Predictive control of grid-connected multilevel inverter with output LCL filter.
Multilevel inverters are enabling technology for high voltage high power applications mainly after 1990s when they become mature technology , . However multilevel inverters have several advantages which are interesting even for low power applications. The quality of the grid current and reliability of the inverter are, among others, important parameters for grid connected photovoltaic (PV) inverter. Multilevel inverters with their lower voltage stresses, lower du/dt, lower switching frequency, lower harmonic distortion and better electromagnetic compatibility with reduced requirements for output filter  are interesting alternative for PV inverter.
The high performance output filter of the PV inverter is required to fulfill the grid code standards. Usually some higher-order filter such as LCL filter are used in PWM inverters -. Despite the high attenuation of higher-order harmonics produced by the PWM inverter, the LCL filter may become instable when excited on resonance frequencies. Thus it is required to use a damping technique to suppress those resonances. In order to increase the efficiency of the PV inverter it is advisable to use the active damping of the LCL filter , , .
Modern fast control systems such as DSP allow using of advanced control techniques in digital form. These modern control techniques often incorporate the model of the control system , ,  allowing online calculations of the control law. One of the model based control techniques is model predictive control (MPC). The optimal control move is computed on-line by solving an open-loop optimization problem at each sampling time in opposite to the pre-computed control law such as PI control where the closed-loop performance is considered . Advanced control techniques based on the model of the system are widely used in many areas such as sensorless control of an induction motor , grid connected converters , sensorless control of PMSM  and many other . The advantage of the model predictive control is that it can stabilize unstable systems such as the LCL filter.
For model control it is required to know the exact state of the system in every sampling instant. One can either measure all state variables, but this approach requires several sensors. To avoid this problem it is possible to use observers and estimators , .
II. SYSTEM MODEL
The controlled system consists of a grid-connected 15-level one-phase cascade inverter with the LCL output filter . The required mathematical model depends on the used control set. The MPC can be implemented with continuous or discrete control set. The continuous control set was chosen. This control technique requires using of the modulator. But on the other hand it has fixed switching frequency when compared to the discrete control set. This approach does not require the exact switching model of the inverter. The time delay of the inverter is negligible due to the high frequency PWM. However, it is necessary to focus on the discrete model of the LCL filter.
A. LCL Filter Model
The LCL filter is a dynamic system of the third order (Fig. 1). Due to the possible resonance of the LCL filter it is necessary to use a damping technique. The active damping was chosen to suppers the resonance of the LCL filter at the resonant frequency defined by (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
The design of the LCL filter for the cascade inverter is a complex task. The design guide can be found e.g. in .
For easier control design and tuning it is desirable to control the PV inverter in rotating reference frame d-q. The continuous time state space description of the LCL filter in d-q reference frame is defined by (2):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
The discrete time model [A.sub.Ddq], [B.sub.Ddq], [E.sub.Ddq], [C.sub.Ddq], [D.sub.Ddq] is derived using the forward Euler rule applied to (2).
III. MODEL PREDICTIVE CONTROL
Model predictive controller (MPC) is a controller with a system model and a feedback. The system model is used to predict the system state. The MPC is the most widely used advanced control technique nowadays.
The principle of the MPC is shown in Fig. 2.
At time instant kT the online optimization problem is solved on prediction horizon [N.sub.P] and control horizon [N.sub.C] and the series of optimal control moves is calculated. only the first control move is used and the whole optimization problem is solved again at time (k+1)T.
The significant advantage of the MPC controller is that it can stabilize the unstable LCL filter without implementation of any further active damping technique.
A. Prediction Model
The prediction model is used in the augmented form (3) . The addition of the integrator ensures zero steady state error for step changes of the reference value of the grid current [I.sub.g]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
System response in time for series of control moves u(k) without external error is defined by (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
The control goal is to minimise the control error on prediction horizon [N.sub.p] based on the system state x(k) at time kT and the reference value r(k) at time kT. The goal of the MPC is to minimise the cost function. The cost function in matrix form can be expressed by (5)
J (k) = [([[??].sub.k] + S[DELTA][u.sub.k] - [r.sub.k]).sup.T] Q ([[??].sub.k] + S[DELTA][u.sub.k] - [r.sub.k]) + + [DELTA][u.sup.T.sub.k]R[DELTA][u.sub.k], (5)
where [r.sub.k]--is vector of reference values on Np, [[??].sub.k]--is vector of the free system response, S[DELTA][u.sub.k]--is vector of the system response to control moves, Q--is weight positive-definite diagonal state matrix, R--is weight positive-definite diagonal control move matrix.
