14th International Power Electronics and Motion Control Conference, EPE-PEMC 2010
Interconnection and Damping Assignment Passivity-Based Current Control of Grid-Connected PWM Converter with LCL-Filter Matthias B¨ottcher, J¨org Dannehl∗ , Friedrich W. Fuchs∗∗ Institute for Power Electronics and Electrical Drives / University of Kiel, Kiel, Germany, email:
[email protected], ∗
[email protected], ∗∗
[email protected]
Abstract—Interconnection and Damping Assignment Passivity-Based Control of grid-connected PWM converters with LCL-filter is designed and analyzed. Dynamic performance, resonance damping and robustness against grid impedance variations are studied by simulation and show good results. Experimental tests on a 22 kW induction motor drive prove the simulation results. Index Terms—IDA-PBC, PWM converter, LCL-filter, robustness against grid impedance.
I. I NTRODUCTION Grid-connected PWM converters offer bi-directional power transfer with low harmonic current distortion. Typical applications are in the growing field of distributed generation systems (DGS), regenerative motor drives and active filters [1], [2]. Standards set stringent limits on the grid current harmonic distortion which can be devided into two parts. The first issue is related to low frequency harmonics due to grid voltage background distortion. It is a matter of disturbance rejection capability of the current controller [3]. Typically harmonic compensators like resonant controllers are employed for low frequency harmonic rejection [4]. High control bandwidth is required for good harmonic rejection. Otherwise stability problems may occur [5]. Due to this reason stability problems may also occur when current control bandwidth suffers from grid impedance variations [5]. The second issue of grid current harmonic distortion is related to harmonics at the PWM frequency of the converter and multiples. LCL grid filters are used for mitigation of PWM harmonics due to their good filtering characteristic above its resonance frequency, especially in high power applications when the switching frequency is very limited by the losses [6], [7], [8]. On the other hand its resonance may trigger undesired oscillations or even instability. Active resonance damping is usually preferred as it is more flexibel and efficient compared with passive damping. Manifold solutions have been proposed differing in complexity, number of sensors, and performance [9], [10], [11], [12], [13], [14], [15]. Statebased control approaches have also attracted attention due to the possibility of superior performance [16], [17], [18], [19], [20], [21], [22], [23], [24]. Other approaches based 978-1-4244-7855-2/10/$26.00 ©2010 IEEE
on nonlinear control methods entered also into the area of control of power electronics and drives [25], [26], [27], [28], [29]. Especially the passivity-based control (PBC) is a promising control technique, which offers considerable advantages in robustness compared to linear control methods [30]. The recently developed interconnection and damping assignment passivity-based control (IDA-PBC) is the most auspicious variant [30]. First applications in the field of power electronics can be found in literature [31], [32], [33], [34], [35]. In this paper the IDA-PBC is applied to the gridconnected PWM converter with LCL-filter. Robustness against grid disturbance with contemporaneously good behaviour regarding dynamic performance and active resonance damping is shown. The paper is organized as follows. Section II starts with a survey of PBC with the focus on IDA-PBC. In section III the system is modeled in equations of port-controlled Hamiltonian systems with dissipation (PCHD) as base for the control design in section IV. After analyzing the controlled system regarding dynamic behaviour, active damping and robustness by means of simulations results in section V and experimental results in section VI the conclusion is added in section VII. II. PBC - S URVEY The recent past has shown a trend towards nonlinear control techniques [30]. Here the system property of passivity [36], which is targeted on an interpretation of the energy flow between system and ambience, is the key for most of these developments [30]. According to notation of Lyapunov passivity implies asymptotic stability [36]. Based on the aim to impose passivity on a closed loop system in order to stabilize it the term passivity-based control (PBC) has been introduced in [37]. The principle is firstly to allocate a potential energy function with a strict local minimum in the desired equilibrium point to the system in Euler-Lagrange (EL) equations (energy shaping). The second step is to impose an additional damping in order to reach asymptotic stability (injection damping). This technique, known as classical PBC in literature, has been applied very successfully to mechanical systems in EL equations [30]. The application to electrical
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and electromechanical systems has necessitated a new mathematic system description [38]. This has entailed the introduction of the port-controlled Hamiltonian systems with dissipation (PCHD) in [38]. Each dissipassive system [36] can be described in PCHD form by means of matrices J(x) and G(x), which comprise the interconnection structure, dissipation matrix R(x) and energy function H(x), which represents a Lyapunov function [38]: d x = [J(x) − R(x)] ∇x H(x) + G(x)u dt y = GT (x)∇x H(x)
(1) (2)
Fig. 1. control
System consisting of PWM converter with LCL-filter and
J(x) = −JT (x)
(3)
B. Energy Function
R(x) = RT (x) ≥ 0
(4)
In the first step the total system energy has to be expressed in terms of the state variables. The following T definition of the state vector x = [x1 . . . x6 ] turns out to be advantageous: T x = Lf g igd Lf g igq Lf c icd Lf c icq Cf uCf d Cf uCf q (12)
Requirements are [38]:
Based on the PCHD form the interconnection and damping assignment passivity-based control (IDA-PBC) has been developed and introduced in [39]. The basic idea is to impose a desired PCHD form to the closed loop system with a control action β(x). The mathematic key is the so-called matching equation [39], whereas index d denotes the desired system: [J(x) − R(x)] ∇x H(x) + G(x)β(x) !
