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Sliding mode controller for the single-phase grid-connected photovoltaic system
Applied Energy 83 (2006) 1101–1115
APPLIED ENERGY www.elsevier.com/locate/apenergy
Sliding mode controller for the single-phase grid-connected photovoltaic system Il-Song Kim
*
LG Chem/Research Park, Mobile Energy R&D, 104-1, Moonji-Dong, Yuseong-Gu, Daejeon 305-380, Republic of Korea Received 1 August 2005; received in revised form 9 November 2005; accepted 12 November 2005 Available online 27 January 2006
Abstract A sliding mode controller for the single-phase grid-connected photovoltaic system has been proposed in this paper. Contrary to the conventional controller, the proposed system consists of maximum power point tracker (MPPT) controller and sliding mode current controller only. The proposed MPPT controller generates current reference directly from the solar array power information and the current controller uses the sliding mode technique for the tight regulation of current. The new MPPT controller does not require the measurement of the voltage derivative which can be a cause of divide-by-zero singularity problems. The sliding mode controller has been constructed based on a time-varying sliding surface to control the sinusoidal inductor current and solar array power simultaneously. The proposed system can avoid the current overshoot and make optimal design for the system components. The structures of a proposed system are simple, but they show the robust tracking property against modeling uncertainties and parameter variations. The mathematical modeling is developed and the experimental results verify the validity of the proposed controller. 2005 Elsevier Ltd. All rights reserved. Keywords: Photovoltaic power systems; Sliding mode controller; Maximum power point tracker
1. Introduction In recent years the need for renewable energy has become more pressing. Among them, the photovoltaic system (PV) such as solar cell is the most promising energy. The PV *
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0306-2619/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2005.11.004
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Nomenclature A Cn es(t) fs I iL Iph Ipk Iref isa isa avg Isat Ln Np Ns PE P E avg Pref Psa P sa avg q RL Rs Rsh S1–S4 T u(t) ueq un V VEP Vmp Vsa V sa avg a g1,g2 k h r Df1,Df2
deviation factor from the ideal p–n junction diode, dimensionless nominal input dc link capacitor in lF grid voltage in V inverter switching frequency in Hz solar cell output current in A inductor current in A light generated current in A peak value of the current in A current reference in A solar array current in A average solar array current in V cell reverse saturation current in A nominal output filter inductor in mH number of parallel modules, dimensionless number of series modules, dimensionless power transferred to the grid in W average power of the grid in W power reference in W solar array power in W average solar array power in W electronic charge: 1.6022 · 1019 C output filter inductor resistance in X series resistance in X shunt resistance in X. full-bridge inverter switch cell temperature in K control input, dimensionless equivalent input, dimensionless nonlinear control input, dimensionless solar cell output voltage in V peak grid voltage in V maximum power voltage in V solar array voltage in V average solar array voltage in V sliding mode controller gain, dimensionless boundary values of the uncertainties, dimensionless Boltzmann’s constant: 1.3807 · 1023 J/K grid voltage phase angle in sliding surface, dimensionless modeling uncertainties, dimensionless
energy is free, abundant and distributed through the earth. Among the PV energy applications, they can be divided into two categories: one is stand-alone system and the other is grid-connected system. Stand-alone system requires the battery bank to store the PV
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energy and suitable for low-power system. Grid-connected system does not require the battery bank and has become the primary PV application for high power applications. The main purpose of the grid-connected system is to transfer maximum solar array energy into grid with a unity power factor. The output power of PV cell is changed by environmental factors, such as illumination and temperature. Since the characteristic curve of a solar cell exhibits a nonlinear voltage– current characteristic, a controller named maximum power point tracker (MPPT) is required to match the solar cell power to the environmental changes. Many algorithms have been developed for tracking maximum power point of a solar cell [1–4]. Among them, the most commonly used methods are perturb and observe (P & O) and incremental conductance algorithm. The P & O method measures the derivative of power (Dp) and the derivative of voltage (Dm) to determine the movement of the operating point. If the sign of (Dp/Dm) is positive, the reference voltage is increased by some amount of value and vice versa. The other method, the incremental conductance method can track the maximum power point voltage more accurately than P & O, by comparing the incremental conductance and instantaneous conductance of a PV array. The configuration of a single-phase grid-connected photovoltaic system is shown in Fig. 1. It consists of solar array, input capacitor Cn, single-phase inverter, filter inductor Ln, and grid voltage es(t). The solar cells are connected in a series–parallel configuration to match the required solar voltage and power rating. The direct current (DC) link capacitor maintains the solar-array voltage at a certain level for the voltage source inverter. The single-phase inverter with filter inductor converts a DC input voltage into an AC sinusoidalvoltage by means of appropriate switch signals to make the output current in phase with the utility voltage and so obtain a power factor of unity. A typical controller configuration of the single-phase grid-connected photovoltaic system consists of a MPPT controller, voltage controller and current controller [5]. The MPPT controller detects the power slope from the solar-array voltage and current information, and generates the reference voltage. The voltage controller controls the solararray voltage to follow the reference voltage using the proportional-integral (PI) controller. The output of the voltage controller becomes the DC value of the reference current. The current controller controls the inductor current to follow the reference current using a hysteresis or predictive controller. The hysteresis controller has a fast response time, but it has an irregular switching frequency. The predictive controller has a constant switching frequency and good current control, but it requires exact information for the circuit parameters. As the current controller has a cascade configuration with the voltage controller,
isa Solar array
Cn
S1
+ vsa _
.
S3
RL
. S2
Ln
iL
Grid Vo ltage
~
es(t)
S4
Fig. 1. Typical configuration of single stage grid-connected PV system.
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the inductor current overshoots when the reference voltage is changed. This is caused by the integrator of the voltage controller. This current peak can stress the power device in the inverter and cause a failure in the switching devices. The additional drawbacks of the conventional system are the tedious tuning for the PI-gain selection and the requirement of exact knowledge for the circuit parameters. The new controller has been proposed to overcome the above problems. It consists of a MPPT controller and current controller. The new MPPT controller generates a reference power instead of a reference voltage. This reference power is used for the current reference directly by the power-balance relation. For the tight regulation of the inductor current, a sliding mode controller has been used for the controller current. The sliding mode controller has a robust control property under the presence of parameter variations and can achieve a tight regulation of the states for all operating points [6–8]. The mathematical modeling is evolved and the experimental result verifies the worthwhileness of the proposed controller. 2. Solar cell/array modeling The electrical equivalent-circuit of a solar cell is shown in Fig. 2(a). It is composed of a light-generated current source, diode, series resistance, and parallel resistance. The characteristic equation for the current and voltage of a solar cell is given as follows: n h q i o V þ IR s I ¼ I ph I sat exp ðV þ IRs Þ 1 ð1Þ AkT Rsh where I is the solar-cell output current (A), V is the solar cell output voltage (V), Iph is the light-generated current (A), Isat is the cell reverse saturation current (A), q is the electronic charge = 1.6022 · 1019 C, A is the dimensionless deviation factor from the ideal p–n junction diode, k is Boltzmann’s constant = 1.3807 · 1023 J/K4, T is the cell temperature (K), Rs is the series resistance (X), and Rsh is the shunt resistance (X). The equivalent circuit for solar cells arranged in Np-parallel and Ns-series is shown in Fig. 2(b) and the mathematical equation relating the array current to the array voltage becomes: q V sa I sa Rs N p V sa I sa Rs I sa ¼ N p I ph N p I sat exp þ þ 1 ð2Þ AkT N s Np Rsh N s Np where Np represents the number of parallel modules. Note that each module is composed of Ns cells connected in series; NpIph corresponds to the short-circuit current of the solar array. The detailed parameters for the simulation and experiment are shown in Table 1. Ns Rs Np
Np Rs
+ I Rsh
V
N p I ph
Ns
.. .. . .