After derivation of (5) dJ/du the optimal control move vector [u.sub.k] on prediction horizon [N.sub.P] is (6)
[u.sup.*.sub.k] = -[([s.sup.T]QS + R).sup.-1] [[s.sup.T]Q ([[??].sub.k] - [r.sub.k])]. (6)
Equation (6) in more compact form is (7)
[u.sup.*.sub.k] =-Mx (k) + [Gr.sup.k], (7)
with matrixes M and G:
M = [([S.sup.T] QS + R).sup.-1] [S.sup.T] QF, (8)
G = [([S.sup.T] QS + R).sup.-1] [S.sup.T] Q. (9)
Because only the first control move is applied just the first element of the vector uk is considered (10)
u* (k) = -[K.sub.x]x(k) + [K.sub.x]x (k) + [K.sub.r]r, (10)
with matrixes [K.sub.x] and [K.sub.r]:
[K.sub.x] = [m.sup.T.sub.1], (11)
[K.sub.r] = [[[C.sub.d] [(I - [A.sub.d] + [B.sub.d] [K.sub.x]).sup.-1] [B.sub.d]].sup.-1]. (12)
IV. KALMAN OBSERVER
The MpC controller needs to know the exact state of the system. If this information is not available it needs to be observed. The Kalman observer is suitable for stochastic systems (e.g. deterministic system with process noise).
The dynamical system is described by (13):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
It is a classical representation of a discrete time-invariant system with measured error ([A.sub.D], [B.sub.D], [C.sub.D], [D.sup.D], [E.sup.D]) influenced by process noise ([G.sub.D], [H.sub.D]) and sensor noise ([K.sub.D]).
The auto-covariance process noise matrix [Q.sub.K] and the auto-covariance sensor noise matrix [R.sub.K] are defined. The output of the Kalman observer is the vector of observed system state [x.sub.k] (k) defined by (14)
[x.sub.k] (k) = [x.sub.p] (k) + Ke(k). (14)
The static Kalman gain matrix K is defined by (15)
K (k) = [P.sub.p] (k) [C.sup.T.sub.D] [[[C.sub.D][P.sup.p] (k) [C.sup.T.sub.D] + [R.sub.K]].sup.-1]. (15)
The auto-covariance matrix of the prediction [P.sub.P](k) is calculated in the previous step (16)
[P.sub.p] (k + 1) = [A.sub.D][P.sub.k] (k) [A.sup.T.sub.D] + [G.sub.D][Q.sub.K][G.sup.T.sub.D] = (16)
and the auto-covariance matrix of the state estimate is (17)
[P.sub.k] (k) = [I - K (k) [C.sub.D]] [P.sub.p] (k). (17)
To tune the Kalman observer it is needed to change the auto-covariance matrixes [Q.sub.K] and [R.sub.K]. The sensor noise auto-covariance matrix [R.sub.K] can be calculated based on the sensor measurement of the random variable. However the process noise auto-covariance matrix [Q.sub.K] cannot be easily calculated and often needs to by tune by trail-error method.
In the real system only the grid current [I.sub.g] is measured and the inverter current [I.sub.s] and the capacitor voltage [V.sub.C] are observed.
V. THE LABORATORY SETUP
The controlled dynamic system consists of a one phase grid-connected cascade H-bridge inverter created by three H-bridge inverters connected in series at its output . The H-bridge inverters are supplied by three galvanically isolated voltage sources [U.sub.A] = 120 V, [U.sub.B] = 60 V and [U.sub.C] = 30 V. The switching frequency is 5 kHz and the opposition disposition modulation technique was used. The output LCL filter parameters are described in Table I. The system is controlled in the rotating reference frame d-q. The control structure of the system is shown in Fig. 6
The one-phase grid voltage [V.sub.g] is sensed and the virtual two-phase system in stationary reference frame ([V.sub.g[alpha]], [V.sub.g[beta]]) is created. The grid phase is detected by a PLL circuit. The one-phase grid current at the output of the LCL filter is sensed. Than the virtual two-phase system in stationary reference frame ([I.sub.g[alpha]], [I.sub.g[beta]]) is created.
The grid current in the stationary reference frame is converted to rotating reference frame ([I.sub.gd], [I.sub.gq]). Because only one state variable of the LCL filter (the output current [I.sub.g]) is measured, the Kalman observer is used to reconstruct the missing two state variables: the inverter side current Is and the capacitor voltage [V.sub.c]. The full state of the LCL filter is fed to the MPC controller which creates the compensating inverter voltages in the rotating reference frame ([V.sub.sd], [V.sub.sq]). Next the voltages are summed with the grid voltage and are transformed into the stationary reference frame. Then only the [V.sub.s[beta]] voltage is used to control the modulator of the multilevel inverter.