= [Jd (x) − Rd (x)] ∇x Hd (x)
(5)
There are different ways of solving the matching equation being associated with the design procedure [30].
Then the energy function H(x) can be formulated by: h 2 i x x2 x2 x2 x2 x2 H(x) = 12 Lf1g + Lf2g + Lf3c + Lf4c + C5f + C6f (13) Thus the gradient of the energy function yields: T ∇x H(x) = igd igq icd icq uCf d uCf q
(14)
III. S YSTEM M ODELING A. Differential Equations
C. PCHD Modeling
The system consists of PWM converter, which is connected to the grid by LCL-filter, and control as shown in Fig. 1, whereas system parameters are listed in table I in the appendix. So the control plant basically contains LCL-filter dynamic, PWM and computation delay. The equivalent circuit model of the LCL-filter with space vectors is depicted in Fig. 2. If the resistances are neglected the resonance frequency can be calculated by: q Lf g +Lf c 1 fres = 2π (6) Lf g Lf c Cf
In the second step the PCHD equations of the system according to (1) and (2) have to be derived. Considering the system equations (7), (8) and (9), whereas grid voltage is neglected by treating it as disturbance variable, and taking the requirements (3) and (4) into account expressions for R, J and G are obtained:
The system equations in grid voltage oriented dq-coordinates with angular grid frequency ωg are: dq dq dq d dq Lf g dt ig = −Rf g idq (7) g − jωg Lf g ig − uCf + ug − → − → − → −−→ −→ dq dq dq d dq (8) Lf c dt ic = −Rf c idq c − jωg Lf c ic + uCf − uc − → − → − → −−→ −→ dq dq d dq Cf dt uCf = idq (9) g − ic − jωg Cf uCf − → → −−→ − −−→ Input vector u and output vector y are defined as follows: T u = ucd ucq (10) T y = icd icq (11)
PWM and computation delay can be modeled as a firstorder delay. In this paper both is neglected for control design and thus for plant modeling. T3-21
0
ωg Lf g
0
0
−1
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−1
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ωg Lf c
1
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−ωg Lf c
0
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−1
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−1
−ωg Cf
−ωg Lf g 0 J= 0 1 0
1 ωg Cf 0
0
(15) R=
Rf g
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Rf g
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G=
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(16)
T (17)
errors, which can be minimized by tuning of the control parameters, are eliminated by an integral action. Taking the requirements (3) and (4) into account the matrices Jd and Rd are chosen with undefined parameters as well: Fig. 2.