_ a
. ..
I ph
Isa
+
b
.. .
Ns Rsh Np
Vsa
_
Fig. 2. Electrically equivalent solar cell/array circuit. (a) Single cell circuit. (b) Solar array circuit (Ns-series, Npparallel).
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Table 1 Parameters for simulation and experiment Solar array (simulator) Ns Np Iph T Vmp
60 1 2A 300 K 38 V
Inverter Cn Ln RL fs Output voltage
1000 lF 5 mH 0.5 X 10 kHz 25 Vpeak/60 Hz
Sliding mode controller gain a
3
The voltage–current curve of the solar array shows a highly non-linear characteristic around the maximum-power point. 3. System modeling The dynamic model of a single-phase photovoltaic system can be obtained from the configuration of Fig. 1: Vsa and iL are the solar-array voltage and inductor current, respectively, and es(t) is the grid voltage and is given as follows: es ðtÞ ¼ V EP sin xt
ð3Þ
where x is the angular frequency given by 2pf, where f = grid frequency (60 Hz). The circuit parameters Cn and Ln correspond to their nominal values which are known exactly. The switch status of the single-phase inverter can be represented by the input c, defined as follows: þ1 ! S 1 ; S 4 : on; S 2 ; S 3 : off c¼ ð4Þ 1 ! S 1 ; S 4 : off; S 2 ; S 3 : on When the switch input c = 1, the state equation can be written as 1 ðiL þ isa Þ þ Df1 Cn 1 i_L ¼ ðV sa RL iL es ðtÞÞ þ Df2 . Ln m_ sa ¼
ð5Þ
where Df1 and Df2 represent the modeling uncertainties caused by the noise and measurement errors, and are bounded by the known values g1 and g2, respectively. When the switch input c = 1, the state equation is given as 1 ðiL þ isa Þ þ Df1 Cn 1 i_L ¼ ðV sa RL iL es ðtÞÞ þ Df2 . Ln m_ sa ¼
ð6Þ
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Combining (5) and (6), the state-space averaged model of the single-phase grid-connected photovoltaic system shown in Fig. 1 can be derived as follows: 1 V_ sa ¼ ðiL uðtÞ þ isa Þ þ Df1 Cn _iL ¼ 1 ðV sa uðtÞ RL iL es ðtÞÞ þ Df2 Ln
ð7Þ
where u(t) is defined as the average value of c for the switching period. The state equation given in (7) is a non-linear system and can be expressed in the general form as x_ ¼ f ðx; tÞ þ gðxÞ uðtÞ þ Dd, where the vectors x, f(x, t), g(x) and Dd are given as follows: " isa # " i # L msa Df1 Cn Cn x¼ ; f ðx; tÞ ¼ RL iL es ðtÞ ; gðxÞ ¼ msa ; Dd ¼ . ð8Þ iL Df2 Ln L n
The linear controller, such as PI, requires exact information on the circuit parameters for the desired performance when controlling the non-linear system. However, in a grid connected photovoltaic inverter system, the electrolytic capacitor is used for the input capacitor and it is known that its actual capacitance value has a 50% tolerance from its nominal value and deteriorates year-by-year [9]. The capacitance value determines the amount of ripple in the solar-array voltage. The higher capacitance shows the lower voltage ripple. The inductance also has a variation from its nominal value. All of these uncertainties make it difficult to use the linear state-controller for the single-phase grid-connected photovoltaic system. Linear controllers are known not to perform well under the presence of disturbances and unknown parameter variations [6,7]. 4. Background on the sliding mode control Sliding mode control is a kind of non-linear control which is robust in the presence of parameter uncertainties and disturbances. It is able to constrain the system status to follow trajectories which lie on a suitable surface in the sliding surface. The equilibrium state is constructed so that the system restricted to the manifold has a desired behavior. Consider a non-linear system of the form, x_ ¼ f ðx; tÞ þ gðxÞ uðtÞ þ Dd
ð9Þ
The main steps for sliding mode controller design can be summarized, by using an equivalent control concept, as follows [6]: The first step is the selection of a switching surface r(x, t) = 0 (where x is the system’s state vector) that provides the desired asymptotic behavior in steady state. Obtaining the equivalent control ueq by applying the invariance condition _ tÞ ¼ 0 with uðtÞ ¼ ueq rðx; tÞ ¼ 0 and rðx;
ð10Þ
The existence of the equivalent control ueq assures the feasibility of a sliding motion over the switching surface r(x, t) = 0. On the other hand, beside describing the average dynamical behavior of the power stage over the switching surface, the equivalent control enables one to obtain the sliding domain, given by
I.-S. Kim / Applied Energy 83 (2006) 1101–1115
minfu ; uþ g < ueq < maxfu ; uþ g
1107
ð11Þ
+
where u and u are the control values for r(x, t) < 0 and r(x, t) > 0, respectively. The sliding domain is the state plane region where sliding motion is ensured. Finally, selecting a non-linear control input un to ensure that Lyapunov stability criteria, i.e., rr_ < 0. 5. Design of the sliding mode controller The design of the sliding mode controller starts from the design of the sliding surface. Usually, the sliding surface is constructed by the linear combination of state variable errors that are defined as the differences between the state variables and their references. Therefore, in this case, the sliding surface can be designed with errors of the solar-array voltage and inductor current in a single-phase grid-connected photovoltaic system. The reference solar-array voltage is a DC voltage which is generated from the MPPT, but the solar-array voltage is oscillating due to the sinusoidal inductor current, which results in an undesirable sliding mode performance. For this reason, no attempt has been made to apply the sliding mode control for a grid-connected photovoltaic system. The main purpose of a grid-connected photovoltaic system is to transfer the maximum solar-array power into the grid with a power factor of unity. Therefore, the sliding surface should be designed to control the inductor current and solar-array power simultaneously. This requirement can be achieved by selecting a sliding surface only using the errors of the inductor current. If the reference inductor current is expressed as a function of the solararray power, then the sliding surface can control both the inductor current and the solararray power simultaneously. The mathematical expression for the peak reference current is Eq. A.6 of the Appendix. The proposed time-varying sliding surface is defined by: rðx; tÞ ¼ iL I ref ¼ iL
2 P ref sin xt V EP
ð12Þ
