2nd Renewable Energy for Developing Countries - REDEC 2014 - November 26-27, Beirut - Lebanon http://www.redeconf.org/
Sliding-Mode Current Control Design For a GridConnected Three-Level NPC Inverter F. Sebaaly1,2,3, H. Vahedi2, H. Y. Kanaan1,2, N. Moubayed3 and K. Al-Haddad2 1
Saint-Joseph University, Faculty of Engineering – ESIB, Mar Roukoz, Mkalles, Lebanon 2 Ecole de Technologie Supérieure, Université du Québec, Montreal, Canada 3 Lebanese University, Faculty of Engineering – Branch I, Tripoli Campus, Tripoli, Lebanon Emails:
[email protected],
[email protected],
[email protected],
[email protected] Generating electricity from renewable sources such as photovoltaic panels, wind turbines and even hybrid systems commits the design of accurate controllers. The robustness of sliding mode controller in face of system variations was the cause for the integration of this technique in AC drive systems. [9-10] Grid connected inverters ask for a current control that improves the dynamic characteristics of the system and to generate current waveform with unity power factor. The immunity to parameter variations and sensitivity to disturbances and different operation points should characterize the current controller. Sliding mode controller has been chosen for these purposes. Studies on sliding mode controller applied on 3L-NPC inverters have focused on reducing the chattering problem of this technique which leads to variable and high frequency switching on the converter. Severe electromagnetic noise and high power losses can result from chattering problem. In [11], a space vector (α,β) sliding mode controller was introduced as a current controller for the 3L-NPC inverter. This paper deals with the best selection of switching states in order to improve sliding mode existence. ISMC controllers (integral sliding mode controller) were introduced to reduce tracking error and THD (Total Harmonic Distortion)[12-13]. For AC tracking systems a multi resonant sliding mode controller with a LCL filter was introduced in order to reduce the steady state error [14]. And finally authors in [15] has introduced an integral resonant SMC; a detailed comparison between the three methods shows the performance of this technique in decreasing the THD of the output current by adding multiple resonant terms to the sliding surfaces. However these methods proposed for grid connected system require the introduction of LCL filter in order to comply with system requirements. The sliding mode controller proposed in this paper deals with chattering problem considering Gao et al theory that imposes fixed switching frequency with a PWM modulation[16]; and in the same time it guarantees the performance of the system in steady state and dynamic states, reduces the tracking error and shows a smooth output waveforms This paper provides a sliding mode controller applied to a grid connected 3-level NPC inverter. Section II introduces the configuration and the mathematical model of the 3Linverter in d-q frame, while section III details the design of
Abstract—Three-level neutral point clamped (NPC) inverter topologies are becoming more and more the interest of studies, especially in grid connected systems, due to their advantages compared to other multilevel inverters. Synchronization with an AC source remains a challenge ring current injection to the grid. In order to enhance the performance and the immunity of such grid connected inverters, a sliding mode current controller based on Gao’s reaching law has been designed in this paper, and then applied to a three-wire three-level NPC inverter to have a unity power factor system. A PI regulator has been also employed to deal with the split DC-capacitors voltage unbalance problem. Robustness towards external disturbances was verified through simulations using MATLAB. Keywords—Sliding-mode control, neutral point clamped inverter, grid synchronization.
I. INTRODUCTION Since the whole world is fighting today against the persistent increasing of the CO2 emission caused by the highly developed industries; renewable energy resources have started to play their vital role in producing electricity and research are intensified in order to comply with pollution consequences. Recent years describe the highly integration of power electronic components in transmission, generation and lot of industries applications. Multilevel inverters have described well their path into the new developed technology that contributes today in the electricity generation. From the well-known neutral point clamped inverter to the cascaded H-bridges and the flying capacitor to the newly introduced CSC in 2013 in [1], multilevel inverters equipped with significant advantages (high power application, low output harmonic voltage….) continue to be a challenging subject to the researchers[2-6]. Three-level neutral point clamped inverter is one of the most promising topology between multilevel inverter especially in grid connected systems. This three leg topology ensures the possibility of integration of multiple lower DC sources with lower DC link capacitors compared to others [7-8]. For this fact, this topology is commonly used in grid-connected inverters as three-phase three-wire or four-wire network.
