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Passivity based control applied to power converters Passivity based control applied to power converters Marco Liserre [email protected] Marco Liserre [email protected] Passivity based control applied to power converters 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) Marco Liserre [email protected] Passivity based control applied to power converters 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 EnergyBased Control for an n-H-Bridges Multilevel Active Rectifier” IEEE Transactions on Industrial Electronics, June 2005, vol. 52, no. 3. Marco Liserre [email protected] Passivity based control applied to power converters 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) Marco Liserre [email protected] Passivity based control applied to power converters 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) Marco Liserre [email protected] Passivity based control applied to power converters The Eulero-Lagrange formulation Passivity-based control has been firstly developed on the basis of EuleroLagrange 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 Marco Liserre [email protected] Passivity based control applied to power converters 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? Marco Liserre [email protected] Passivity based control applied to power converters Dissipativity definition dissipativity definition Marco Liserre [email protected] Passivity based control applied to power converters Passivity definition Marco Liserre [email protected] Passivity based control applied to power converters 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 Marco Liserre [email protected] Passivity based control applied to power converters 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 Marco Liserre [email protected] Passivity based control applied to power converters 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) Marco Liserre [email protected] Passivity based control applied to power converters 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 Marco Liserre [email protected] Passivity based control applied to power converters 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 Marco Liserre [email protected] Passivity based control applied to power converters 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. Marco Liserre [email protected] Passivity based control applied to power converters 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) Marco Liserre [email protected] Passivity based control applied to power converters 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 Marco Liserre [email protected] Passivity based control applied to power converters 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 ! Marco Liserre [email protected] Passivity based control applied to power converters 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 Function P(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. Marco Liserre [email protected] Passivity based control applied to power converters 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 Additionally for 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 Marco Liserre [email protected] Passivity based control applied to power converters Brayton-Moser Equations For the class of topologically complete networks it is possible to construct the mixedpotential function directly. For this class it is known that the mixed potential is of the form: P( iL ,vC ) R( iL ) G( vC ) N( iL ,vC ) 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 Marco Liserre [email protected] Passivity based control applied to power converters Brayton-Moser Equations The components of the Mixed-Potential Function can be analysed in more detail as follows: P(iL ,vC ) PR (iL ) PE (iL ) PG ( vC ) PJ ( vC ) PT (iL ,vC ) R G N PR is the Dissipative Current Potential PG is the Dissipative Voltage Potential Marco Liserre [email protected] Passivity based control applied to power converters Brayton-Moser Equations The dissipative current and voltage potentials