One-phase equivalent circuit model of the LCL-filter
Jd =
IV. C ONTROLLER D ESIGN In this application parameterized IDA, introduced as a basic design procedure in [30], is most suited for control design. This means that energy function as well as matrices of the desired PCHD system are described by undefined parameters at first. The energy function is fixed to a particular class in order to restrict the number of solutions of the matching equation. The control action is obtained by evaluation of the matching equation, whereas fixed expressions for some of the undefined parameters are exposed on the one hand. The remaining ones turn out to be degrees of freedom for controller tuning on the other hand. A. Setting up the Matching Equation In the first step of control design the matching equation according to (5) has to be set up. The control action T has to be derived by means β (x) = [βd (x) βq (x)] of solving this equation with desired energy function Hd (x) and matrices Jd and Rd . The energy function has to represent a Lyapunov function [38], which can be achieved by complying with the class that the energy function of the control plant belongs to. Furthermore it must have a local minimum in the desired equilibrium T point x0 = [x the requirement of 01 . .∗. x06 ] .∗ Especially [x03 x04 ] = Lf c icd Lf c icq has to be satisfied, whereas i∗cd and i∗cq are the reference values of the converter current. In order to achieve this aim x01 , x02 , x05 and x06 must capture explicit values that have to be calculated since the state variables are physically associated with each other by the LCL-filter. This would entail large mathematic efforts, especially concerning the solution of the matching equation, and complexity of the control would increase considerably. In order to simplify the control design procedure the energy function of the desired PCHD closed loop system is chosen with x01 = x02 = x05 = x06 = 0, whereas γ1 , γ2 and γ3 are undefined parameters: Hd (x) = 12 γ1 x21 + γ1 x22 + γ2 (x3 − x03 )2 (18) +γ2 (x4 − x04 )2 + γ3 x25 + γ3 x26 The closed loop system is not able to reach the equilibrium point of this simplified energy function since it includes the requirement of zero grid current and zero capacitor voltage during a desired converter current flow. Nevertheless the system aims for the state of minimal total energy according to (18). Thus the control variable is aimed to approach the reference value. Steady state
0
J12
J13
J14
J15
J16
−J12
0
J23
J24
J25
J26
−J13
−J23
0
J34
J35
J36
−J14
−J24
−J34
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J45
J46
−J15
−J25
−J35
−J45
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J56
−J16
−J26
−J36
−J46
−J56
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Rd =
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(19)
(20)
B. Solving the Matching Equation Combining (13), (15), (16) and (17) with (18), (19) and (20) the matching equation (5) comprises six equations. The control action can be derived from the equations of the third and fourth row since they contain x03 and x04 and thus the reference values i∗cd and i∗cq of the converter current. At first the equations of the remaining rows can be used for determination of undefined parameters by means of comparison of coefficients. With redefinition r2 = r and J34 = J this leads to the following difference between interconnection and dissipation matrix of the PCHD closed loop system: Jd − Rd = −Rf g Lf g γ1 ω − γg 1
0 0 1 Lf g γ1 0
ωg γ1 −Rf g Lf g γ 1
0
0
−1 Lf g γ 1
0
0
0
0
−1 Lf g γ 1
0 1 Lf c γ 2 ωg Cf Lf g γ 1
0
−r
J
1 Lf c γ2
0
−J
−r
0
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−1 Lf c γ 2
0
0
1 Lf g γ 1
0
−1 Lf c γ 2
−ωg Cf Lf g γ 1
0
(21) Now the control action arises from row three and four of the matching equation: βd = (rγ2 Lf c − Rf c ) icd − rγ2 Lf c i∗cd L + 1 − γγ21 Lff gc uCf d
(22)
+ (ωg Lf c − Jγ2 Lf c ) icq + Jγ2 Lf c i∗cq βq = (rγ2 Lf c − Rf c ) icq − rγ2 Lf c i∗cq L + 1 − γγ12 Lff gc uCf q
(23)
− (ωg Lf c − Jγ2 Lf c ) icd − Jγ2 Lf c i∗cd Obviously γ2 can be defined as a fixed expression, whereas γ2 = L1f c turns out to be most beneficial for simplification. Finally with redefinition of γ1 = γ this yields the control action for d- and q-component with
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tuning parameters r, γ and J: βd = − r (i∗cd − icd ) − Rf c icd + (1 − Lf g γ) uCf d (24) + J i∗cq − icq + ωg Lf c icq βq = − r i∗cq − icq − Rf c icq + (1 − Lf g γ) uCf q (25) − J (i∗cd − icd ) − ωg Lf c icd Now the energy function of the PCHD closed loop system is given by: h Hd (x) = 12 γx21 + γx22 + L1f c (x3 − x03 )2 i L γ L γ + L1f c (x4 − x04 )2 + Cf gf x25 + Cf gf x26 (26) C. Extension by Integrator In order to eliminate steady state errors an integrator can be added without losing the property of passivity [30]. Here the control action of (24) and (25) is extended as follows, whereas k is a further design parameter, which can be used for controller tuning [33]: Z t βd0 = βd − k · (i∗cd − icd ) dt (27) 0 Z t βq0 = βq − k · i∗cq − icq dt (28) 0
