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Analysis and design of sliding mode controller gains for boost power factor corrector
ISA Transactions 52 (2013) 638–643
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Analysis and design of sliding mode controller gains for boost power factor corrector Abdelhalim Kessal a,n, Lazhar Rahmani b a b
Mohammed el bachir el ibrahimi University, Bordj Bou Arréridj, Algeria Automatic Laboratory of Sétif (LAS), Ferhat Abbas University, Algeria
art ic l e i nf o
a b s t r a c t
Article history: Received 26 March 2013 Accepted 3 May 2013 Available online 2 June 2013 This paper was recommended for publication by Jeff Pieper
This paper presents a systematic procedure to compute the gains of sliding mode controller based on an optimization scheme. This controller is oriented to drive an AC–DC converter operating in continuous mode with power factor near unity, and in order to improve static and dynamic performances with large variations of reference voltage and load. This study shows the great influence of the controller gains on the global performances of the system. Hence, a methodology for choosing the gains is detailed. The sliding surface used in this study contains two state variables, input current and output voltage; the advantage of this surface is getting reactions against various disturbances—at the power source, the reference of the output, or the value of the load. The controller is experimentally confirmed for steadystate performance and transient response. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: PFC Power factor Sliding mode control Switching frequency
1. Introduction Usually, traditional PID controllers are used for the control of power converters [1–3]. Simple models of converters are generally obtained from signals averaging and linearization techniques; these models may then be used for control design [4,5]. On the other hand, PID controllers failed to satisfactorily perform constrained specifications under large parameter variations and load disturbances [2]. Another choice for controlling power converter is to use the sliding control techniques. Sliding mode control (SMC) of variable structure systems such as power converters is particularly interesting because of its natural robustness, its capability of system order reduction, and suitability for the nonlinearity aspect of power converters [5–7]. However, despite being a popular research subject, SMC is still rarely applied in practical AC–DC converters. It is mainly due to the fact that no systematic procedure is available for the design of SMC in practical applications [8]. For example, the influence of the controller gains on the closed loop system performances for a given application is not properly clarified, and most of the previous works are limited to the study of the influence of these parameters only on the existence and stability of sliding mode [9,10]. In other cases an empirical approach is adopted for selecting these gains of SMC; computer simulation and experiments were performed to study the effect of the various control gains on the response of the output voltage [10]. Therefore in this paper, analysis and design of
n
Corresponding author. Tel.: +213 66404 8090; fax: +213 35 674543. E-mail address: [email protected] (A. Kessal).
SMC for power factor corrector (PFC) are studied. After studying and analyzing different existing solutions for sliding mode control of PFC, a control mode that allows a direct control of the voltage of boost converter is proposed. The performances of the controller in terms of robustness and dynamic response will be improved. Most literature works are concerned with the study of hitting, existence and stability conditions of the SMC. The contribution of this paper goes beyond this direction by involving the study of the influence of control parameters on system performances. In this context, an optimization algorithm is developed in order to choose the controller parameters based on a predefined specification for a given real application. Accordingly, this paper is oriented in the application of the sliding modes for control of the bench of the power factor corrector (PFC). Principle of control by sliding modes is described briefly. Thereafter, the application of this principle for the control of the bench of PFC will be evoked. Based on the choice of the sliding surface, various modes of control will be studied. Then a mode of control based on a sliding surface utilizing all the variables of state are studied; this is in order to improve the performances of the closed loop. The important concepts associated to this type of control such as the convergence conditions, existence, or stability of the sliding mode, are considered carefully. This paper proposes a systematic analysis, design and digital implementation of the proposed controller, composed by linear controller in the DC voltage loop and sliding mode controller in the current loop. This controller is verified by detailed MATLAB/ Simulink based on simulations through the use of a continuous time plant model and a discrete time controller. Design is comprehensive in the sense that it accounts for sampling effects,
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.05.002
A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643
639
Fig. 1. Boost converter circuit.
computation delays, hardware filtering for antialiasing, and software filtering for measurement noise reduction, where necessary. Real-time implementation is done on an experimental prototype using the dSPACE DS1104 controller board. This controller is experimentally compared for steady-state performance and transient response over the entire range of input and load conditions for which the system is designed. The paper is organized as follows. In Sections 2 and 3, a description of converter, and a design and analysis of controllers are given. The experimental setup is detailed in Section 4. Section 5 presents the obtained results with discussions.