can be calculated as follows: iL PR ( iL ) vR ( i' L )di' L 0 vC PG ( vC ) iG ( v'C )dv'C 0 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. P(iL ,vC ) PR (iL ) PE (iL ) PG ( vC ) PJ ( vC ) PT (iL ,vC ) R G N PE is the total supplied power to the voltage sources E PJ is the total supplied power to the current sources J Marco Liserre [email protected] Passivity based control applied to power converters Brayton-Moser Equations P(iL ,vC ) PR (iL ) PE (iL ) PG ( vC ) PJ ( vC ) PT (iL ,vC ) R G N PT is the internal power circulating across the dynamic elements and is represented by: PT ( x ) iLT vC In this representation denotes the interconnection matrix and it is determined by KVL and KCL Marco Liserre [email protected] Passivity based control applied to power converters Brayton-Moser Equations Finally the expression of the mixed-potential function P(iL ,vC ) PR (iL ) PE (iL ) PG ( vC ) PJ ( vC ) PT (iL ,vC ) R G N can be rewritten as follows: P( x ) PD ( x ) PT ( x ) PF ( x ) PD(x)= PR(x)- PG(x) is the Dissipative Potential PF(x)= PJ(x)- PE(x) is the Total Supplied Power Marco Liserre [email protected] Passivity based control applied to power converters Brayton-Moser Equations The dynamic behaviour of topologically complete networks is governed by the following differential equations : diL P( iL ,vC ) L( iL ) dt iL dvC P( iL ,vC ) C( vC ) dt vC 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 Marco Liserre [email protected] Passivity based control applied to power converters Brayton-Moser Equations The previous equations can be expressed in a more compact way as follows: P( x ) P( x ) P( x ) Q( x )x x P( x ) ... x x x 1 n T with the state vector xRn = R+ defined as T iL x vC and with the diagonal square matrix Q(x) R(+)x(+) defined as 0 L(iL ) Q( x ) 0 C ( v ) C Marco Liserre [email protected] Passivity based control applied to power converters Brayton-Moser Equations When a circuit contains only linear passive inductors and capacitors, then the diagonal matrices L(iL) Rx and C(vC) Rx are of the form: L1 0 L( iL ) . 0 0 L2 . 0 0 0 0 0 . . 0 L C1 0 0 0 0 C 0 0 2 C( vC ) . . . . 0 0 0 C The Brayton-Moser equations are closely related to the co-Hamiltonian H*(iL, vC) (that represents the total co-energy stored in the network). If the co-Hamiltonian is known, then the matrices L(iL) and C(vC) can be easily found as follows L(iL ) i2L H * (iL , vC ), C(vC ) v2C H * (iL , vC ) Marco Liserre [email protected] Passivity based control applied to power converters 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 P(u, x) (1 u) P0 ( x) uP1 ( x), u 0, 1 The Switched Mixed-Potential Function is consistent with the individual MixedPotential Functions Marco Liserre [email protected] Passivity based control applied to power converters 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 PT ( x ) iLT vC and in particular in the interconnection matrix which becomes a function of u, (u) Marco Liserre [email protected] Passivity based control applied to power converters 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 μ u Average Discrete State Model z Marco Liserre x Model [email protected] Passivity based control applied to power converters 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}) μi ui Discrete State Model zi Marco Liserre Average xi Model [email protected] Passivity based control applied to power converters Passivity Based Control – Procedure 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 BraytonMoser equations can be rewritten as: PT ( z ) PD ( z ) PF ( z ) Qz z z z f(z) constant 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 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: Q PT ( ) PD ( ) PF ( ) f(ξ) Marco Liserre constant 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) [email protected] Passivity based control applied to power converters Passivity Based Control – Procedure being z = z − ξ the average state errors, it is possible to write: Qz Qz Q PT ( z ) P ( z ) P ( ) P ( ) ( z ) D ( z ) T ( ) D ( ) z z Assuming that the first two derivatives