Cost-Function Based Predictive Voltage Control of Two ... - IEEE Xplore

Cost-Function based Predictive Voltage Control of Two-Level Four-Leg Inverters using Two Step Prediction Horizon for Standalone Power Systems V. Yaramasu and B. Wu

M. Rivera, J. Rodriguez and A. Wilson

Department of Electrical and Computer Engineering Ryerson University Toronto, ON M5B 2K3 Canada Email: [email protected]

Department of Electronics Engineering Universidad T´ecnica Federico Santa Mar´ıa Valpara´ıso, 2390123 Chile Email: [email protected]

Abstract—This paper presents a cost-function based predictive voltage control strategy with a prediction horizon of two samples to effectively control the output voltage of three-phase fourleg inverter used for the standalone power systems. The threephase inverter with an additional leg is developed to deliver symmetrical sinusoidal three-phase voltages irrespective of the arbitrary consumer load profiles. The proposed controller uses the discrete model of the inverter and RLC filter for twostep prediction of output voltage for each switching state of the inverter. The control method chooses a switching state that minimizes the error between the output voltage and its reference. The proposed controller offers an excellent reference tracking with less voltage harmonic distortion for balanced, unbalanced and nonlinear loading conditions. The feasibility of the proposed control scheme has been verified by MATLAB/Simulink.

I. I NTRODUCTION The standalone power systems are an alternative solution to power-up remotely located consumers where the expansion of electrical grid is prohibitive and expensive. The hybrid power system combines two or more power generation units such as photovoltaic (PV), solar thermal, wind energy conversion system (WECS), mini/micro hydro, fuel-cell and biomass and so on to overcome inherent limitations in either; and two or more energy storage systems such as battery banks and flywheel to provide continuous and high-quality energy flow to the consumers [1]. The load could be a single home or several homes or large communities or islands. The other examples of standalone systems include satellite earth stations; broadcasting stations; military, medical and telecommunication equipments; aircraft and ship power supply networks; and large scale computer systems. The voltage source inverter in standalone power system needs to provide symmetrical and robust three-phase sinusoidal voltages irrespective of the arbitrary consumer load profiles. In order to supply such loads, a three-phase four-wire system is used with the neutral point accomplished by an additional transformer or by the inverter. The three-phase inverter with an additional fourth (neutral) leg and output RLC filter is proved to be the best candidate to provide transformerless neutral connection and symmetrical sinusoidal voltages to the loads [2]–[5].

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Three different output variables for four-leg converter such as voltage, current or power can be controlled depending upon the application. The current and power control techniques are used for grid connected distributed generation [6], active power filters [7], [8], active front-end rectifiers [9] and control of two or more motors from an inverter [10]. The voltage control technique is used for standalone distributed generation [11], uninterruptible power supplies (UPS) [12] and dynamic voltage regulators [13]. The voltage control using hysteresis regulators [14]; open-loop feed forward controllers [15]; linear PID controllers in stationary (αβ0) [16], synchronous (qd0) [17], [18] and natural (abcn) [3] reference frame with external voltage and internal current control loops; and pole placement controllers [19], [20] are being analyzed before. The tuning of PI controller is a trade-off between robustness and transient performance and is very empirical. All these abstruse voltage control techniques use even more complicated modulation stage in order to generate switching signals for the inverter. Various carrier-less modulation schemes such as hysteresis [21], [22], flux vector [23] and selective harmonic elimination (SHE) [24]; and carried-based sinusoidal pulse width modulation (SPWM) [4], [5], [25]–[27] and three-dimensional space vector modulation (3D-SVM) [28]–[31] methods are analyzed before. The hysteresis and flux vector techniques use complicated switching tables. The calculation of switching angles and their digital implementation for SHE is quite complex. Compared to the SPWM, 3D-SVM offers many advantages: good dc-link utilization, lower switching frequency and minimum output distortion [28]. Despite its benefits, the 3D-SVM is very complicated, time consuming and non-intuitive for software and hardware implementation [26], [32]. The cost-function based finite control set model predictive control (FCS-MPC) has found recent application in power electronics [33], [34]. This method appears as an attractive alternative to the classical control methods, due to its simple concept, fast dynamic response, and easy inclusion of nonlinearities & constraints in the design of controller [35]. Moreover, this scheme does not require internal current control loops and modulators and thus greatly reduces the complexity.