where Pref is the reference solar-array power, which is given by the MPPT controller and VEP is the peak grid-voltage. The next step is to design a control input which satisfies the sliding mode existence law. The control input is chosen to have the structure: uðtÞ ¼ ueq ðtÞ þ un ðtÞ
ð13Þ
where ueq(t) is an equivalent control-input that determines the system’s behavior on the sliding surface and un(t) is a non-linear switching input, which drives the state to the sliding surface and maintains the state on the sliding surface in the presence of the parameter variations and disturbances. The equivalent control-input is obtained from the invariance condition and is given by the following condition _ tÞ ¼ 0 ) uðtÞ ¼ ueq ðtÞ. rðx; tÞ ¼ 0 and rðx;
ð14Þ
Rewriting the above equation gives _ tÞ rðx;
or or ðf ðt; xÞ þ gðxÞ ueq ðtÞÞ þ ¼ 0. ox ot
ð15Þ
The necessary condition for the existence of a sliding motion of r(x, t) is represented by the transversality condition and expressed as follows:
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Lg r
or gðxÞ 6¼ 0 oxT
ð16Þ
where Lgr denotes the directional derivative of the scalar function r with respect to the vector field g(x). If the transversality condition is satisfied, then substituting (12) into (15) gives a result as follows: _ tÞ ¼ rðx;
1 2P ref x cos xt ðV sa ueq ðtÞ RL iL es ðtÞÞ ¼ 0. Ln V EP
ð17Þ
Therefore, the equivalent control-input is given as ueq ðtÞ ¼
RL iL þ es ðtÞ þ 2P ref Ln x cos xt=V EP . V sa
ð18Þ
The nonlinear switching input un(t) can be chosen as follows: un ðtÞ ¼ a sgnðrÞ.
ð19Þ
If (18) and (19) are substituted into (13), the range of a ensuring rr_ < 0 can be determined as follows: 2P ref x cos xtÞ rr_ ¼ rði_L V EP V sa es ðtÞ 2P ref ðueq ðtÞ þ un ðtÞÞ þ Df2 x cos xt ¼r Ln Ln V EP V sa ¼r a sgn ðrÞ þ Df2 < 0. ð20Þ Ln From this result, the range of switching gain is given as a>
Ln Ln jDf2 j ¼ g. V sa V sa 2
ð21Þ
From (18), (19) and (21), the control input u(t) = ueq(t) + un(t) is given as follows: uðtÞ ¼
RL iL þ es ðtÞ þ 2P ref Ln x cos xt=V EP g2 Ln sgnðrÞ . V sa
ð22Þ
Then, the control input is compared with the pulse-width modulation (PWM) ramp voltage and generates the appropriate switching pattern of the inverter. 6. MPPT controller design The MPPT controller takes Vsa and isa as inputs to detect the power slope and generates Pref to track the maximum-power point. Due to the oscillating solar-array voltage and current, the average values are used for calculation of the solar-array power P sa avg . The average values are obtained using half-cycle data of the utility-grid frequency. A simple maximum-power point updating algorithm is given as follows: P ref > P sa avg ) P ref ¼ Hold previous value ð23Þ P ref 6 P sa avg ) P ref ¼ P ref þ D where D means a shift step from the previous value. The value of Pref is reset periodically to compensate for any environmental changes of the solar array. The main advantage of
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Isa a
b
Pref2 B
Pref1 A
Vsa
Fig. 3. Operating point movement for the power reference change.
Current controller
Pref
MPPT
2 VEP
Grid voltage
PWM S1,S4 Single-
I ref Sliding u (t )
γ
phase inverter S2,S3 with L
Mode Controller
iL
es (t )
vsa _ avg Average
es (t )
isa _ avg Calculation
.
. Isa
vsa
sin(ω t )
θ
PLL
Fig. 4. Overall controller configuration of the proposed system.
this controller is that it does not require the measurement of the voltage derivative which can be a cause of divide-by-zero singularity problems. The operating point movement of the solar array is shown in Fig. 3 when the reference power has been changed from Pref1 to Pref2. For a reference power Pref1, there can be two intersection points (A, a) between the solar-array characteristic curve and the reference power. Point ‘A’ is stable and will be the desired operation point. On the other hand, point ‘a’ would be clearly unstable if the system operated permanently in this mode of operation. When the reference power changed to Pref2, the operating point moved from point ‘A’ to point ‘B’ in a similar manner. Fig. 4 shows the proposed controller configuration. It consists of a MPPT controller, sliding mode controller, and PWM generator. The MPPT controller tracks the maximum power point using average values of the solar-array voltage and current. The sliding mode controller controls the inductor current to follow the reference current by means of the sliding surface. The PWM generator generates the switching pattern according to the control input. 7. Simulation and experiment results Fig. 5 shows the simulation result of the proposed controller using the parameters shown in Table 1. The peak grid voltage is 25 V and the frequency is 60 Hz. For a reference power of Pref = 50 W, the peak reference current is obtained as 4 A from Eq. (A.6). In
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I.-S. Kim / Applied Energy 83 (2006) 1101–1115 5A 0 -5A 1
iL
iref
u
ueq
0 -1 46V 44V
vsa
42V 1.6A 1.2A
isa
0.8A 0.06
0.07
0.08
0.09 [sec]
Fig. 5. Simulation waveform of the voltage, control input and current.
this figure, the inductor current tracks the reference current with switching ripple. The control input is composed of the equivalent input and the switching input. The switching input shows a chattering value over the equivalent input. As can be seen in this figure, the solararray voltage and current oscillate at the frequency of 120 Hz which is twice the grid frequency. The average solar-array voltage is 44.2 V and average solar-array current is 1.2 A. Therefore, it can be seen that the solar-array power is exactly controlled to the given reference power of Pref = 50 W considering the power loss caused by the inductor resistance. The solar array simulator (SAS) has been used to simulate the photovoltaic array following Eq. (2). It consists of the adjustable current source and series connected diode string as can be seen in Fig. 6(a). This SAS can simulate the change in voltage–current characteristics according to the temperature and illumination level variations by adjusting the value of the current source and the number of series cells in a diode string. The DB is a blocking diode to prevent a reverse flowing current. The measured voltage–current characteristic for Iph = 2 A is shown in Fig. 6(b). This curve is plotted from the automated test
DB
Isa + 2.5
vsa RL
Iph Set
(0 ~ ∞)
Isa [A]
..