are linear functions of z and the last two derivatives are linear functions of ξ, yields: Qz PT ( z ) PD ( z ) PT ( z ) PD ( z ) ( z ) ( z ) ( z ) ( z ) The previous expression represents the error dynamics and it could be obtained from Qz PT ( z ) PD ( z ) PF ( z ) z z z by simply replacing the variable z with the error variable z derivative of PF Marco Liserre and eliminating the [email protected] Passivity based control applied to power converters Passivity Based Control – Procedure The next step is to add a damping term to the error dynamics to ensure asymptotic stability PT ( z ) PD ( z ) PV ( z ) Qz z z z 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: PV ( z ) PVr ( iL ) PVg ( vC ) The injected dissipation together with the dissipative potential of the system, gives the Total Modified Dissipation Potential PM PM ( z ) PD ( z ) z z PV ( z ) Marco Liserre [email protected] Passivity based control applied to power converters Passivity Based Control – Procedure Subtracting Qz PT ( z ) PD ( z ) PV ( z ) z z z Qz PT ( z ) PD ( z ) PF ( z ) z z z from the controller dynamics are obtained PF ( z ) PT ( ) PD ( ) PV ( z ) Q z z Marco Liserre [email protected] Passivity based control applied to power converters 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 1 2 1 L RS ( )C with 0 << 1, then for all (i Qz tend to zero as t → ∞ L, v 1 2 1 C) the solutions of PT ( z ) PD ( z ) PV ( z ) z z z where closed-loop resistance matrix RS is Marco Liserre 2 PM ( z ) RS ( iL ) 2 iL [email protected] Passivity based control applied to power converters Passivity Based Control – Procedure THEOREM 2 (G-Stability) If GP is a positive-definite and constant matrix, and 1 2 1 C GP T ( )L with 0 << 1, then for all (i Qz tend to zero as t → ∞. L, v 1 2 1 C) the solutions of PT ( z ) PD ( z ) PV ( z ) z z z where closed-loop conductance matrix GP is Marco Liserre 2 PM ( z ) GP ( vC ) 2 vC [email protected] Passivity based control applied to power converters Passivity Based Control – Procedure 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: Series Damping (damping on inductor currents) PM ( z ) PD ( z ) z z PVr ( iL ) Parallel Damping (damping on capacitor voltages) PM ( z ) PD ( z ) z z PVg ( vC ) 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 Marco Liserre [email protected] Passivity based control applied to power converters Passivity Based Control – Procedure Finally, if n is the number of minimum phase states it is possible to modify n equations of the + differential equations in PF ( z ) PT ( ) PD ( ) PV ( z ) Q z z 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: PF ( z ) PT ( ) PD ( ) PV ( z ) 0 zi i i zi and +-n differential equations: Qi Marco Liserre PF ( z ) PT ( ) PD ( ) PV ( z ) zi i i zi Controller Equations [email protected] Passivity based control applied to power converters Passivity Based Control – Procedure Controller Implementation At the beginning initial values of the n control inputs have to be set Using these values the differential equations Qi PF ( z ) PT ( ) PD ( ) PV ( z ) zi i i zi 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 PF ( z ) PT ( ) PD ( ) PV ( z ) 0 zi i i zi which solutions are the set of values for the control inputs to be applied in the next switching period. Marco Liserre [email protected] Passivity based control applied to power converters PBC of an H-bridge +DC T1 L T3 a i R C e b T2 T4 -DC grid The passivity-based controller will be designed by inspection, identifying the potential functions Marco Liserre [email protected] Passivity based control applied to power converters PBC of an H-bridge L PR 0 +DC z1 T1 T3 z22 PG 2R a i R C e b T2 z2 PD PR PG T4 -DC PJ 0 PE ez1 PF PJ PE z22 PD 2R PF ez1 PT z1 z2 Marco Liserre [email protected] Passivity based control applied to power converters PBC of an H-bridge L 2 2 +DC z1 T1 T3 a i R C e b T2 T4 -DC z2 z PD 2R PF e z1 PT z1 z2 P z2 e z1 P z2 z1 z2 R Marco Liserre 2 2 z P z1 z2 e z1 2R [email protected] Passivity based control applied to power converters PBC of an H-bridge +DC z1 