCAS 99) • 2002 Brayton-Moser formulation (Jeltsema and Scherpen, Am. Control Conf.) • 2002 application to multilevel converters (Cecati, Dell’Aquila, Liserre, Monopoli, IECON 2002)

Contribution of my research group on the topic • collaborations with: • University of L’Aquila (Prof. Cecati) • University of Delft (Prof. Scherpen) • main papers: • C. Cecati, A. Dell’Aquila, M. Liserre, V. G. Monopoli “A passivity-based multilevel active rectifier with adaptive compensation for traction applications” IEEE Transactions on Industry Applications, Sep./Oct. 2003, vol. 39, no. 5. • A. Dell’Aquila, M. Liserre, V. G. Monopoli, P. Rotondo “An Energy-Based Control for an n-H-Bridges Multilevel Active Rectifier” IEEE Transactions on Industrial Electronics, June 2005, vol. 52, no. 3.

Basic idea of the Passivity-based approach • The basic idea of the PBC is to use the energy to describe the state of the system • Since the main goal of any controller is to feed a dynamic system through a desired evolution as well as to guarantee its steady state behavior, an energy-based controller shapes the energy of the system and its variations according to the desired state trajectory • If the controller is designed aiming at obtaining the minimum energy transformation, optimum control is achieved • The PBC offers a method to design controllers that make the system Lyapunov-stable • The “energy approach” is particularly suitable when dealing with: • electromechanical systems as electrical machines • grid connected converters (non-linear model)

The introduction of damping • The control objective is usually achieved through an energy reshaping process and by injecting damping to modify the dissipation structure of the system • From a circuit theoretic perspective, a PBC forces the closed-loop dynamics to behave as if there are artificial resistors — the control parameters — connected in series or in parallel to the real circuit elements • When the PBC is applied to grid connected converters, harmonic rejection is one of the main task, hence the passive damping can be substituted by a dynamic damping (i.e. virtual inductive and capacitive elements should be added) • The point of view is always the energy reshaping (i.e. the energy associated to the harmonics)

The Eulero-Lagrange formulation • Passivity-based control has been firstly developed on the basis of Eulero-Lagrange formulation • One of the major advantages of using the EL approach is that the physical structure (e.g., energy, dissipation, and interconnection), including the nonlinear phenomena and features, is explicitly incorporated in the model, and thus in the corresponding PBC • This in contrast to conventional techniques that are mainly based on linearized dynamics and corresponding proportional-integral–derivative (PID) or lead–lag control

The Passivity Based Controller design • In the context of EL-based PBC designs for power converters, two fundamental questions arise: • which variables have to be stabilized to a certain value in order to regulate the output(s) of interest toward a desired equilibrium value? In other words, are the zero-dynamics of the output(s) to be controlled stable with respect to the available control input(s), and if not, for which state variables are they stable? • where to inject the damping and how to tune the various parameters associated to the energy modification and to the damping assignment stage?

Dissipativity definition dissipativity definition

Passivity definition

Definitions • Supply Rate: speed of the energy flow from a source to the system • Storage function: energy accumulated in a system • Dissipative systems: systems verifying dissipation inequality: “Along time trajectories of dissipative systems the following relationship holds: energy flow ≥ storage function” (In other words, dissipative systems can accumulate less energy than that supplied by external sources) • The basic idea of PBC is to shape the energy of the system according to a desired state trajectory, leaving uncontrolled those parts of the system not involved in energy transformations, this result can be obtained only working on “strictly passive” systems

Feedback systems decomposition • dividing the system into simpler subsystems, each one identifying those parts of the system actively involved in energy transformations • each subsystem has to be passive introducing energy balances, expressed in terms of the Eulero-Lagrange equations passivity invariance

Feedback systems decomposition • The full order model describing the system is divided into simpler subsystems identifying those parts actively involved in energy transformation • Hence, energy balances, expressed in terms of the Eulero-Lagrange equations (based on the variational method and energy functions expressed in terms of generalised coordinates), are introduced • The system goes in the direction where the integral of the Lagrangian is minimized (Hamilton's principle)