2.0
100
1.5
75
power
1.0
_
50
0.5
0
Automated Au tomated Test Equipment (a)
125
current
(b)
25
0
5
10
15
20
25
Vsa [V]
Fig. 6. Configuration of the solar array simulator.
30
35
40
45
0 50
Psa [w]
Rs
I.-S. Kim / Applied Energy 83 (2006) 1101–1115
1111
equipment (ATE) which is connected to the SAS by varying the resistance of the ATE. The maximum power is 70 W and the maximum power point occurs at 38 V, as can be seen in Fig. 6(b). As the maximum output voltage is limited only to 48 V, which is lower than the grid voltage, the inverter output is connected to the grid voltage through a step-up transformer in the experiment. To verify the performance of the proposed sliding mode controller in a grid connected photovoltaic system, an experimental configuration has been set up, as shown in Fig. 7. A digital signal processor (DSP), type TMS320C31, is used to implement the proposed controller including the MPPT algorithm. The sensor board collects analog signals for the control algorithm. The four-channel simultaneous sampling analog to digital (A/D) converters are used for the analog-sensor-data acquisition. The field programmable gate array (FPGA) is used for the PWM generation of each power switch. The PWM signals are applied to the gate driver of each switch. The four-channel digital-to-analog (D/A) converter is used to display the control variable via an oscilloscope. The control program is compiled in a PC and downloaded to the DSP via an emulator. The software is executed by interrupt routine that is called every 0.1 ms. Whenever the grid voltage crosses the zerocross point, the average solar-array voltage and average solar-array current are updated using the half-cycle data. The experimental waveforms of the proposed controller are shown in Figs. 8 and 9. The grid voltage, control input, inductor current, solar-array voltage and current waveforms are shown. The inductor current is exactly controlled to equal the reference current. It can be seen that the inductor current is in phase with the utility voltage and the unity power-factor transmission to the grid is achieved (see Fig. 9). The solar-array voltage and current waveforms coincide exactly with the simulation result. Fig. 10 shows the tracking performance comparison between the conventional PI + predictive controller and the proposed sliding mode controller when the inductance Vsa DSP TMS320C31 40MHz
A/D (4-Ch.)
Sensor Board
isa
es (t ) iL
Emulator S1
FPGA (PWM Gen.)
Gate Driver
S2 S3 S4
D/A PC
(4-Ch.)
Psa Pref
u
σ
Fig. 7. Experimental system configuration.
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Fig. 8. Experimental waveforms of the proposed system.
Fig. 9. Experimental waveforms of the current and voltage waveforms.
Ln changes from 2.5 to 5 mH. Due to the oscillating solar-array voltage, the PI + predictive controller has an oscillating current error and this error is larger when the inductance is changed from the nominal value. However, the sliding mode controller shows a tight regulation of the inductor current regardless of changes in Ln. Therefore it can be concluded that the robust tracking performance is achieved using the sliding mode controller under the parameter-variation environment. Using the average solar-array voltage and current, the MPPT tracking performance is shown in Fig. 11. The MPPT updates Pref periodically by the algorithm Eq. (23). If P sa avg
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Fig. 10. Tracking performance for a change in Ln. (a) PI + predictive controller. (b) Sliding mode controller.
Fig. 11. MPPT tracking performance.
is equal to or larger than Pref, the value of Pref increases by a certain amount. Otherwise, Pref is returned to the previous value. If Pref is exceeds the maximum solar-array power, there can be a distortion in the inductor current waveform. The maximum Pref is found at 64 W and the solar-array voltage oscillates from 37 to 42 V. This corresponds to the
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maximum-power point in Fig. 6(b). It can be seen that the proposed algorithm tracks the maximum power exactly in this figure. 8. Conclusions A sliding mode controller for a single-phase grid-connected photovoltaic system has been proposed. The system consists of a new MPPT controller and current controller. The current reference is directly generated from the MPPT controller. The sliding mode controller was used as a current controller for the tight regulation of the inductor current. The proposed system can avoid current overshoot and so contribute to the optimal design of power devices. The proposed system is simple and is predicted to have superior performances under the parameter variation environments. Appendix The power transferred to the grid is given as the product of grid voltage and inductor current and is expressed as follows: V EP I pk P E ¼ es ðtÞ iL ¼ V EP I pk sin2 xt ¼ ð1 cos 2xtÞ ðA:1Þ 2 where es ðtÞ ¼ V EP sin xt, iL ¼ I pk sin xt, x ¼ 2p . The average power of the grid is given as T follows: Z T =2 1 V EP I pk V EP I pk ð1 cos 2xtÞ dt ¼ . ðA:2Þ P E avg ¼ T =2 0 2 2 Assuming loss-less power transmission from the solar array to the grid, the following relation always holds P sa
avg
¼ PE
avg
¼
V EP I pk . 2
ðA:3Þ
Then, the peak inductor current value is given as follows: I pk ¼
2 P sa avg . V EP
ðA:4Þ
Therefore, the inductor current reference becomes I ref ¼
2 P sa avg sin xt. V EP
ðA:5Þ
The MPPT controls the P sa avg to follow the Pref by periodically updating the reference voltage, and then the above equation holds: I ref ¼
2 P ref sin xt. V EP
ðA:6Þ
References [1] Koutroulis E, Kalaitzakis K, et al. Development of a microcontroller-based, photovoltaic maximum power point tracking control system. IEEE Trans Power Electronics 2001;16(1):P46–54.
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[2] Valenciaga F, Puleston PF, Battiaiotto PE. Power control of a photovoltaic array in a hybrid electric generation system using sliding mode techniques. IEE Proc-Control Theory Appl 2001;148(6):P448–55. [3] Noguchi T, Togashi S, Nakamoto R. Short-current pulse-based maximum power point tracking method for multiple photovoltaic and converter module system. IEEE Trans Industrial Electronics 2002;49(1):P217–23. [4] Hohm DP, Ropp ME. Comparative study of maximum power point tracking algorithm using an experimental, programmable, maximum power point tracking test bed. PESC 2000:P1699–702. [5] A. Kotsopolos et al., A predictive control scheme for DC voltage and AC current in grid-connected photovoltaic inverters with minimum dc link capacitance, IECON’2001, P1994–P1969. [6] Biel D, Fossas Enric, et al. Application of sliding-mode control to the design of a buck-based sinusoidal generator. IEEE Trans Industrial Electronics 2001;48(3):P563–71. [7] Carpita M, Marchesoni M. Experimental study of a power conditioning system using sliding mode control. IEEE Trans Power Electronics 1996;11(5):P731–42. [8] Liang TJ, Kuo YC, Chen JF. Single-stage photovoltaic energy conversion system. IEE Proc-Elec Power Appl 2001;148(4):P339–44. [9] Kasa N, lida T, Iwamoto H. Maximum power point tracking with capacitor identifier for photovoltaic power system. IEE Proc Electr Power Appl 2000;147(6):P497–502.