L T1 T3 a i R C e b T2 T4 z2 Lz1 z2 e LKT z2 Cz2 z1 LKC R -DC Lz1 z2 e L1 2 e z2 Cz2 z1 R C2 Marco Liserre 2 R 1 controller [email protected] Passivity based control applied to power converters PBC of an H-bridge: damping injection +DC z1 L T1 T3 a i R C e b T2 T4 -DC Lz1 z2 e z2 Cz2 z1 R Lz1 z2 e RDS z1 z2 RDS z12 z22 PV ( z ) 2 2RDP z22 RDS z12 z22 PM ( z ) 2R 2 2RDP R( z1 ) RDS 1 1 G( z2 ) R RDP z2 z2 Cz2 z1 R RDP Marco Liserre [email protected] Passivity based control applied to power converters PBC of an H-bridge: damping injection Lz1 z2 e +DC z1 T1 L T3 a i R C e b T2 z2 z2 Cz2 z1 R T4 -DC Lz1 z2 e RDS z1 z2 z2 Cz2 z1 R RDP L1 2 e RDS z1 2 z2 C 2 1 R RDP Marco Liserre [email protected] Passivity based control applied to power converters PBC of an H-bridge: control variables +DC z1 L T1 L1 2 e RDS z1 T3 a i R C e b T2 T4 z2 2 z2 C 2 1 R RDP -DC 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? Marco Liserre [email protected] Passivity based control applied to power converters PBC of an H-bridge: zero-dynamics 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 Switching function in case of direct control Switching function in case of indirect control 2 1.5 1.5 1 0.5 0.5 Funzione di switching s Funzione di switching s 1 0 -0.5 0 -0.5 -1 -1 -1.5 -2 0 0.002 0.004 0.006 0.008 0.01 Tempo [s] 0.012 0.014 0.016 0.018 1 2 eR R L V d 0.02 -1.5 0 0.05 0.1 0.15 0.2 0.25 Tempo [s] 0.3 0.35 0.4 0.45 0.5 2 Rz1 Lz1 RLCz1 e RCe CR Lz1 e A stable system can be obtained only by indirectly controlling the dc voltages through the ac current i* Marco Liserre [email protected] Passivity based control applied to power converters PBC of an H-bridge A stable system can be obtained only by indirectly controlling the dc voltages through the ac current i* This means that as i i*, vc ξ2 Vd From the power balance it results that dc voltage reference 2Vd2 Id E load conductance 1 R grid voltage amplitude controller and i* I sin t d Lz*1 2 e RDS z1 C 2 2 z*1 GDP z2 reference reference voltage Vd power balance current i* and load conductance θ Marco Liserre ODE internally algebraic switching function µ generated ξ2 [email protected] Passivity based control applied to power converters PBC of an H-bridge Since it is possible to control directly the grid current, z1 1 (e Rz1 ) z2 L average KVL on the a.c. side 1 2 e Rz Lz 1 * 1 RDS z1 predictive action + damping injection The d.c. voltage is controllable only indirectly, through an internal variable ξ2 z2 1 z1 z2 C average KCL on the d.c. side 2 Then it is necessary to estimate the d.c. load Marco Liserre 1 ˆ G z1* 2 DP C differential eq. + z2 damping injection ˆ ˆ c2 ˆ c1 2 z2 [email protected] Passivity based control applied to power converters PBC of an H-bridge: damping tuning 1 2 1 L RDS ( )C 1 2 L 1 C RDS 1 RDS 1 L C L +DC T1 T3 RDS a i R C e GDP b T2 T4 -DC 1 2 1 C GDP ( )L Marco Liserre T 1 2 1 C 1 L GDP GDP 1 C L [email protected] Passivity based control applied to power converters Passivity Control of the Multilevel Converter The use of the passivity-based control (PBC) properly fits stability problems related to this type of converter Two approaches for the PBC design have been considered the first is developed considering the overall multilevel converter the second is developed by splitting the system into n subsystems and controlling them independently As regards the second, the partition of the multilevel converter is done on the basis of energy considerations The main advantage of the second approach is the separate control of the different DC-links and a flexible loading capability Marco Liserre [email protected] Passivity based control applied to power converters Mathematical model of the system The converter is controlled with a discrete switching function ui (i=1,2,...,n) for each Hbridge T11 R L T31 + a C1 R1 T31 T41 vc1 _ T1i T3i + T3i T4i T1n T3n ui T1i T 3i i n 1 x1 (e Rx1 ) ui x2,i L i 1 x2,i 1 ui x1 i x2,i Ci 1 equation e Ri vci _ n equations n+1 equations Marco Liserre Ci + b T3n T4n Cn Rn vcn _ [email protected] Passivity based