128

PV Array

4L-Inverter

DC/DC Converter-1

3 ∼ Linear

AC Filter AC PCC

4

DC PCC

4

4

Wind Turbine AC/DC Converter-1

PMSG

Gear Box

DC/DC Converter-2

+

Flywheel Storage

AC/DC Converter-2

2

3 ∼ Non-Linear

DC Loads

4

+

Figure 1.

1 ∼ Linear

Battery Banks

Standalone hybrid power system with four-leg inverter.

P

Sv

Su

AC Filter

Sn

Sw

iu

u

Rf

Lf

iou

Unknown Load

v vdc

w n Su

Sv

Sw

vou

Sn

Cf

vnN

N Figure 2.

Two-level four-leg inverter topology.

This digital control technique has successfully been applied to a wide range of power converters, drives and energy systems applications [36]–[48]. The above FCS-MPC methods usually consider one step horizon due to the reduced number of switching states and the reduced order of the load models. But in case of complex systems and special applications such as voltage control in standalone systems, a higher prediction horizon is necessary to improve the control performance [33], [49], [50]. The output voltage control for two- and three-level four-leg inverters using one step prediction has been analyzed in [51], [52]. In [53] two-step prediction for three-phase UPS system is presented. In this paper, the concept of FCS-MPC with two-step prediction has been extended to four-leg inverter to improve the output voltage regulation under unbalanced and nonlinear loading conditions. This controller predicts the behavior of the output voltage in terms of the measured voltages, currents and possible switching states of the inverter. The controller then selects a switching state that minimizes the error between the measured and desired output voltage.

II. F OUR -L EG I NVERTER M ODEL The typical standalone hybrid power system which consists of PV arrays, PMSG WECS, flywheel for short-term energy storage, battery banks for long-term energy storage, threephase four-leg inverter, RLC filter and arbitrary loads is shown in Fig. 1. The DC/DC converter-1 and AC/DC converter-1 perform maximum power point tracking to extract maximum possible energy from the sun and wind respectively. The power generation and storage units provide constant dc-link voltage to the four-leg inverter. The loads are unknown and can be of single-phase or three-phase, balanced or unbalanced, linear or non-linear nature. The power converter topology for fourleg inverter with output RLC filter is shown in Fig. 2. The connection format is similar to the conventional three-phase inverter with fourth leg connected to the neutral point of the load. As shown in Fig. 3, the fourth leg increases switching states from 8 (23 ) to 16 (24 ) and thus offers control flexibility and improved output voltage quality [15]. The voltage in any leg x of the inverter, measured from the negative point of the dc-link (N ) can be expressed in terms

129

1110 γ

III. M ODEL P REDICTIVE VOLTAGE C ONTROL S CHEME

β

1100

The proposed model predictive voltage control scheme is shown in Fig. 4. This method uses the inherent discrete nature of the four-leg inverter and RLC filter to predict the output voltage vector for a predefined horizon time k + N , in terms of the measured voltage and current in time k, and selects a switching state based on the minimization of cost (quality) function for each sampling time.

α

0110 0100 1010

1000 0010 1111

A. Discrete-Time Model for Predictive Control

0000 1101

The cost function requires the predicted output voltage vector vo [k + N ] in discrete-time form. For this reason, the space-state system in (9) can be represented in discrete-time as follows:       vo [k + N ] v [k + N − 1] v[k + N − 1] , =Φ o +Γ i [k + N ] i [k + N − 1] io [k + N − 1] (11) where,

0111 1011

0101

1001 0011 Figure 3.