2. Mathematical model of boost converter The basic circuit diagram of the DC–DC converter with front end solid state input power factor conditioner used in the proposed scheme is shown in Fig. 1. The power circuit is that of an elementary step-up converter. When the boost switch Sw is turned on (u ¼1), the inductor current builds up, and energy is stored in the magnetic field of the inductor, whereas the boost diode D is reverse biased, and the capacitor supplies power to the load. This is the first mode operation. As soon as the boost switch is turned off (u ¼0), the power circuit changes mode, and the stored energy in the inductor, together with the energy coming from the input AC source, is pumped to the output circuitry (capacitor–load combination). This is mode 2 of the circuit. Then the state space model for the boost PFC in continuous current mode can be found by the circuit analysis. The output voltage and inductor current dynamics are governed by the variable structure real switched system. 8 < C dvo ¼ ð1−uÞiL −io dt ð1Þ : L didtL ¼ vin −ð1−uÞvo In order to obtain a sinusoidal input current in phase with the input voltage, the control unit should act in such a way that vin sees a resistive load equal to the ratio of vin and iL. This has been done by comparing the actual current passing through the inductor with a current reference, which is derived from vin and has an amplitude determined by the output voltage controller.
Fig. 2. Boost converter circuit governed by sliding mode controller.
inductor current via sliding mode that replace classical hysteresis current control (Fig. 2). This control of the output voltage of AC–DC converter meets the criteria of stability and existence of sliding mode. However, it is difficult to determine the gains of the voltage loop since sliding mode is a highly nonlinear method [2]. Furthermore, since SMC is only applied to current regulation, the voltage loop will be more sensitive to high frequencies phenomena and to uncertainties in the reference current. In order to improve the performances of the controller, a control mode based on a sliding surface which involves output voltage will be treated. Let (Vequ, Iequ) be the desired equilibrium point, where Vequ is the output voltage, and Iequ is the inductor current peak at equilibrium point. The input current peak IL can be expressed as [13] IL ¼
π ð1−αÞ Vo 2 R
S ¼ λ1 ðvo −V ref Þ þ λ2 ðiL −iref Þ
The control objectives of the PFC are twofold: regulate the output voltage vo to a reference voltage Vref and give the input current iL a rectified sine waveform in phase with the rectified voltage vin. The design of sliding mode controller for PFC starts with the choice of sliding surface. As it is shown in [11], it is clear that direct surface vo−Vref can tend to zero only if the current increases continuously. Usually, a cascade control structure is used, which leads to solve the control problem using two control loops [12]: an outer voltage loop which generates the reference current from voltage error and an inner current loop which controls the
ð3Þ
where λ1 and λ2∈R+. The control by current imposes the average power passed to the load with the ideal PFC pre-regulators [13]. P¼
V SM I ref ¼ vo io 2
ð4Þ
The reference current peak depends on the operating point; it can be taken as V SM I ref io vo ¼ ¼ 2V ref ð1−cos 2ωtÞ Rð1−cos 2ωtÞ
ð5Þ
2vo V ref io ¼ ð1−cos 2ωtÞ V SM Rð1−cos 2ωtÞ
ð6Þ
I ref ¼ 3. Design of sliding mode controller
ð2Þ
equilibrium point becomes ðV equ ; I equ ¼ π=2 V equ So, the ð1−αÞ=R , and the sliding surface shall be given according to the expression
Sliding surface coefficients (λ1, λ2) should be chosen such that the sliding mode exists at least around the desired equilibrium point, and the dynamics of the system will reach the surface and lead toward the equilibrium point. 3.1. Existence condition The existence condition of sliding mode implies that both S_ and _S will tend to zero (when t-∞), which means that the system dynamics remains on the sliding surface. The existence condition of the sliding mode is SS_ o0 (when S-0); achieving this