APPLIED ENERGY www.elsevier.com/locate/apenergy
Sliding mode controller for the single-phase grid-connected photovoltaic system Il-Song Kim
*
LG Chem/Research Park, Mobile Energy R&D, 104-1, Moonji-Dong, Yuseong-Gu, Daejeon 305-380, Republic of Korea Received 1 August 2005; received in revised form 9 November 2005; accepted 12 November 2005 Available online 27 January 2006
Abstract A sliding mode controller for the single-phase grid-connected photovoltaic system has been proposed in this paper. Contrary to the conventional controller, the proposed system consists of maximum power point tracker (MPPT) controller and sliding mode current controller only. The proposed MPPT controller generates current reference directly from the solar array power information and the current controller uses the sliding mode technique for the tight regulation of current. The new MPPT controller does not require the measurement of the voltage derivative which can be a cause of divide-by-zero singularity problems. The sliding mode controller has been constructed based on a time-varying sliding surface to control the sinusoidal inductor current and solar array power simultaneously. The proposed system can avoid the current overshoot and make optimal design for the system components. The structures of a proposed system are simple, but they show the robust tracking property against modeling uncertainties and parameter variations. The mathematical modeling is developed and the experimental results verify the validity of the proposed controller. 2005 Elsevier Ltd. All rights reserved. Keywords: Photovoltaic power systems; Sliding mode controller; Maximum power point tracker
1. Introduction In recent years the need for renewable energy has become more pressing. Among them, the photovoltaic system (PV) such as solar cell is the most promising energy. The PV *
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0306-2619/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2005.11.004
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Nomenclature A Cn es(t) fs I iL Iph Ipk Iref isa isa avg Isat Ln Np Ns PE P E avg Pref Psa P sa avg q RL Rs Rsh S1–S4 T u(t) ueq un V VEP Vmp Vsa V sa avg a g1,g2 k h r Df1,Df2
deviation factor from the ideal p–n junction diode, dimensionless nominal input dc link capacitor in lF grid voltage in V inverter switching frequency in Hz solar cell output current in A inductor current in A light generated current in A peak value of the current in A current reference in A solar array current in A average solar array current in V cell reverse saturation current in A nominal output filter inductor in mH number of parallel modules, dimensionless number of series modules, dimensionless power transferred to the grid in W average power of the grid in W power reference in W solar array power in W average solar array power in W electronic charge: 1.6022 · 1019 C output filter inductor resistance in X series resistance in X shunt resistance in X. full-bridge inverter switch cell temperature in K control input, dimensionless equivalent input, dimensionless nonlinear control input, dimensionless solar cell output voltage in V peak grid voltage in V maximum power voltage in V solar array voltage in V average solar array voltage in V sliding mode controller gain, dimensionless boundary values of the uncertainties, dimensionless Boltzmann’s constant: 1.3807 · 1023 J/K grid voltage phase angle in sliding surface, dimensionless modeling uncertainties, dimensionless
energy is free, abundant and distributed through the earth. Among the PV energy applications, they can be divided into two categories: one is stand-alone system and the other is grid-connected system. Stand-alone system requires the battery bank to store the PV
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energy and suitable for low-power system. Grid-connected system does not require the battery bank and has become the primary PV application for high power applications. The main purpose of the grid-connected system is to transfer maximum solar array energy into grid with a unity power factor. The output power of PV cell is changed by environmental factors, such as illumination and temperature. Since the characteristic curve of a solar cell exhibits a nonlinear voltage– current characteristic, a controller named maximum power point tracker (MPPT) is required to match the solar cell power to the environmental changes. Many algorithms have been developed for tracking maximum power point of a solar cell [1–4]. Among them, the most commonly used methods are perturb and observe (P & O) and incremental conductance algorithm. The P & O method measures the derivative of power (Dp) and the derivative of voltage (Dm) to determine the movement of the operating point. If the sign of (Dp/Dm) is positive, the reference voltage is increased by some amount of value and vice versa. The other method, the incremental conductance method can track the maximum power point voltage more accurately than P & O, by comparing the incremental conductance and instantaneous conductance of a PV array. The configuration of a single-phase grid-connected photovoltaic system is shown in Fig. 1. It consists of solar array, input capacitor Cn, single-phase inverter, filter inductor Ln, and grid voltage es(t). The solar cells are connected in a series–parallel configuration to match the required solar voltage and power rating. The direct current (DC) link capacitor maintains the solar-array voltage at a certain level for the voltage source inverter. The single-phase inverter with filter inductor converts a DC input voltage into an AC sinusoidalvoltage by means of appropriate switch signals to make the output current in phase with the utility voltage and so obtain a power factor of unity. A typical controller configuration of the single-phase grid-connected photovoltaic system consists of a MPPT controller, voltage controller and current controller [5]. The MPPT controller detects the power slope from the solar-array voltage and current information, and generates the reference voltage. The voltage controller controls the solararray voltage to follow the reference voltage using the proportional-integral (PI) controller. The output of the voltage controller becomes the DC value of the reference current. The current controller controls the inductor current to follow the reference current using a hysteresis or predictive controller. The hysteresis controller has a fast response time, but it has an irregular switching frequency. The predictive controller has a constant switching frequency and good current control, but it requires exact information for the circuit parameters. As the current controller has a cascade configuration with the voltage controller,
isa Solar array
Cn
S1
+ vsa _
.
S3
RL
. S2
Ln
iL
Grid Vo ltage
~
es(t)
S4
Fig. 1. Typical configuration of single stage grid-connected PV system.
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the inductor current overshoots when the reference voltage is changed. This is caused by the integrator of the voltage controller. This current peak can stress the power device in the inverter and cause a failure in the switching devices. The additional drawbacks of the conventional system are the tedious tuning for the PI-gain selection and the requirement of exact knowledge for the circuit parameters. The new controller has been proposed to overcome the above problems. It consists of a MPPT controller and current controller. The new MPPT controller generates a reference power instead of a reference voltage. This reference power is used for the current reference directly by the power-balance relation. For the tight regulation of the inductor current, a sliding mode controller has been used for the controller current. The sliding mode controller has a robust control property under the presence of parameter variations and can achieve a tight regulation of the states for all operating points [6–8]. The mathematical modeling is evolved and the experimental result verifies the worthwhileness of the proposed controller. 2. Solar cell/array modeling The electrical equivalent-circuit of a solar cell is shown in Fig. 2(a). It is composed of a light-generated current source, diode, series resistance, and parallel resistance. The characteristic equation for the current and voltage of a solar cell is given as follows: n h q i o V þ IR s I ¼ I ph I sat exp ðV þ IRs Þ 1 ð1Þ AkT Rsh where I is the solar-cell output current (A), V is the solar cell output voltage (V), Iph is the light-generated current (A), Isat is the cell reverse saturation current (A), q is the electronic charge = 1.6022 · 1019 C, A is the dimensionless deviation factor from the ideal p–n junction diode, k is Boltzmann’s constant = 1.3807 · 1023 J/K4, T is the cell temperature (K), Rs is the series resistance (X), and Rsh is the shunt resistance (X). The equivalent circuit for solar cells arranged in Np-parallel and Ns-series is shown in Fig. 2(b) and the mathematical equation relating the array current to the array voltage becomes: q V sa I sa Rs N p V sa I sa Rs I sa ¼ N p I ph N p I sat exp þ þ 1 ð2Þ AkT N s Np Rsh N s Np where Np represents the number of parallel modules. Note that each module is composed of Ns cells connected in series; NpIph corresponds to the short-circuit current of the solar array. The detailed parameters for the simulation and experiment are shown in Table 1. Ns Rs Np
Np Rs
+ I Rsh
V
N p I ph
Ns
.. .. . .