The structure of the MPC controller is shown in Fig. 4. The Fig.4 is described by (10).
The parameters of the MPC controller and the Kalman observer are shown in Table II.
A. Simulation Results
First the performance of the controller (Fig. 6) was simulated using MATLAB/Simulink. The grid voltage was set to 120 V/50 Hz. The simulation results for step change of [I.sub.gq] are shown in Fig. 7 and Fig. 8. The [I.sub.gd] current reference was held constant and the [I.sub.gq] reference was stepped.
The initial oscillation of current is caused by a virtual two phase generator with second order filter. If the control system was used for a three-phase system this oscillation
Even though there is no additional active or passive damping technique used, the LCL filter is stable. This is the natural consequence of the MPC controller.
B. Measurement Results
The laboratory model of the one-phase grid-connected 15-level cascade H-bridge inverter with the output LCL filter was built with parameters shown in Table I. The system was controlled by RT-Lab with DAQ card. The master computer and the DAQ card have no hardware ability to generate the multilevel PWM. Thus the Texas Instrument DSP TMS320F28335 was used to generate the PWM. Even though the DSP had the sampling time of 10 [micro]s, the PWM modulator had low resolution. The other parameters were the same as in the simulation.
The measured results are shown in Fig. Felektr9 and Fig. 10. The measurements are unfiltered real results from the LEM current sensor output sensed by the computer DAQ card. The [I.sub.gd] current reference was held constant and the [I.sub.gq] reference was stepped. The measurements show two major results.
The first one is that the LCL filter is stable even for step changes of the grid current. Only one current sensor (grid current [I.sub.g]) and one voltage sensor (grid voltage [V.sub.g]) are needed to stabilize the LCL filter. This is the significant
The second one is the difference between the reference and the actual values of currents. The MPC controller relays on the dynamical model of the system. The dynamical model of the LCL filter is defined by (2). However after connection of the LCL filter to the grid, the dynamical model is influenced by the grid inductance, which is now a part of the system. This will result in a steady state error in the grid current which can be seen in Fig. 9 and Fig. 10. The scaling of the system matrix [A.sub.Cdq] does not help. For the model predictive control it is necessary to know the exact parameters of the controlled system. The identification of the grid parameters needs to be added to the control system. This complicates the design of the controller but allows using of the multilevel inverter without tuning for any particular grid connection.
The paper presents model predictive control technique for the one-phase grid-connected cascade inverter with the output LCL filter. The suggested control technique requires only measuring of the grid voltage and the grid current which minimizes sensor costs. The control technique also provides natural active damping for the LCL filter which enables to increase the overall efficiency of the system. The biggest disadvantage of the proposed control technique is a need for exact system model, mainly the LCL filter and grid model. To ensure zero steady state error it is needed to add the grid parameters identification to the control system. It will be a part of the next work.
Manuscript received October 21, 2014; accepted February 22, 2015.
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Marek Pastor (1), Jaroslav Dudrik (1)
(1) Department of Electrical Engineering and Mechatronics, Faculty of Electrical Engineering and Informatics, Technical University of Kosice, Letna 9, 042 00 Kosice, Slovakia email@example.com
TABLE I. PARAMETERS OF THE LCL FILTER AND INVERTER. Parameter Symbol Value Apparent power S 1.2 kVA Switching frequency [f.sub.SW] 5 kHz Inverter side inductor [L.sub.S] 2.11 mH Grid side inductor [L.sub.G] 1.03 mH Capacitor C 9.14 [micro]F Resistance of [L.sub.G] [R.sub.G] 33 m[ohm] Resistance of [L.sub.S] [R.sub.S] 63 m[ohm] First H-bridge DC link [U.sub.A] 120 V Second H-bridge DC link [U.sub.B] 60 V Third H-bridge DC link [U.sub.C] 30 V TABLE II. PARAMETERS OF THE PI CONTROLLER. Parameter Symbol Value Prediction horizon of the MPC [N.sub.P] 10 Control horizon of the MPC [N.sub.C] 3 Matrix Q of the MPC Q 10*[I.sub.20] Matrix R of the MPC R 0.3*[I.sub.6] Auto-covariance process [Q.sub.K] [MATHEMATICAL noise matrix EXPRESSION NOT REPRODUCIBLE IN ASCII] Auto-covariance sensor [R.sub.K] 0.1*[I.sub.2] noise matrix [MATHEMATICAL EXPRESSION NOT Kalman gain K REPRODUCIBLE IN ASCII] Sampling period T 200 [micro]s
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|Author:||Pastor, Marek; Dudrik, Jaroslav|
|Publication:||Elektronika ir Elektrotechnika|
|Date:||Jun 1, 2015|
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