control applied to power converters PBC of an H-bridge Since it is possible to control directly the grid current, z1 1 (e Rz1 ) z2 L average KVL on the a.c. side 1 2 e Rz Lz 1 * 1 RDS z1 predictive action + damping injection The d.c. voltage is controllable only indirectly, through an internal variable ξ2 z2 1 z1 z2 C average KCL on the d.c. side 2 Then it is necessary to estimate the d.c. load Marco Liserre 1 ˆ G z1* 2 DP C differential eq. + z2 damping injection ˆ ˆ c2 ˆ c1 2 z2 [email protected] Passivity based control applied to power converters PBC of an n-H-bridge converter It is not possible a simple extension of the PBC of one H-bridge active rectifier to n-bridge active rectifier In fact the PBC of an H-bridge active rectifier needs one algebraic equation and one differential equation Marco Liserre [email protected] Passivity based control applied to power converters PBC of an n-H-bridge converter It is not possible a simple extension of the PBC of one H-bridge active rectifier to n-bridge active rectifier In fact the PBC of an H-bridge active rectifier needs one algebraic equation and one equation 1 differential 1 ˆ G * z z 2 e Rz Lz 1 1 RDS z1 2 C * 1 2 DP 2 Thus “a simple extension” of this control needs n algebraic equations and n differential equations. However this is not possible since the n H-bridges are connected in series on the grid side and the ac voltage equation results in only one algebraic equation Marco Liserre [email protected] Passivity based control applied to power converters PBC of an n-H-bridge converter It is not possible a simple extension of the PBC of one H-bridge active rectifier to n-bridge active rectifier In fact the PBC of an H-bridge active rectifier needs 1 1 ˆ G z equation e Rz1 equation Lz1* RDS z1 and 2 one differential z1* one algebraic 2 DP 2 2 C Thus “a simple extension” of this control needs n algebraic equations and n differential equations. However this is not possible since the n Hbridges are connected in series on the grid side and the ac voltage equation results in only one algebraic equation We have proposed two PBC approaches: Marco Liserre 1. one algebraic eq. plus n differential eq. (with ξ2,1= .. = ξ2,n) 2. n algebraic eq. (based on n virtual KVL’s) plus n differential eq. (with ξ2,1≠ .. ≠ ξ2,n) [email protected] Passivity based control applied to power converters Passivity-based control approach 1 n 1 e Rz1 Lz1* RDS z1 i 2 i 1 1 ˆ G z z1* 2 2 DP 2,i for i 1, 2,...., n C ˆ ˆ c ˆ c z 2 1 2 2 Indirect control of output voltages i* I d sin t n Id 2Vd2 i 2n Vd2 Id E i 1 E ˆi i ˆi c2,i ˆi c1,i 2,i z2,i ˆ ˆ c2 ˆ c1 2 z2 1 2 ... n & 2,1 2,2 ... 2, n 2 reference voltage Vd and load power balance reference current i* conductance θ equal for all the bridges Marco Liserre internally algebraic switching function µ generated ξ2, equal ODE for all the bridges [email protected] Passivity based control applied to power converters PBC 2: Model formulation in subsystems R1 a1 L1 a1 H-Bridge 1 + Load 1 1 e i1 i R1 1 e L1 H-Bridge 1 + Load 1 i1 a2 a2 Ri ai Li e H-Bridge i + Load i i e ii ai+1 Rn i 1 ai+1 Rn an in an+1 H-Bridge n + Load n H-Bridge i + Load i ii Ln n e Li i e n i 1 an Marco Liserre i1 i2 .... in Ri ai n e Ln in H-Bridge n + Load n an+1 [email protected] Passivity based control applied to power converters PBC 2: Model formulation in subsystems R1 a1 L1 R1 a1 H-Bridge 1 + Load 1 1 e i1 i 1 e L1 H-Bridge 1 + Load 1 i1 a2 a2 Ri ai Li e H-Bridge i + Load i i e ii ai+1 Rn Ln i 1 in an+1 H-Bridge n + Load n H-Bridge i + Load i ii ai+1 Rn Ln n KVL Li x1 i e Ri x1 ui x2,i Ci x2,i ui x1 i x2,i an n e Li i e n i 1 an Marco Liserre i1 i2 .... in Ri ai ne in H-Bridge n + Load n an+1 [email protected] Passivity based control applied to power converters Passivity-based control approach 2 Indirect control of each output voltage achieved via the separate control of each bridge leading to n passivity-based controllers related through i i Vd2,i i n 2 Vd ,i i i 1 supervisor Marco Liserre 1 i e Ri z1 Li z1* RDS ,i z1 i 2,i 1 i z1* ˆi 2,i GDP,i z2,i 2,i Ci