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A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643
inequality guarantees the existence of the sliding mode around the switching surface. The model of PFC can be written in a state space where the equilibrium point is the origin, whether x1 ¼ vo −V ref and x2 ¼ iL −iref where iref ¼ I ref jsin ωtj, so
the surface is given by vin =L − λ10 ðx1 þ V ref Þ=λ2 RC ueq ¼ 1− x1 þ V ref =L − λ10 ðx2 þ iref Þ=λ2 C
8 < C dx1 ¼ ð1−uÞðx2 þ iref Þ−ðx1 þV ref Þ R dt
Replacing the equivalent control (12) in the state space model (7) and from S¼ 0, the dynamic of x1 at the sliding regime is given as 2 vin iref − x1 λ10 =λ2 − x1 þ V ref =R dx1 ð13Þ ¼ dt C x1 þ V ref −L iref − λ10 x1 =λ2 λ10 =λ2
ð7Þ
: L dx2 ¼ vin −ð1−uÞðx1 þ V ref Þ dt
Replacing iref by its value from (6) in the expression of the commutation surface (3), the surface becomes
2vo V ref jsin ωtj S ¼ λ1 vo −V ref þ λ2 iL − V SM Rð1−cos 2ωt Þ
2vo V ref jsin ωtj V SM Rð1−cos 2ωtÞ 2λ2 V ref jsin ωtj vo −λ2 iL −λ1 V ref ¼ λ1 − V SM Rð1−cos 2ωtÞ ¼ λ1 ðvo −V ref Þ þ λ2 iL −λ2
ð8Þ
Introducing the Lyapunov function V ¼ 1=2x21 , its derivative is V_ ¼ x1 x_ 1 , so vin ðλ1 =λ2 Þ þ ðv0 þ V ref Þ=R 2 _ ð14Þ V ¼ −x1 Cðx1 þ V ref Þ−Lðiref − λ10 x1 =λ2 Þðλ10 =λ2 Þ The condition for V_ to be negative is 0 LV 2ref =Rvin ðλ10 =λ2 Þ−CV ref λ10 x1 λ1 4 0⇒x1 4 CðV ref þ x1 Þ−L iref − λ2 λ2 C þ Lðλ10 =λ2 Þ2 ð15Þ
This equation can be written in the coordinate system (x1, x2): 2λ2 V ref jsin ωtj 2λ2 V ref jsin ωtj x1 þ λ1 − V S ¼ λ1 − V SM Rð1−cos 2ωtÞ V SM Rð1−cos 2ωtÞ ref þλ2 x2 þ λ2 iref −λ1 V ref ¼ λ01 x1 þ λ2 x2
ð9Þ
where λ01 ¼ λ1 −ð2λ2 V ref jsin ωtj=V SM Rð1−cos 2ωtÞÞ The state of the switch (u∈{0,1}) imposes two signs of the derivative of the sliding surface; replacing u in the state system (7), the boundaries of the sliding area are deduced by 8 < −x1 −λ10 RC
: x1 ð−
λ01
RC
V ref RC
þ λ2 vLin 4 0
− λL2 Þ þ x2
λ01 C
ðS_ 4 0Þ
þ λ2 ðvLin −
V ref 0 vref L Þ−λ1 ð RC
−
iref C Þo0
ðS_ o 0Þ
ð10Þ
To ensure that the sliding mode exists at least around the equilibrium point (x1 ¼x2 ¼0), the following condition must be satisfied: λ′1 RCvin o λ2 V ref L
ð11Þ
3.2. Stability condition To ensure stability, the system dynamics during sliding mode is directed to the desired equilibrium point. The goal is to determine the dynamic of x1 and x2 when the sliding regime is achieved. Taking the state space model in (7) and the commutation surface in (8), from S_ ¼ 0, the equivalent average control that must be applied to the system in order that the system state slides along
ð12Þ
Based on the sliding region defined by (10) and the existence condition (11), the condition given in (15) is always satisfied along the sliding region of the commutation surface. According to the theorem of Lyapunov stability, the system is globally asymptotically stable. 3.3. Controller parameters and system performance Inequality (11) provides only general information concerning the existence of sliding mode. On the other hand, performances of the closed loop system are influenced by the choice of parameters of the controller, especially when the system presents large variations around nominal. To choose these parameters, the size of the sliding part in the switching surface must be taken into consideration. Really, sliding condition is only satisfied on a subpart of the surface and not on the entire surface as shown in (10). Therefore, the controller parameters must be carefully chosen to ensure that the system dynamic will intercept the commutation surface in the sliding part. For this, precautions against unwanted behaviors that cause an overshoot response must be taken (Fig. 3). The two points A(x1A,x2A) and B(x1B,x2B) are assumed which are respectively the crossing of system dynamic together with the commutation surface, and the boundary of the sliding parcel (Fig. 3). Supposing at t¼ 0, the output voltage is Vini (initial voltage value) and the input current is null, since the surface S will be negative, then u ¼1 such that the space vector becomes 8 < C dx1 ¼ −ðx1 þV ref Þ R dt ð16Þ : L dx2 ¼ vin dt
Fig. 3. Sliding mode with and without overshoot.