_ a
. ..
I ph
Isa
+
b
.. .
Ns Rsh Np
Vsa
_
Fig. 2. Electrically equivalent solar cell/array circuit. (a) Single cell circuit. (b) Solar array circuit (Ns-series, Npparallel).
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Table 1 Parameters for simulation and experiment Solar array (simulator) Ns Np Iph T Vmp
60 1 2A 300 K 38 V
Inverter Cn Ln RL fs Output voltage
1000 lF 5 mH 0.5 X 10 kHz 25 Vpeak/60 Hz
Sliding mode controller gain a
3
The voltage–current curve of the solar array shows a highly non-linear characteristic around the maximum-power point. 3. System modeling The dynamic model of a single-phase photovoltaic system can be obtained from the configuration of Fig. 1: Vsa and iL are the solar-array voltage and inductor current, respectively, and es(t) is the grid voltage and is given as follows: es ðtÞ ¼ V EP sin xt
ð3Þ
where x is the angular frequency given by 2pf, where f = grid frequency (60 Hz). The circuit parameters Cn and Ln correspond to their nominal values which are known exactly. The switch status of the single-phase inverter can be represented by the input c, defined as follows: þ1 ! S 1 ; S 4 : on; S 2 ; S 3 : off c¼ ð4Þ 1 ! S 1 ; S 4 : off; S 2 ; S 3 : on When the switch input c = 1, the state equation can be written as 1 ðiL þ isa Þ þ Df1 Cn 1 i_L ¼ ðV sa RL iL es ðtÞÞ þ Df2 . Ln m_ sa ¼
ð5Þ
where Df1 and Df2 represent the modeling uncertainties caused by the noise and measurement errors, and are bounded by the known values g1 and g2, respectively. When the switch input c = 1, the state equation is given as 1 ðiL þ isa Þ þ Df1 Cn 1 i_L ¼ ðV sa RL iL es ðtÞÞ þ Df2 . Ln m_ sa ¼
ð6Þ
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Combining (5) and (6), the state-space averaged model of the single-phase grid-connected photovoltaic system shown in Fig. 1 can be derived as follows: 1 V_ sa ¼ ðiL uðtÞ þ isa Þ þ Df1 Cn _iL ¼ 1 ðV sa uðtÞ RL iL es ðtÞÞ þ Df2 Ln
ð7Þ
where u(t) is defined as the average value of c for the switching period. The state equation given in (7) is a non-linear system and can be expressed in the general form as x_ ¼ f ðx; tÞ þ gðxÞ uðtÞ þ Dd, where the vectors x, f(x, t), g(x) and Dd are given as follows: " isa # " i # L msa Df1 Cn Cn x¼ ; f ðx; tÞ ¼ RL iL es ðtÞ ; gðxÞ ¼ msa ; Dd ¼ . ð8Þ iL Df2 Ln L n
The linear controller, such as PI, requires exact information on the circuit parameters for the desired performance when controlling the non-linear system. However, in a grid connected photovoltaic inverter system, the electrolytic capacitor is used for the input capacitor and it is known that its actual capacitance value has a 50% tolerance from its nominal value and deteriorates year-by-year [9]. The capacitance value determines the amount of ripple in the solar-array voltage. The higher capacitance shows the lower voltage ripple. The inductance also has a variation from its nominal value. All of these uncertainties make it difficult to use the linear state-controller for the single-phase grid-connected photovoltaic system. Linear controllers are known not to perform well under the presence of disturbances and unknown parameter variations [6,7]. 4. Background on the sliding mode control Sliding mode control is a kind of non-linear control which is robust in the presence of parameter uncertainties and disturbances. It is able to constrain the system status to follow trajectories which lie on a suitable surface in the sliding surface. The equilibrium state is constructed so that the system restricted to the manifold has a desired behavior. Consider a non-linear system of the form, x_ ¼ f ðx; tÞ þ gðxÞ uðtÞ þ Dd
ð9Þ
The main steps for sliding mode controller design can be summarized, by using an equivalent control concept, as follows [6]: The first step is the selection of a switching surface r(x, t) = 0 (where x is the system’s state vector) that provides the desired asymptotic behavior in steady state. Obtaining the equivalent control ueq by applying the invariance condition _ tÞ ¼ 0 with uðtÞ ¼ ueq rðx; tÞ ¼ 0 and rðx;
ð10Þ
The existence of the equivalent control ueq assures the feasibility of a sliding motion over the switching surface r(x, t) = 0. On the other hand, beside describing the average dynamical behavior of the power stage over the switching surface, the equivalent control enables one to obtain the sliding domain, given by
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minfu ; uþ g < ueq < maxfu ; uþ g
1107
ð11Þ
+
where u and u are the control values for r(x, t) < 0 and r(x, t) > 0, respectively. The sliding domain is the state plane region where sliding motion is ensured. Finally, selecting a non-linear control input un to ensure that Lyapunov stability criteria, i.e., rr_ < 0. 5. Design of the sliding mode controller The design of the sliding mode controller starts from the design of the sliding surface. Usually, the sliding surface is constructed by the linear combination of state variable errors that are defined as the differences between the state variables and their references. Therefore, in this case, the sliding surface can be designed with errors of the solar-array voltage and inductor current in a single-phase grid-connected photovoltaic system. The reference solar-array voltage is a DC voltage which is generated from the MPPT, but the solar-array voltage is oscillating due to the sinusoidal inductor current, which results in an undesirable sliding mode performance. For this reason, no attempt has been made to apply the sliding mode control for a grid-connected photovoltaic system. The main purpose of a grid-connected photovoltaic system is to transfer the maximum solar-array power into the grid with a power factor of unity. Therefore, the sliding surface should be designed to control the inductor current and solar-array power simultaneously. This requirement can be achieved by selecting a sliding surface only using the errors of the inductor current. If the reference inductor current is expressed as a function of the solararray power, then the sliding surface can control both the inductor current and the solararray power simultaneously. The mathematical expression for the peak reference current is Eq. A.6 of the Appendix. The proposed time-varying sliding surface is defined by: rðx; tÞ ¼ iL I ref ¼ iL
2 P ref sin xt V EP
ð12Þ