D. Controller Tuning There are four tunable parameters within the extended control action, r, γ, J and k. In this paper analysis of their influence on the closed loop system and tuning is done empirically by means of a simulation model in MATLAB/Simulink, which just contains a simplified control plant dynamic according to (7), (8) and (9) and the control action (27) and (28). The most important parameter turns out to be r, since it represents the gain in the feedback of the control deviation. According to requirement (4) it must have a positive algebraic sign. It can be stated that the closed loop system is stable for r > 0 which means that passivation is already achieved by positive feedback of the control deviation. A compromise between fast tracking response and overshoot has lead to a value of r = 4.5. The feedback of the coupled current components disposes the steady state deviations in the actual values of the currents, which would appear without this feedback due to couplings. The bigger parameter J is chosen the faster these deviations are compensated but also the higher is the overshoot. In order to avoid overshoot in the actual value of one component when the reference value of the other component changes J is defined as zero. Thus only the actual currents are fed back with gain ωg Lf c . Reasonably the term g = (1−Lf g γ) has to be positive. g = 1 turns out to be a stability bound since bigger values lead to instability. Fig. 3 points out grafically the influence of this term on the dynamic behaviour. Reference and actual values of the converter currents, whereas a step is given for the d-component reference, as well as converter voltages are depicted with comparison of g = 1 (left) and g = 0.5 (right). Steady state deviations between reference and actual current at the beginning stem from the grid
Fig. 3. Simulation results with simplified model - Influence of tuning parameter γ; waveforms of reference values ( ) and actual values ( ) of d-components (a, b) and q-comp. (c,d) of converter currents and dcomp. ( ) and q-comp. ( ) of converter voltages (e, f); parameters: r = 4.5, J = 0, k = 0, g = 1 (left) and g = 0.5 (right).
current, which represents a disturbance variable. Only allocation of g = 1 fully compensates this disturbance theoretically. The actuating variable oscillates with the resonance frequency of the LCL-filter in a damped envelope oscillation with the frequency of the grid voltage. For g > 1 these oscillations would increase and thus cause instability. g = 0.5 is chosen as a reasonable value in order to have a safety margin regarding model uncertainty of grid-side inductance. It becomes clear from Fig. 3 that an integral action is required for steady state accuracy. k = 1500 is chosen as a compromise between a rise time as short as possible and an adequate overshoot. V. S IMULATION R ESULTS A simulation model of the system as depicted in Fig. 1 is built in MATLAB/Simulink, whereas grid, LCL-filter and switching converter are implemented in PLECS. Simulation results are shown in Fig. 4 to Fig. 7. Waveforms of grid and converter current as well as capacitor voltage and reference voltage of the PWM, which represents the actuating variable, are depicted with a step in the d- and q-component of the converter current. Fig. 4 shows the dynamic behaviour of the closed loop system when a switching and control frequency of 5 kHz is used. The step responses of the d- and q-component of the converter current are quite similar. A rise time of approximately five sample periods, that is 1 msec, can be stated with an overshoot of about 4.5 A, which is 22.5 % relating to the step height of 20 A. In the waveforms of the q-components there are slight oscillations with
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a frequency of 150 Hz, which is visible especially in the capacitor voltage. This is due to aliasing in the capacitor voltage, since this quantity is not sampled on its average value. This effect is diminished by using a bigger switching respectively control frequency as becomes clear in Fig 5, whereas a switching frequency of 10 kHz is taken. Regarding the converter current it can be stated a lower overshoot due to a shorter delay time, caused by the PWM, which is not respected for control design. Oscillations with the resonance frequency of the LCLfilter according to (6) are obvious from the capacitor voltage during the transient effect of the converter current. This phenomenon is also visible in the waveforms of grid current and converter voltage, but with less intensity. Concerning the waveform of the control variable, which is the converter current, a good active damping of the resonance can be stated. Fig. 6 shows the analysis of robustness against grid impedance variations. In comparison with Fig. 4 the only difference is the present grid-side inductance of the system, which is twice as high as the nominal one which is used for tuning. This means that a grid inductance equal to the grid-side filter inductance is regarded. For control design this additional inductance is not respected. Concerning the converter current step the simulation results show that the rise time is lengthened by one sample period. Overshoot stays approximately the same. Thus the control is very robust against grid impedance variations.