0001

Switching vectors for four-leg inverter in αβγ coordinates.

of switching states as, vxN = Sx vdc ,

x = u, v, w, n,

(1)

by using (10) and (12),    φ 1 − cos(q) φ12 Φ = 11 = φ21 φ22 (1/p) sin q

and hence, the voltage applied to the output RLC filter, in terms of these inverter voltages is: vyn = vyN − vnN = (Sy − Sn ) vdc ,

y = u, v, w.

vo i



γ Γ = 11 γ21

(2)

The differential equations for the output filter, in terms of voltage and current vectors is described as follows: di − Rf i, = v − Lf dt dvo , = io + Cf dt

where,

(4)

0 A= −1/Lf

  1/Cf 0 , B= −Rf /Lf 1/Lf

(12)

 p sin(q) , cos(q)

(13)

 −p sin(q) , 1 − cos(q)

(14)

 p=

Lf , Cf

Ts q= . Lf Cf

4L-Inverter i[k]

AC filter Rf Lf

(15)

Load io [k]

u, v, w

(5) vdc [k]

vo [k]

(6)

Cf

n

(7) 8

(8)

S[k]

Minimization of cost function g with (24)

The system in (3) and (4) can be represented in space state form as follows:       v˙o vo v = A + B , (9) ˙i i io 

  γ12 cos(q) = γ22 −(1/p) sin q

(3)

where the voltage and current vectors are defined as,  T v = vun vvn vwn , T  io = iou iov iow , T  i = iu iv iw ,  T vo = vou vov vow .

where,

Γ = A−1 (Φ − I2x2 )B,

Φ = eTs A ,

16N

vo [k + N ]

vo∗ [k + N ]

Prediction of output voltage vo

 −1/Cf . (10) 0

with (11) and (12)

vdc [k] vo [k]

io [k] i[k]

Figure 4. Block diagram of model predictive voltage control scheme for the four-leg inverter.

130

B. Two-Step Prediction Horizon

C. Minimization of Cost Function

The FCS-MPC with a prediction horizon of N = 1 is used in many power electronics applications due to the simplified mathematical modeling and reduced computational burden [34]. For one-step prediction horizon, the discrete-time model for predictive control can be expressed as follows using (11):       vo [k + 1] v [k] v[k] , (16) =Φ o +Γ i [k + 1] i [k] io [k]

As shown in Fig. 4, the cost function requires output voltage vector vo [k+N ], and the reference voltage vector vo ∗ [k+N ]. To calculate the future reference voltage vector for N step prediction model, the Lagrange extrapolation method of order 4 can be used as follows:

from which,

The cost function for N step prediction horizon can be defined as follows:

vo [k + 1] = φ11 vo [k] + φ12 i [k] + γ11 v[k] + γ12 io [k]. (17) The 16 switching states at sampling time k, are used to predict the inverter voltage v[k] as shown in Fig. 5a. These 16 predictions for v along with measured variables vo , i and io at sampling time k are used to predict future (at sampling time k + 1) behavior of vo . These 16 predictions for vo are used by the cost function as shown in Fig. 4. There is no need to calculate i [k + 1] with one-step predictive control. For applications where robust control performance is needed, such as the one in this paper, the FCS-MPC algorithm requires a prediction horizon greater than 1. For prediction horizon of N =2, the number of feasible switching states K become: (18) K = 16N = 162 = 256. For two-step prediction horizon, the discrete-time model for predictive control can be expressed as follows using (11):       vo [k + 1] v[k + 1] vo [k + 2] =Φ +Γ , (19) i [k + 2] i [k + 1] io [k + 1] from which, vo [k + 2] = φ11 vo [k + 1] + φ12 i [k + 1]+ γ11 v[k + 1] + γ12 io [k + 1].