where Pref is the reference solar-array power, which is given by the MPPT controller and VEP is the peak grid-voltage. The next step is to design a control input which satisfies the sliding mode existence law. The control input is chosen to have the structure: uðtÞ ¼ ueq ðtÞ þ un ðtÞ
ð13Þ
where ueq(t) is an equivalent control-input that determines the system’s behavior on the sliding surface and un(t) is a non-linear switching input, which drives the state to the sliding surface and maintains the state on the sliding surface in the presence of the parameter variations and disturbances. The equivalent control-input is obtained from the invariance condition and is given by the following condition _ tÞ ¼ 0 ) uðtÞ ¼ ueq ðtÞ. rðx; tÞ ¼ 0 and rðx;
ð14Þ
Rewriting the above equation gives _ tÞ rðx;
or or ðf ðt; xÞ þ gðxÞ ueq ðtÞÞ þ ¼ 0. ox ot
ð15Þ
The necessary condition for the existence of a sliding motion of r(x, t) is represented by the transversality condition and expressed as follows:
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Lg r
or gðxÞ 6¼ 0 oxT
ð16Þ
where Lgr denotes the directional derivative of the scalar function r with respect to the vector field g(x). If the transversality condition is satisfied, then substituting (12) into (15) gives a result as follows: _ tÞ ¼ rðx;
1 2P ref x cos xt ðV sa ueq ðtÞ RL iL es ðtÞÞ ¼ 0. Ln V EP
ð17Þ
Therefore, the equivalent control-input is given as ueq ðtÞ ¼
RL iL þ es ðtÞ þ 2P ref Ln x cos xt=V EP . V sa
ð18Þ
The nonlinear switching input un(t) can be chosen as follows: un ðtÞ ¼ a sgnðrÞ.
ð19Þ
If (18) and (19) are substituted into (13), the range of a ensuring rr_ < 0 can be determined as follows: 2P ref x cos xtÞ rr_ ¼ rði_L V EP V sa es ðtÞ 2P ref ðueq ðtÞ þ un ðtÞÞ þ Df2 x cos xt ¼r Ln Ln V EP V sa ¼r a sgn ðrÞ þ Df2 < 0. ð20Þ Ln From this result, the range of switching gain is given as a>
Ln Ln jDf2 j ¼ g. V sa V sa 2
ð21Þ
From (18), (19) and (21), the control input u(t) = ueq(t) + un(t) is given as follows: uðtÞ ¼
RL iL þ es ðtÞ þ 2P ref Ln x cos xt=V EP g2 Ln sgnðrÞ . V sa
ð22Þ
Then, the control input is compared with the pulse-width modulation (PWM) ramp voltage and generates the appropriate switching pattern of the inverter. 6. MPPT controller design The MPPT controller takes Vsa and isa as inputs to detect the power slope and generates Pref to track the maximum-power point. Due to the oscillating solar-array voltage and current, the average values are used for calculation of the solar-array power P sa avg . The average values are obtained using half-cycle data of the utility-grid frequency. A simple maximum-power point updating algorithm is given as follows: P ref > P sa avg ) P ref ¼ Hold previous value ð23Þ P ref 6 P sa avg ) P ref ¼ P ref þ D where D means a shift step from the previous value. The value of Pref is reset periodically to compensate for any environmental changes of the solar array. The main advantage of
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Isa a
b
Pref2 B
Pref1 A
Vsa
Fig. 3. Operating point movement for the power reference change.
Current controller
Pref
MPPT
2 VEP
Grid voltage
PWM S1,S4 Single-
I ref Sliding u (t )
γ
phase inverter S2,S3 with L
Mode Controller
iL
es (t )
vsa _ avg Average
es (t )
isa _ avg Calculation
.
. Isa
vsa
sin(ω t )
θ
PLL
Fig. 4. Overall controller configuration of the proposed system.
this controller is that it does not require the measurement of the voltage derivative which can be a cause of divide-by-zero singularity problems. The operating point movement of the solar array is shown in Fig. 3 when the reference power has been changed from Pref1 to Pref2. For a reference power Pref1, there can be two intersection points (A, a) between the solar-array characteristic curve and the reference power. Point ‘A’ is stable and will be the desired operation point. On the other hand, point ‘a’ would be clearly unstable if the system operated permanently in this mode of operation. When the reference power changed to Pref2, the operating point moved from point ‘A’ to point ‘B’ in a similar manner. Fig. 4 shows the proposed controller configuration. It consists of a MPPT controller, sliding mode controller, and PWM generator. The MPPT controller tracks the maximum power point using average values of the solar-array voltage and current. The sliding mode controller controls the inductor current to follow the reference current by means of the sliding surface. The PWM generator generates the switching pattern according to the control input. 7. Simulation and experiment results Fig. 5 shows the simulation result of the proposed controller using the parameters shown in Table 1. The peak grid voltage is 25 V and the frequency is 60 Hz. For a reference power of Pref = 50 W, the peak reference current is obtained as 4 A from Eq. (A.6). In
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iL
iref
u
ueq
0 -1 46V 44V
vsa
42V 1.6A 1.2A
isa
0.8A 0.06
0.07
0.08
0.09 [sec]
Fig. 5. Simulation waveform of the voltage, control input and current.
this figure, the inductor current tracks the reference current with switching ripple. The control input is composed of the equivalent input and the switching input. The switching input shows a chattering value over the equivalent input. As can be seen in this figure, the solararray voltage and current oscillate at the frequency of 120 Hz which is twice the grid frequency. The average solar-array voltage is 44.2 V and average solar-array current is 1.2 A. Therefore, it can be seen that the solar-array power is exactly controlled to the given reference power of Pref = 50 W considering the power loss caused by the inductor resistance. The solar array simulator (SAS) has been used to simulate the photovoltaic array following Eq. (2). It consists of the adjustable current source and series connected diode string as can be seen in Fig. 6(a). This SAS can simulate the change in voltage–current characteristics according to the temperature and illumination level variations by adjusting the value of the current source and the number of series cells in a diode string. The DB is a blocking diode to prevent a reverse flowing current. The measured voltage–current characteristic for Iph = 2 A is shown in Fig. 6(b). This curve is plotted from the automated test
DB
Isa + 2.5
vsa RL
Iph Set
(0 ~ ∞)
Isa [A]
..