Feedback systems decomposition • This formulation highlights active, dissipative and workless forces i.e. the active parts of the system (those which energy can be modified by external forces), those passive (i.e. dissipating energy, e.g. thermal energy), and those parts which do not contribute in any form to control actions and can be neglected during controller design • Because of the energy approach, it is quite straightforward to obtain fast response under condition that the control "moves" the minimum amount of energy inside the system • Moreover, because global stability is ensured by passivity properties, a simple a effective controller can be designed

Eulero-Lagrange formulation • The eulero-lagrange formulation is particularly suited for the control of electromechanical systems as electrical motors • In fact different subsystems are related by their ability to transform energy, therefore it is a good thing to define energy functions for each one, expressed in terms of generalised coordinates qi. • In electric motor case: qm mechanical position (for mechanical subsystems) qe electric charge (for electrical subsystems) • Using variational approch we can introduce Lagrangian equations of the system and apply Hamilton's principle. This method highlights subsystems interconnections and their various energies: dissipated, stored and supplied

Eulero-Lagrange formulation induction motor formulation The mechanical subsystem does not take an active part in control actions, i.e. it doesn't produce energy but only transforms and dissipates the input energy, for design purposes its contribution can be considered as an external disturbance for the electrical subsystem and the controller has to compensate for this disturbance, in order to maintain electrical equation balance. In “passivity terms”, it defines a passive mapping around the electrical subsystem, it can be neglected during controller design and the attention can be focused on the electrical subsystem.

Eulero-Lagrange formulation The electrical subsystem is simply passive, then its evolution can be corrupted by any external disturbance leading to instability. Therefore, in order to obtain global stability, it is an important step of the approach to make it strictly passive by means of the addition of a suitable dissipative term (damping injection)

Passivity-based control of the H-bridge converter • PBC has been successfully applied to d.c./d.c. converters, active rectifiers and multilevel topologies • Particularly the single-phase Voltage Source Converter (VSC) also called H-bridge or full bridge can be used as universal converter due to the possibility to perform dc/dc, dc/ac or ac/dc conversion • Moreover it can be used as basic cell of the cascade multilevel converters • In the following it will be reviewed the application of the PBC to H-bridge single phase inverters (one-stage and multilevel) using the Brayton-Moser formulation which is the most suitable for the converter control

Passivity-based control of the H-bridge converter • Control of one H-bridge-based active rectifier G. Escobar, D. Chevreau, R. Ortega, E. Mendes, “An adaptive passivity-based controller for a unity power factor rectifier”, IEEE Trans. on Cont. Syst. Techn., vol. 9, no. 4, July 2001, pp. 637 –644 • Control of two (multilevel) H-bridge-based active rectifier C. Cecati, A. Dell'Aquila, M. Liserre and V. G. Monopoli, "A passivity-based multilevel active rectifier with adaptive compensation for traction applications", IEEE Trans. on Ind. Applicat., vol. 39, Sept./Oct. 2003 pp. 1404-1413 the two dc-links are not independent ! • Control of n (multilevel) H-bridge-based active rectifier A. Dell’Aquila, M. Liserre, V. G. Monopoli, P. Rotondo “An Energy-Based Control for an n-H-Bridges Multilevel Active Rectifier” IEEE Transactions on Industrial Electronics, June 2005, vol. 52, no. 3. the n dc-links are independent !

Brayton-Moser Equations • Brayton and Moser, introduced in 1964 a scalar function of the voltages across capacitors and the currents through inductors in order to characterize a given network • This function was called the Mixed-Potential FunctionP(iL, vC) and it allows to analyze the dynamics and the stability of a broad class of RLC networks • These equations can be considered an effective alternative to Euler-Lagrange formulation • This formulation has a main advantage over the counterpart in case of power converter control: it allows the controllers to be implemented using measurable quantities such as voltages and currents.