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2nd Renewable Energy for Developing Countries - REDEC 2014 - November 26-27, Beirut - Lebanon http://www.redeconf.org/
the sliding mode technique proposed to control the current and finally section IV presents and discusses all the simulations results obtained that shows the performance of this inverter in both the steady and transient state.
vabc =
2 −1 −1 vDC −1 2 −1 uabc 3 −1 −1 2
(2)
where: II. THREE LEVEL NEUTRAL POINT CLAMPED INVERTER MODELING The three level neutral point clamped inverter was firstly introduced by Nabae in 1981[17].Its schematic diagram is shown in Fig. 1. This inverter is made of three legs; each leg consists of four switches and two clamping diodes paralleled to the switches. It is supplied by a DC voltage source which is connected to two capacitors; the upper one and the lower one which are connected to the clamping diodes; the voltage at the terminals of each capacitor is equal to the half of the DC voltage. The balancing of these two voltages is a main drawback in the control of this structure. It is according to the switching states that this inverter injects an AC current directly to the grid. The switching states (for leg a) of this topology are presented in Table 1.It is identified that in each leg the switches and (or and ) work in a complementary way. This inverter adds a third level to the output voltage which the zero level is; allowingby this a lower stress on the switches and a smoother waveform compared to the conventional two levels inverter.
iabc
: Grid currents vector
vabc
: Inverter output voltages vector
vgabc
: Grid voltages vector
uabc vDC
: Switching position vector denoted in table I.
L R
: Line inductors : Line resistors
: DC voltage
In order to design an accurate controller, a transformation in the dq frame is applied to the model. Then, the transformation matrix turns out to be: 2π 2π cos(θ ) cos(θ − ) cos(θ + ) 2 3 3 2π 2π 3 ) − sin(θ + ) − sin(θ ) − sin(θ − 3 3
T =
(3)
where:
θ = ωt + ϕ θ is the angular position of dq reference frame with respect to the stationary frame and ; f is the grid frequency:
TL
TABLE I SWITCHING STATES OF 3L-NPC INVERTER
L Output Voltage
Switching states
+1 0 −1
1 0 0
1 1 0
0 1 1
0 0 1
+vDC / 2 0 −vDC / 2
diabc = − Riabc + vabc − vgabc dt
didq dt
= − Aidq + vDC udq − vgdq
(5)
where:
R −ω L A = R wL III. SLIDING-MODE CONTROLLER Designing a sliding-mode controller generally requires the satisfaction of different steps: First determining a switching function s(x) for that the sliding mode of the system on the switching surface will stay stale. This stability is assumed when s(x) = 0 ; assuming the existence and the reaching conditions are other the second step in the tasks of the
Fig. 1 presents a typical configuration of the three level three wires NPC grid connected inverter. The dynamical model of the system in abc frame can be written as follow: L
(4)
The model of the system in the dq frame is then given by equation (5)
Fig.1. Grid connected three-wire 3L-NPC inverter.
Switching Cases
diabc = T [ − Riabc + vabc − vgabc ] dt
(1)
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controller and finally the determination of the control lawu(x).
.
A. Sliding surfaces
.
sq Based on the state space model of the system presented in equation 5and on the choice of an optimal control vector that assumes the best switching sequence, the first step in designing the controller is to define the sliding surfaces.
B. System stability and reaching conditions The stability of the system in sliding mode is ensured with the below conditions:
sd = i d −idref sq = i q −iqref
. 1 [ Rsd + ε d sgn( sd ) + qd sd + Ridref + L i dref ] L (10) . 1 = − [ Rsq + ε q sgn( sq ) + qq sq + Riqref + L i qref ] L
sd = −
.
sd s d < 0
(6)
(11)
.
sq s q < 0
where , are the reference currents imposed on the system to track. The design of the controller is to ensure the best tracking of the currents with their references in presence of disturbances. The existence of an ideal sliding mode over a surface = [ ] = 0 is with the equivalent control vector which is the solution of = 0. The chattering in the system is the main cause for the difficult application of the sliding mode system in practical uses. This consequence is due to the substantially discontinuity switching characteristic of this technique. In this paper, the reaching law approach used was presented by Gao et al. in 1995 and detailed in [16].This approach not only reduces the system chattering but also assumes the satisfaction of the three modes: reaching mode, sliding mode and steady state mode of the system starting from any point and within a finite time; with the judicious choice of the parameters it permits the calculation of the bandwidth of the sliding mode and it is applicable with a multiple input system. It is defined as follows by equation (7).
According to equation (11), this system stability is ensured with the following conditions:
qd , q q > 0 .