(20)

From the above it can be noted that the prediction vo at sampling time k + 2 requires the prediction of all the four variables vo , i, v and io at sampling time k + 1. The output voltage vo [k + 1] can be predicted using (17). The inverter current i [k + 1] can be obtained from (16) as follows:

vo ∗ [k + N ] = 4vo ∗ [k + N − 1] − 6vo ∗ [k + N − 2]+ (23) 4vo ∗ [k + N − 3] − vo ∗ [k + N − 4]

g=

N 

αn ||vo [k + n]∗ − vo [k + n]||,

(24)

n=1

where αn is an arbitrary weighting factor. For one- and two-step predictive control, 16 and 256 predictions are compared with the reference during each sampling time. The output voltage vector equals its reference when g = 0. Therefore, objective of the cost function considered in this paper is to achieve g value close to zero. The voltage vector that minimizes the cost function is chosen and then applied at the next sampling instant. Additional constraints such as switching frequency reduction, current limitation and spectrum shaping can also be included in this cost function g. IV. S IMULATION R ESULTS To validate the proposed control scheme, a simulation model for the three-phase four-leg inverter with the parameters as indicated in Table I, has been developed using MATLABSimulink. The phase sequence of the three phases is considered to be 0, 2 π/3 and 4π/3. Three different loading conditions are considered: balanced loads, unbalanced loads and non-linear loads. All the results are presented in peak per-unit system to simplify the analysis. The base values for voltage and current are given in Table I. The phase-u output voltage with 1- and 2-step MPC for balanced loading condition is shown in Fig. 6. As shown in Fig. 6b, the 2-step MPC tracks reference very well compared Table I F OUR - LEG I NVERTER AND L OAD PARAMETERS

i [k + 1] = φ21 vo [k] + φ22 i [k] + γ21 v[k] + γ22 io [k]. (21) The 256 switching states at sampling time k + 1, are used to predict the inverter voltage v[k + 1] as shown in Fig. 5b. The consumer maintains unknown load profile, and this leads to unknown mathematical model for the output current io . In such cases Lagrange extrapolation method of order 4 [36], [54] can be used to extrapolate current vector io into future in terms of the present and past values: io [k + 1] = 4 io [k] − 6 io[k − 1] + 4 io[k − 2] − io [k − 3]. (22) Compared to one-step prediction, this method considerably increases the amount of calculations per sample. However, with the development of faster and more powerful digital signal processors, the industry standard discrete-time digital implementation is possible and reachable.

131

Variable

Description

Value

vdc Cdc An fn fo∗ vo∗ ioB Lf Rf Cf R L Ts

dc-link voltage dc-link capacitor dc-link noise amplitude dc-link noise frequency Reference output frequency Peak reference output voltage Peak base load current Filter inductor Filter resistor Filter capacitor Load resistance Load inductance Sampling time

515 [V ] 1000 [μF ] 2 [V ] 100 [Hz] 50 [Hz] 311 [V ][1.0 pu] 15.52 [A][1.0 pu] 2 [mH] 0.05 [Ω] 80 [μF ] 20 [Ω] 3 [mH] 50 [μs]

v[k + 1]

v[k]

v[k + 2]

v[k + 1]

v[k]

v[k + 2] 1

1

1

16

16

256 k+1

k

k+2

k+1

k

(a) Figure 5.

k+2

(b)

Four-leg inverter output voltages using: (a) one-step prediction, (b) two-step prediction.

vou (pu)

vo (pu)

1

1

0

0

−1

vou

vov

vow

−1 (a)

vou (pu)

(a)

io (pu) iou

1

0

iov

iow

0 −1 ∗ vou

2-step

1-step

0.5 0

−1 6π

(b)

in (pu)

−0.5 6.2π

6.4π

(b)

6.6π

6.8π

t(s)

0

7π

π

2π

3π

4π (c)

5π

6π

7π t(s)

Figure 6. Simulation results for 1- and 2-step MPC with balanced loads: (a) phase-u output voltage, (b) zoom of phase-u output voltage.