Brayton-Moser Equations Topologically Complete Networks = networks which state variables form a complete set of variables Complete Set of Variables = set of variables that can be chosen independently without violating Kirchhoff’s laws and determining either the current or voltage (or both) in every branch of the network Additionallyfor Topologically Complete Networks it is possible to define two subnetworks One subnetwork has to contain all inductors and current-controlled resistors The other has to contain all capacitors and voltage controlled resistors

For the class of topologically complete networks it is possible to construct the mixed-potential function directly.For this class it is known that the mixed potential is of the form: Brayton-Moser Equations R(iL) is the Current Potential(Content) and is related with the current-controlled resistors and voltage sources G(vC) is the Voltage Potential (Co-content) and is related with the voltage-controlled resistors and current sources N(iL,vC) is related to the internal power circulating across the dynamic elements

Brayton-Moser Equations The components of the Mixed-Potential Function can be analysed in more detail as follows: PR is the Dissipative Current Potential PG is the Dissipative Voltage Potential

Brayton-Moser Equations The dissipative current and voltage potentials can be calculated as follows: PE is the total supplied power to the voltage sources E PJ is the total supplied power to the current sources J In case of linear resistor PR is half the dissipated power expressed in terms of inductor current, and PG is half the dissipated power expressed in terms of capacitor voltages.

Brayton-Moser Equations PT is the internal power circulating across the dynamic elements and is represented by: In this representation  denotes the interconnection matrix and it is determined by KVL and KCL

Brayton-Moser Equations Finally the expression of the mixed-potential function can be rewritten as follows: PD(x)= PR(x)- PG(x)isthe Dissipative Potential PF(x)= PJ(x)- PE(x)isthe Total Supplied Power

The dynamic behaviour of topologically complete networksis governed by the following differential equations : Brayton-Moser Equations iL = (iL1 , . . . , iL)T are the currents through the  inductors vC= (vC1 , . . . , vC )T are the voltages across the  capacitors. These differential equations correspond with Kirchhoff’s voltage and current laws

Brayton-Moser Equations The previous equations can be expressed in a more compact way as follows: with the state vector xRn = R+ defined as and with the diagonal square matrix Q(x) R(+)x(+) defined as

Brayton-Moser Equations When a circuit contains only linear passive inductors and capacitors, then the diagonal matrices L(iL)  Rx and C(vC)  Rx are of the form: The Brayton-Moser equations are closely related to the co-Hamiltonian H*(iL, vC) (that represents the total co-energy stored in the network). Ifthe co-Hamiltonian is known, then the matrices L(iL) and C(vC) can be easily found as follows

Switched Brayton-Moser Equations For a circuit with one or more switches it is possible to obtain a single Switched Mixed-Potential Function by properly combining the individual mixed-potential functions associated to each operating mode. u=0 P0(x) u=1 P1(x) Then it is possible to obtain one Switched Mixed-Potential Function parameterized by u as The Switched Mixed-Potential Function is consistent with the individual Mixed-Potential Functions

Switched Brayton-Moser Equations It is worth to notice that the only difference between each individual Mixed-Potential Function and the Switched Mixed-Potential Function will be in the term and in particular in the interconnection matrix  which becomes a function of u, (u)

μ z u x Average State Model When the switching frequency is sufficiently high, it is possible to prove that the average state model of a circuit with a single switch can be derived from the discrete model by only replacing the discrete variable u{0,1} with the continuously varying duty-cycle variable μ[0,1]. Additionally, to show that the model is a state average model, the state vector x is replaced by the state average vector z Discrete Model Average State Model

μi zi ui xi Average State Model The former result can be extended to circuits with multiple switches. In this case the matrix  (u) assumes as many configurations as the possible combinations of the status of the switches are (e. g. for an H-bridge converter  is a mono-dimensional matrix and may assume three distinct values {-1,0,1}) Discrete Model Average State Model

To design a Passivity Based controller the average co-energy function H*(z) and the dissipative potential PD(z) have to be modified. To this purpose the Brayton-Moser equations can be rewritten as: Passivity Based Control – Procedure The first two derivative terms are still function of z, in the sense that the partial derivatives of PT(z) and PD(z) are still dependent on z The third term is constant meaning that the partial derivative of PF(z) is not dependent on z anymore f(z) constant The following step is to rewrite the previous equations by replacing the state variables z with an auxiliary system of variables ξ which represent the desired state trajectories for inductor currents and capacitor voltages: The first two derivative terms are still function of ξ The third term is constant and is obviously equal to the partial derivative of PF(z) f(ξ) constant

being z = z − ξ the average state errors, it is possible to write: Passivity Based Control – Procedure Assuming that the first two derivatives are linear functions of z and the last two derivatives are linear functions of ξ, yields: The previous expression represents the error dynamics and it could be obtained from by simply replacing the variable z with the error variable z and eliminating the derivative of PF