2.0
100
1.5
75
power
1.0
_
50
0.5
0
Automated Au tomated Test Equipment (a)
125
current
(b)
25
0
5
10
15
20
25
Vsa [V]
Fig. 6. Configuration of the solar array simulator.
30
35
40
45
0 50
Psa [w]
Rs
I.-S. Kim / Applied Energy 83 (2006) 1101–1115
1111
equipment (ATE) which is connected to the SAS by varying the resistance of the ATE. The maximum power is 70 W and the maximum power point occurs at 38 V, as can be seen in Fig. 6(b). As the maximum output voltage is limited only to 48 V, which is lower than the grid voltage, the inverter output is connected to the grid voltage through a step-up transformer in the experiment. To verify the performance of the proposed sliding mode controller in a grid connected photovoltaic system, an experimental configuration has been set up, as shown in Fig. 7. A digital signal processor (DSP), type TMS320C31, is used to implement the proposed controller including the MPPT algorithm. The sensor board collects analog signals for the control algorithm. The four-channel simultaneous sampling analog to digital (A/D) converters are used for the analog-sensor-data acquisition. The field programmable gate array (FPGA) is used for the PWM generation of each power switch. The PWM signals are applied to the gate driver of each switch. The four-channel digital-to-analog (D/A) converter is used to display the control variable via an oscilloscope. The control program is compiled in a PC and downloaded to the DSP via an emulator. The software is executed by interrupt routine that is called every 0.1 ms. Whenever the grid voltage crosses the zerocross point, the average solar-array voltage and average solar-array current are updated using the half-cycle data. The experimental waveforms of the proposed controller are shown in Figs. 8 and 9. The grid voltage, control input, inductor current, solar-array voltage and current waveforms are shown. The inductor current is exactly controlled to equal the reference current. It can be seen that the inductor current is in phase with the utility voltage and the unity power-factor transmission to the grid is achieved (see Fig. 9). The solar-array voltage and current waveforms coincide exactly with the simulation result. Fig. 10 shows the tracking performance comparison between the conventional PI + predictive controller and the proposed sliding mode controller when the inductance Vsa DSP TMS320C31 40MHz
A/D (4-Ch.)
Sensor Board
isa
es (t ) iL
Emulator S1
FPGA (PWM Gen.)
Gate Driver
S2 S3 S4
D/A PC
(4-Ch.)
Psa Pref
u
σ
Fig. 7. Experimental system configuration.
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Fig. 8. Experimental waveforms of the proposed system.
Fig. 9. Experimental waveforms of the current and voltage waveforms.
Ln changes from 2.5 to 5 mH. Due to the oscillating solar-array voltage, the PI + predictive controller has an oscillating current error and this error is larger when the inductance is changed from the nominal value. However, the sliding mode controller shows a tight regulation of the inductor current regardless of changes in Ln. Therefore it can be concluded that the robust tracking performance is achieved using the sliding mode controller under the parameter-variation environment. Using the average solar-array voltage and current, the MPPT tracking performance is shown in Fig. 11. The MPPT updates Pref periodically by the algorithm Eq. (23). If P sa avg
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Fig. 10. Tracking performance for a change in Ln. (a) PI + predictive controller. (b) Sliding mode controller.
Fig. 11. MPPT tracking performance.
is equal to or larger than Pref, the value of Pref increases by a certain amount. Otherwise, Pref is returned to the previous value. If Pref is exceeds the maximum solar-array power, there can be a distortion in the inductor current waveform. The maximum Pref is found at 64 W and the solar-array voltage oscillates from 37 to 42 V. This corresponds to the
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maximum-power point in Fig. 6(b). It can be seen that the proposed algorithm tracks the maximum power exactly in this figure. 8. Conclusions A sliding mode controller for a single-phase grid-connected photovoltaic system has been proposed. The system consists of a new MPPT controller and current controller. The current reference is directly generated from the MPPT controller. The sliding mode controller was used as a current controller for the tight regulation of the inductor current. The proposed system can avoid current overshoot and so contribute to the optimal design of power devices. The proposed system is simple and is predicted to have superior performances under the parameter variation environments. Appendix The power transferred to the grid is given as the product of grid voltage and inductor current and is expressed as follows: V EP I pk P E ¼ es ðtÞ iL ¼ V EP I pk sin2 xt ¼ ð1 cos 2xtÞ ðA:1Þ 2 where es ðtÞ ¼ V EP sin xt, iL ¼ I pk sin xt, x ¼ 2p . The average power of the grid is given as T follows: Z T =2 1 V EP I pk V EP I pk ð1 cos 2xtÞ dt ¼ . ðA:2Þ P E avg ¼ T =2 0 2 2 Assuming loss-less power transmission from the solar array to the grid, the following relation always holds P sa
avg
¼ PE
avg
¼
V EP I pk . 2
ðA:3Þ
Then, the peak inductor current value is given as follows: I pk ¼
2 P sa avg . V EP
ðA:4Þ
Therefore, the inductor current reference becomes I ref ¼
2 P sa avg sin xt. V EP
ðA:5Þ
The MPPT controls the P sa avg to follow the Pref by periodically updating the reference voltage, and then the above equation holds: I ref ¼
2 P ref sin xt. V EP
ðA:6Þ
References [1] Koutroulis E, Kalaitzakis K, et al. Development of a microcontroller-based, photovoltaic maximum power point tracking control system. IEEE Trans Power Electronics 2001;16(1):P46–54.
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[2] Valenciaga F, Puleston PF, Battiaiotto PE. Power control of a photovoltaic array in a hybrid electric generation system using sliding mode techniques. IEE Proc-Control Theory Appl 2001;148(6):P448–55. [3] Noguchi T, Togashi S, Nakamoto R. Short-current pulse-based maximum power point tracking method for multiple photovoltaic and converter module system. IEEE Trans Industrial Electronics 2002;49(1):P217–23. [4] Hohm DP, Ropp ME. Comparative study of maximum power point tracking algorithm using an experimental, programmable, maximum power point tracking test bed. PESC 2000:P1699–702. [5] A. Kotsopolos et al., A predictive control scheme for DC voltage and AC current in grid-connected photovoltaic inverters with minimum dc link capacitance, IECON’2001, P1994–P1969. [6] Biel D, Fossas Enric, et al. Application of sliding-mode control to the design of a buck-based sinusoidal generator. IEEE Trans Industrial Electronics 2001;48(3):P563–71. [7] Carpita M, Marchesoni M. Experimental study of a power conditioning system using sliding mode control. IEEE Trans Power Electronics 1996;11(5):P731–42. [8] Liang TJ, Kuo YC, Chen JF. Single-stage photovoltaic energy conversion system. IEE Proc-Elec Power Appl 2001;148(4):P339–44. [9] Kasa N, lida T, Iwamoto H. Maximum power point tracking with capacitor identifier for photovoltaic power system. IEE Proc Electr Power Appl 2000;147(6):P497–502.