Figure 7. Simulation results for 2-step MPC with balanced loads: (a) output voltages, (b) output currents, (c) neutral current.

to the 1-step MPC. The output voltages, line and neutral currents with 2-step MPC are shown in Fig. 7. Since load is balanced, the neutral current (in = iou +iov +iow ) is observed to be zero as shown in Fig. 7c. For unbalanced loading condition, a step change in phaseu, v, w from 1pu, 0pu, 1pu to 0.5pu, 1pu, 1.5pu respectively is applied at time t = 4π. This is the typical case for most of the standalone power system. The phase-u output voltage with 1- and 2-step MPC is shown in Fig. 8. The 2-step MPC tracks reference with very less error compared to the 1-step MPC. The load step change does not perturb the output voltage control as observed in Fig. 9a. The output voltages, line and neutral currents with 2-step MPC are shown in Fig. 9. The neutral current in which is sinusoidal in nature flows through

the fourth-leg as shown in Fig. 9c. The diode rectifier bridge along with RL load as shown in Fig. 1 is used to simulate non-linear loading condition. A step change from no-load to nonlinear load is applied at time t = 4π. The 1-step MPC generates more ripple than 2-step MPC as shown in Fig. 10. The non-linear current flowing through the all the four-wire do not effect the output voltage control as shown in Fig. 11. A detailed measurement of the output rms voltage error, ve and % total harmonic distortion (THD) of the output voltage are summarized in Table IV for the 1- and 2-step MPC. The 2-step MPC provides high quality power supply in typical standalone power system compared to the 1-step MPC as proved by the significant reduction in ve and %THD.

132

vou (pu)

vou (pu) 1

1

0

0 −1

−1 (a)

vou (pu)

(a)

vou (pu)

0

0

∗ vou

∗ vou

2-step

1-step

2-step

−1

1-step

−1 6.2π

6π

6.4π

(b)

6.6π

6.8π

t(s)

7π

Figure 8. Simulation results for 1- and 2-step MPC with unbalanced loads: (a) phase-u output voltage, (b) zoom of phase-u output voltage.

vo (pu)

vou

1

vov

6.2π

6π

6.4π

(b)

6.6π

6.8π

t(s)

7π

Figure 10. Simulation results for 1- and 2-step MPC with non-linear loads: (a) phase-u output voltage, (b) zoom of phase-u output voltage.

vo (pu)

vow

vou

1

0

vov

vow

0

−1

−1 (a)

io (pu)

iou

1

iov

(a)

io (pu)

iow

iou

1

0

ioviow

0

−1

−1 (b)

in (pu)

(b)

in (pu)

0.5

0.5

0

0

−0.5 0

−0.5 π

2π

3π

4π (c)

5π

6π

7π t(s)

0

Figure 9. Simulation results for 2-step MPC with unbalanced loads: (a) output voltages, (b) output currents, (c) neutral current.

V. C ONCLUSION A finite control set model predictive voltage control strategy for standalone three-phase four-leg inverter with a prediction horizon of two samples has been proposed to improve the output voltage quality compared to the one-step prediction horizon. This algorithm does not need internal current controllers and modulation stages. The proposed predictive control strategy is much simpler and intuitive than the classical methods proposed previously. The algorithm tests each of the 256 possible switching states and then selects a state that

π

2π

3π

4π (c)

5π

6π

7π t(s)

Figure 11. Simulation results for 2-step MPC with non-linear loads: (a) output voltages, (b) output currents, (c) neutral current.

minimizes the cost function. The proposed controller delivers high quality power to the standalone hybrid power system with stiff and symmetrical three-phase sinusoidal voltages and also offers lower values of voltage error in reference tracking and less %THD for output voltages for balanced, unbalanced and nonlinear loading conditions. The proposed control can compensate the effect of uncertainties in the load and dc-link voltage and in consequence the load voltage waveform remains balanced. This compensation has been achieved without any penalty in the transient and steady state operation.

133

Table II C OMPARISON OF O UTPUT V OLTAGE E RROR ev AND %THD FOR 1- AND 2-S TEP MPC M ETHODS .