ˆ ˆ ˆ i i i c2,i i c1,i 2,i z2,i for i 1, 2,...., n for i 1, 2,...., n for i 1, 2,...., n n controllers for n H-bridges [email protected] Passivity based control applied to power converters Passivity-based control approach 2 only changing the parameters of the controllers Marco Liserre [email protected] Passivity based control applied to power converters Harmonic compensation In case of harmonics, the design results in the use of an RLC active damping branch very effective in harmonic rejection The damping is made by a resistance and a band-pass filter energy function includes energy related to harmonics Marco Liserre [email protected] Passivity based control applied to power converters Harmonic compensation bandpass filters Marco Liserre [email protected] Passivity based control applied to power converters Set-up for the multilevel active rectifier VLT 5006 GRID VLT 5006 driving signal e & enable i vc1 vc2 CONTROLLER D ( space card) Rated rms grid voltage Rated power Reference dc bus voltage Ac inductance Dc capacitors Marco Liserre 220 [V] 1 [kW] 400-560 [V] 10 [mH] 2330 [F] [email protected] Passivity based control applied to power converters PBC tuning: voltage error damping GDP GDP = 0.1 GDP= 1 310 310 300 300 290 290 280 280 270 270 260 260 250 250 240 240 3 4 5 6 7 8 GDP = 10 3 4 5 6 7 8 dc-link voltage due to a laod step change Marco Liserre [email protected] Passivity based control applied to power converters PBC tuning: estimate parameter γ γ = 0.01 γ = 0.01 300 Capacitor voltages [V] Estimate R1=R2 [Ohm] 350 300 250 200 150 280 260 240 100 50 220 0 2 4 6 Time [s] 0 2 4 6 Time [s] estimate dc-link load resistance due to a load step change it has a strong influence on the dclink dynamic Marco Liserre [email protected] Passivity based control applied to power converters Steady-state (PBC 1 & 2) grid voltage grid current dc-link voltage dc-link voltage Marco Liserre [email protected] Passivity based control applied to power converters Dynamical test PBC 1 PBC 2 Measured DC voltages [10 V/div] consequent to a load step change from half to full load on both the DC-links (330 F) Marco Liserre [email protected] Passivity based control applied to power converters Dynamical test (PBC 2) dc voltage reference step on one bus dc load steps on the two buses leading to different loads Measured DC voltages [50 V/div] and grid current [4 A/div] (2330 F) Marco Liserre [email protected] Passivity based control applied to power converters Dynamical test for active load (PBC 2) a dc motor has been used as active load Marco Liserre [email protected] Passivity based control applied to power converters Unbalance loads on the two dc-links (PBC 2) Steady-state behavior of PBC2 with full load on DC bus 1 and half load on DC bus 2 multilevel ac voltage [200 V/div] Marco Liserre grid voltage [100V/div] grid current [10 A/div] [email protected] Passivity based control applied to power converters Computational efforts comparison 15 PBC 2 needs p-3 equations more than PBC 1 PBC 2 nr. of control equations 12 However PBC 1 employs a division by the reference current that leads to computational problems 9 PBC 1 7 6 5 4 with p = number of desired levels 5 7 9 11 nr. of desired levels Marco Liserre [email protected] Passivity based control applied to power converters Harmonic compensation R-damping RLC-damping Marco Liserre [email protected] Passivity based control applied to power converters Conclusions Passivity-based theory offers a straightforward approach to design controllers without linearazing the system: physical and intuitive representation of the control problem design method to make the system Lyapunov-stable feedback decomposition useful for electromechanical systems Eulero-Lagrange formulation more suitable for electrical motors Brayton-Moser formulation more suitable for power converters (tuning procedure) Optimal results can be obtained with RLC damping of harmonics (similar to those obtained with generalized integrators – resonant controllers – linear approach) Marco Liserre [email protected]