ε d > Ridref + L idref ε q > Riqref + L iqref
The reaching law condition is verified by choosing the Lyapunov function as:
V =
.
s (t ) = −ε sgn( s (t )) − qs (t ) ε > 0, q > 0
.
sq (t ) = −ε q sgn(s q (t)) − q q sq (t ) We will consider that:
(8)
A simulation for the three phases three-wire grid connected 3L-NPC inverter is accomplished in MATLAB; the sample time of the system is set to 20µs. The system parameters are presented in Table II.
vDC uq − vgq = −ε q sgn( sq ) − qq sq + ω Lid The derivative of equation (6) is given by equation (9) .
.
. d−
i dref
q−
i qref
.
sq = i
(13)
IV. SIMULATION RESULTS
vDC ud − vgd = −ε d sgn( sd ) − qd sd − ω Liq
.
1 2 [s d + s 2q ] 2
By deriving equation (12) and with the conditions on the parameters of equations (11), the convergence of the system to the sliding surface is verified. By decreasing the parameters and , the chattering in the system will be suppressed. However it should be noted that damping the chattering effect will limit the performance of the system.
(7)
From equation (7), we can identify:
sd = i
(12)
.
TABLE II SYSTEM PARAMETERS
(9)
.
DC bus Voltage Switching frequency Capacitance of and Line Inductor Grid Phase to Phase rms Voltage
By getting equation (5) in to equation (9) and taking into consideration the equality in equation (8) we will have:
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2nd Renewable Energy for Developing Countries - REDEC 2014 - November 26-27, Beirut - Lebanon http://www.redeconf.org/
shows the capacitor voltages variations through the 10s of the simulation. The two voltages oscillate around the half of the DC bus voltage with '( = ±0.2( when a current of 5A is injected to the grid and '( = ±0.4( with the presence of disturbances. Fig. 5 illustrates the inverter output voltage; the five levels voltage is obtained in all system variations. The ripple phenomenon in the waveform of the output current is reduced while a smooth waveform is obtained.
The sliding mode controller has three roles in this system:
• To assume the best tracking characteristic for sequence current with its desired current reference. In order to study the performance of the controller in the transient state two perturbations were applied to the system at = 3 and = 7.02 ; the first step response changes the reference current from 5A to 10A at and the second one returns it back to 5A at . • To maintain the quadratic sequence current equal to zero in order to eliminate the reactive power. • To synchronize the phase current injected to the grid with the phase voltage of the grid in order to maintain a unity power factor with a THD less than 5%.
Currents and Current References I d& I dref (A)
20
0 Id -20 0
I q & I qref (A)
For the balancing of the upper and lower capacitor voltages of the 3L-NPC inverter a PI regulator is added. The parameters for the voltage balancing regulator used in this control are given in Table III. The schematic diagram for the controller is represented in Fig. 2.
1
2
3
4
6
7
8
I dref 9
10
0 -20 Iq -40 0
1
2
3
4
5 Time (s)
6
7
8
I qref 9
10
(3.a)
TABLE III VOLTAGE BALANCING REGULATOR
Currents and Current References
3 15
I d& I dref (A)
$% $&
5
10 8 6
I q & I qref (A)
4 2.96
Id 2.97
2.98
2.99
3
3.01
2.97
2.98
2.99
3
3.02
3.03
3.04
3.01 3.02 Time (s)
3.03
3.04
3.05
I dref 3.06
0.5 0 Iq
-0.5 2.96
3.05
I qref 3.06
(3.b)
I d& I dref (A)
Currents and Current References
Fig. 2.Schematic diagram of the whole simulated system.
The results of the simulation on MATLAB show the performance and robustness of the sliding mode controller in steady state and dynamic states. Fig.3 illustrates the direct sequence and quadratic sequences with their references. It can be easily remarked that both of these currents is tracking its reference. The direct current is ascending from 5A to 10A at when a variation is applied to the reference as it is represented inFig. 3.b while Fig. 3.c shows the dynamic response of the system at .In both of these situations the quadratic sequence of the currents remains zeros so does the reactive power. Balancing the DC capacitors voltages of the 3L-NPC inverter is a purpose in the control of this type of inverter. The unbalancing process effects the number of level obtained in the inverter output voltage. For this reason, a PI controller was added to the system in order to keep the voltages at the terminals of the capacitors equal. Fig. 4
10 8 6
I q & I qref (A)
4 6.98
Id 6.99
7
7.01
7.02
6.99
7
7.01
7.02
7.03
7.04
7.05
7.06
7.03 7.04 Time (s)
7.05
7.06
7.07
I dref 7.08
0.5 0 Iq
-0.5 6.98
7.07
I qref 7.08
(3.c) Fig. 3.Simulation results showing the time response of the dq-axis currents compared to their respective references.