Balanced

Unbalanced

Nonlinear

Before Load Step

After Load Step

MPC

1-step

2-step

1-step

2-step

ve (V )

11.08

5.28

11.08

5.28

% THD

4.38

1.99

4.38

1.99 5.57

ve (V )

11.41

5.54

12.00

% THD

4.54

2.15

4.85

2.12

ve (V )

13.23

5.75

14.93

6.20

% THD

5.35

2.08

6.06

2.45

ACKNOWLEDGMENT The authors wish to thank the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) through Wind Energy Strategic Network (WESNet) Project 3.1 and from the Chilean Fund for Scientific and Technological Development FONDECYT through project 1100404, Universidad Tecnica Federico Santa Maria and Basal Project FB0821. R EFERENCES [1] I. Vechiu, H. Camblong, G. Tapia, B. Dakyo, and O. Curea, “Control of four leg inverter for hybrid power system applications with unbalanced load,” Energy Convers. Manage., vol. 48, no. 7, pp. 2119–2128, 2007. [2] R. Zhang, “High performance power converter systems for nonlinear and unbalanced load/source,” Ph.D. dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA, Nov. 1998. [3] G. Kasal and B. Singh, “Voltage and frequency controllers for an asynchronous generator-based isolated wind energy conversion system,” IEEE Trans. Energy Convers., vol. 26, no. 2, pp. 402–416, Jun. 2011. [4] M. Ryan, R. Lorenz, and R. De Doncker, “Modeling of multileg sinewave inverters: a geometric approach,” IEEE Trans. Ind. Electron., vol. 46, no. 6, pp. 1183–1191, Dec. 1999. [5] N.-Y. Dai, M.-C. Wong, F. Ng, and Y.-D. Han, “A FPGA-based generalized pulse width modulator for three-leg center-split and four-leg voltage source inverters,” IEEE Trans. Power Electron., vol. 23, no. 3, pp. 1472–1484, May 2008. [6] E. dos Santos, C. Jacobina, N. Rocha, J. Dias, and M. Correa, “Singlephase to three-phase four-leg converter applied to distributed generation system,” IET Power Electron., vol. 3, no. 6, pp. 892–903, Nov. 2010. [7] V. George and M. Mishra, “Design and analysis of user-defined constant switching frequency current-control-based four-leg DSTATCOM,” IEEE Trans. Power Electron., vol. 24, no. 9, pp. 2148–2158, Sep. 2009. [8] T. Rachmildha, A. Llor, M. Fadel, P. Dahono, and Y. Haroen, “Hybrid direct power control using p-q-r power theory applied on 3-phase 4wire active power filter,” in Proc. IEEE–PESC Conf., Jun. 2008, pp. 1183–1189, Rhodes, Greece. [9] R. Zhang, F. Lee, and D. Boroyevich, “Four-legged three-phase PFC rectifier with fault tolerant capability,” in Proc. IEEE–PESC Conf., vol. 1, Jun. 2000, pp. 359–364, Galway, Ireland. [10] K. Matsuse, N. Kezuka, and K. Oka, “Characteristics of independent two induction motor drives fed by a four-leg inverter,” IEEE Trans. Ind. Electron., vol. 47, no. 5, pp. 2125–2134, Oct. 2011. [11] I. Vechiu, O. Curea, and H. Camblong, “Transient operation of a fourleg inverter for autonomous applications with unbalanced load,” IEEE Trans. Power Electron., vol. 25, no. 2, pp. 399–407, Feb. 2010. [12] J.-K. Park, J.-M. Kwon, E.-H. Kim, and B.-H. Kwon, “Highperformance transformerless online UPS,” IEEE Trans. Ind. Electron., vol. 55, no. 8, pp. 2943–2953, Aug. 2008. [13] S. Naidu and D. Fernandes, “Dynamic voltage restorer based on a fourleg voltage source converter,” IET Gener. Transm. Distrib., vol. 3, no. 5, pp. 437–447, May 2009.

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