Fig. 4. Simulation results - Dynamic performance with fa = 5 kHz; waveforms of grid currents (a, b), reference values ( ) and actual values ( ) of inverter currents (c, d), capacitor voltages (e, f) and reference values of inverter voltages (g, h).
Fig. 5. Simulation results - Dynamic performance with fa = 10 kHz; waveforms of grid currents (a, b), reference values ( ) and actual values ( ) of inverter currents (c, d), capacitor voltages (e, f) and reference values of inverter voltages (g, h).
Fig. 6. Simulation results - Robustness against grid impedance (+200% of Lf g ): Dynamic performance with fa = 5 kHz; waveforms of grid currents (a, b), reference values ( ) and actual values ( ) of inverter currents (c, d), capacitor voltages (e, f) and reference values of inverter voltages (g, h).
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As described in subsection IV-D it has to be valid (1 − Lf g γ) < 1 for stability. So if the grid-side inductance is assumed too small there is no danger of getting unstable due to model uncertainties. In the reverse case this may happen, but prevention is possible by including a reasonable safety margin. In order to constitute effects in the experimental results due to grid harmonics Fig. 7 shows the behaviour at 3% of 5th and 7th harmonic, which are the most important harmonics in the grid. In dq-coordinates 5th and 7th harmonic of same percentage just yield a 6th harmonic in the d-component when they are in phase with the fundamental component. It can be extracted from the waveforms how these grid harmonics affect the grid and converter current as well as capacitor and converter voltage. VI. E XPERIMENTAL R ESULTS Experimental tests have been accomplished on a 22 kW induction motor drive. The control has been implemented with dSPACE. Results with a switching frequency of 5 kHz are shown in Fig. 8, whereas the diagrams are arranged according to the simulation results. In contrast to the simulations a step in the converter current is only made for the q-component. The waveforms are superposed by 6th harmonic because of 5th and 7th harmonic in the grid voltage as shown in the simulation results. Thus comparison of the step response with the theoretic curve is difficult. Nevertheless it can be stated that the rise time is quiet close to the rise time of the simulations. An additional resonant controller would improve the performance.
Fig. 8. Experimental results - Dynamic performance with fa = 5 kHz; waveforms of grid currents (a, b), reference values ( ) and actual values ( ) of inverter currents (c, d), capacitor voltages (e, f) and reference values of inverter voltages (g, h).
VII. C ONCLUSION The basic idea of IDA-PBC with a survey of the field of PBC is presented. A possibility to apply the nonlinear control method of IDA-PBC to the grid-connected PWM converter with LCL-filter is worked out and implemented. Analysis based on simulation and experimental results has been carried out. The control offers a very good robustness against grid impedance variations with contemporaneously good dynamic behaviour and active damping. A PPENDIX TABLE I S YSTEM PARAMETERS Symbol ug ωg Lf g Rf g Cf Lf c Rf c CDC fres fs = fc
Quantity Grid voltage (line-line) Angular grid frequency Grid-side filter inductance Grid-side filter resistance Filter capacitance Converter-side filter inductance Converter-side filter resistance DC link capacitance Resonance frequency Switching frequency = control frequency
Value 400 V 2π50 Hz 0.75 mH 50 mΩ 32 µF 2 mH 50 mΩ 2710 µF 1.2 kHz 5 kHz
R EFERENCES Fig. 7. Simulation results - Behaviour at harmonics in grid voltages (3% of 5th and 7th harmonic): Dynamic performance with fa = 5kHz; waveforms of grid voltages (a, b), grid currents (c, d), reference values ( ) and actual values ( ) of inverter currents (e, f), capacitor voltages (g, h) and reference values of inverter voltages (i, j).
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