The next step is to add a damping term to the error dynamics to ensure asymptotic stability Passivity Based Control – Procedure This injection can be seen as an expansion of the dissipative potential Considering z = (iL, vC)T where iL = (z1 . . . z)T are the error-currents through the inductors vC= (z+1 . . . z+)T are the error-voltages across the capacitors The injected dissipation can be decomposed as follows: The injected dissipation together with the dissipative potential of the system, gives the Total Modified Dissipation Potential PM

Passivity Based Control – Procedure Subtracting from the controller dynamicsare obtained

Passivity Based Control – Procedure Two theorems ensure the stability of the closed loop system. THEOREM 1 (R-Stability) If RS is a positive-definite and constant matrix, and with 0 << 1, then for all (iL, vC) the solutions of tend to zero as t → ∞ where closed-loop resistance matrixRSis

Passivity Based Control – Procedure THEOREM 2 (G-Stability) If GP is a positive-definite and constant matrix, and with 0 << 1, then for all (iL, vC) the solutions of tend to zero as t → ∞. where closed-loop conductancematrixGPis

With these theorems lower bounds are found for the control parameters (RS and/or GP ) • These lower bounds ensure a ”reasonably nice” response in terms of overshoot, settling-time, etc • If just one of these theorems is satisfied, the system is stable. This means there are two damping injection strategies that can be selected: Passivity Based Control – Procedure Series Damping (damping on inductor currents) Parallel Damping (damping on capacitor voltages) Although it is sufficient to use only one of these strategies, the equations could contain both the series damping injection term and the parallel damping injection term

Finally, if n is the number of minimum phase states it is possible to modify n equations of the + differential equations in Passivity Based Control – Procedure To this purpose n minimum phase states have to be found. Consequently the remaining +-n state variables will be indirectly controlled through the control of the n selected states For the n selected variables it is possible to set the derivative of reference value to zero obtaining n algebraic equations: Controller Equations and +-n differential equations:

Controller Implementation Passivity Based Control – Procedure At the beginning initial values of the n control inputs have to be set Using these values the differential equations can be solved to obtain the time evolution of the auxiliary variables for the indirectly controlled variables. The former references are needed to solve the algebraic equations which solutions are the set of values for the control inputs to be applied in the next switching period.

PBC of an H-bridge The passivity-based controller will be designed by inspection, identifying the potential functions

+DC T T 1 3 L a i R C e b T T 2 4 -DC PBC of an H-bridge

+DC T T 1 3 L a i R C e b T T 2 4 -DC PBC of an H-bridge LKT LKC controller

+DC T T 1 3 L a i R C e b T T 2 4 -DC PBC of an H-bridge: damping injection

+DC T T 1 3 L a i R C e b T T 2 4 -DC PBC of an H-bridge: control variables • which variables have to be stabilized to a certain value in order to regulate the output(s) of interest toward a desired equilibrium value? • in other words, are the zero-dynamics of the output(s) to be controlled stable with respect to the available control input(s), and if not, for which state variables are they stable?

The steady-state solution in case of direct control of the dc-voltage or in case of indirect control of the dc-voltage (through the grid current) should be found PBC of an H-bridge: zero-dynamics Switching function in case of direct control Switching function in case of indirect control • A stable system can be obtained only by indirectly controlling the dc voltages through the ac current i*

as i  i*, vc  ξ2  Vd • A stable system can be obtained only by indirectly controlling the dc voltages through the ac current i* • This means that PBC of an H-bridge • From the power balance it results that dc voltage reference load conductance grid voltage amplitude controller and reference voltage Vd and load conductance θ reference current i* switching function µ internally generated ξ2 algebraic power balance ODE

Passivity based control applied to power converters