Grid synchronization is the major purpose of the controller of a grid connected converter. Injecting a current to the grid in phase with the grid voltage with the lower THD possible that confirms with the international standards (less than 5%) through all the system conditions, states and variations
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2nd Renewable Energy for Developing Countries - REDEC 2014 - November 26-27, Beirut - Lebanon http://www.redeconf.org/
controller has been detailed and finally the performance and robustness of the controller were improved with the simulation results obtained. The choice of this method was proved for the utilization of such system as a grid connected photovoltaic inverter since the sliding mode controller introduced responds in excellent way to the system variations while keeping a good synchronization with the grid and produces low THD current.
remains a challenge when designing a controller. The synchronization is done in this system with the PLL technology. Fig. 6 presents the three phase currents and voltages waveforms of the grid. In order to have clear graphs a gain of 10 was multiplied by value of the current. The graphs show the robustness of the controller in tracking its reference especially in transient state while keeping a unity power factor. The value of the THD of the output grid connected current obtained passes from 4% to 2%. All the simulations results obtained prove the choice of this method and the capability of incorporating this system in grid connected system.
Phase a
Gris Currents and Voltages
DC Capacitor Voltages Vdcup
250.4
Phase b
250.5 Vdclow
250.2 Phase C
Vdcup (V) & VdcLow (V)
250.3
250.1 250 249.9
200 Van
0 -200 2.96
2.97
2.98
2.99
3
3.01
3.02
3.03
3.04
ia
3.05
3.06
Vbn
ib
3.05
3.06
Vcn
ic
3.05
3.06
200 0 -200 2.96
2.97
2.98
2.99
3
3.01
3.02
3.03
3.04
200 0 -200 2.96
2.97
2.98
2.99
3
3.01 3.02 Time (s)
3.03
3.04
249.8 Gris Currents and Voltages Phase a
249.7 249.6 249.5 0
1
2
3
4
5 Time (s)
6
7
8
9
10
Phase b
Fig.4.Waveforms of the upper and lower DC voltages. Phase to Phase Voltage and Phase Current
Phase C
Vab (V)
500
0
-500 2.96
2.97
2.98
2.99
3
3.01
3.02
3.03
3.04
3.05
3.06
200 Van
0 -200 6.98
6.99
7
7.01
7.02
7.03
7.04
7.05
7.06
ia
7.07
7.08
Vbn
ib
7.07
7.08
Vcn
ic
7.07
7.08
200 0 -200 6.98
6.99
7
7.01
7.02
7.03
7.04
7.05
7.06
200 0 -200 6.98
6.99
7
7.01
7.02
7.03 7.04 Time (s)
7.05
7.06
Fig. 6.Grid currents and voltages.
10 i a (A)
5
ACKNOWLEDGMENTS
0 -5 -10 2.96
2.97
2.98
2.99
3
3.01 3.02 Time (s)
3.03
3.04
3.05
3.06
7.07
7.08
The authors gratefully thank the GREPCI laboratory of Ecole de Technologie Supérieure, Canada Research Chair in Energy Conversion and Power Electronics, the Agence Universitaire de la Francophonie (AUF), the Lebanese National Council for Scientific Research (CNRS-L), the Research Laboratory of the Lebanese University (LASYS), the Research Council of Saint-Joseph University and CEDRE Project for their financial support.
Phase to Phase Voltage and Phase Current
Vab (V)
500
0
-500 6.98
6.99
7
7.01
7.02
7.03
7.04
7.05
7.06
10
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i a (A)
5 0
[1]
-5 -10 6.98
6.99
7
7.01
7.02
7.03 7.04 Time (s)
7.05
7.06
7.07
7.08
Fig.5.Source voltage and current. [2]
V. CONCLUSION This paper presents a sliding mode controller applied on a grid connected NPC inverter. The modeling of the threelevel inverter has been presented. The design of the
[3]
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2nd Renewable Energy for Developing Countries - REDEC 2014 - November 26-27, Beirut - Lebanon http://www.redeconf.org/
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