A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643
Taking into consideration the initial values, the resolution of this system gives Lðx2 þiref Þ
−
x1 ¼ V ini e
RC:vin
−V ref
ð17Þ
This solution represents the dynamic of the system before intercepting the surface at point A. From S¼0 and from (16) we deduce the coordinate of point A: Lðx2A þiref Þ λ10 − x2A ¼ V ini e RC:vin þ V ref ð18Þ λ2 From the equation of the commutation surface (9) and the limit of sliding zone defined by (10), the coordinate of point B is 0 ! λ1 V ref =RC − iref =C =λ2 − vin −V ref =L λ10 ð19Þ x2B ¼ 2 λ2 λ10 =λ2 þ λ10 =λ2 RC þ 1=C By deduction, the system intercepts the commutation surface in the right part if the controller parameters are selected in such a way that ‖OA‖ o ‖OB‖
ð20Þ
The theory of sliding mode assumes that the hysteresis band shall be null, so frequency approaches infinity. It is obvious that this assumption could not be made owing to the frequency limitation caused by the feature of circuit components and losses. Generally, a hysteresis window is added around the surface to maintain the operating frequency; from Fig. 4, the rise time ton and fall time toff can be expressed as 2Δ t on ¼ þ S
t of f
2Δ ¼ S−
1 f¼ t on þ t of f
ð21Þ
From (9) and (21), the hysteresis band expression is deduced, so
Δ¼
1 2f
terms of steady state error. So the choice of controller parameters must be taken into account to make the closed loop system less sensitive to an error in the current part of the surface. It is supposed that the measured reference current (IrefMEAS) can be expressed as the sum of the expression of reference current given by (4) and an error term (eIref), where e is defined as the error percentage, so that I ref Meas ¼ I ref −eI ref
ð23Þ
Replacing Irefmeas in the expression of sliding surface (3) λ2 eV 2ref λ2 eV ref ðvo −V ref Þ þ λ2 ðiL −I ref Þ þ S ¼ λ10 þ vin R vin R
ð24Þ
The average value of the term λ2 eV 2ref =vin R is constant, so λ2 eV ref ðvo −ðV ref þ Δvo ÞÞ þ λ2 ðiL −ðI ref þ ΔiL ÞÞ ð25Þ S ¼ λ10 þ vin R where Δvo, and ΔiL, are the voltage and the current steady state errors respectively. Error part in the sliding surface is given by 2
ðλ10 þ
λ2 eV ref λ2 eV ref ÞðΔvo Þ þ λ2 ðΔiL Þ ¼ − vin R vin R
ð26Þ
The control u will tend S to zero, so vo -V ref þ Δvo and iL -I ref þ ΔiL ; from the balance of input/output power ðV ref þ Δvo Þio ¼ vin ðI ref þ ΔiL ÞÞ
1 þ 1= λ10 V ref =RC 1=vin −1 þ λ2 =L vin −V ref 1= λ2 =L vin − λ10 V ref =RC
Consequently, from the expression of Δ(λ1,λ2), a limitation on the choice of the controller parameters is maintained. In fact, values of these parameters must ensure that the hysteresis window is greater than the perturbation generated by the converter, and in the same time, the band value should be limited to guarantee the robustness. In addition, the main goal is to regulate the output voltage, but the commutation surface (3) depends on the current errors, so an important optimization study of controller gains should be the analysis of the sensitivity of the controller in front of a measurement or a current reference estimation error. Actually, the current reference is unknown; it can be extracted from the load current (6). However, this latter can be measured or observed through an extended Luenberger observer. In both cases, an error can occur which affects the response of the system in
641
ð27Þ
! ð22Þ
Then, expression of Δvo will be vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 u 2 4eV 2ref Rvin @−λ1 V ref eV ref u λ1 V ref eV ref A Δvo ¼ − − þt þ − − 2 λ2 Rvin Rvin λ2 Rvin Rvin v2 R2 in
ð28Þ
4. Experimental setup In this study, the gains λ1 and λ2 of sliding controller are adjusted employing the off-line iterative genetic algorithm (GA). Thus, GA determines the controller gains which are the most compatible and provide optimum performance. The criteria are based on practical specifications. Thus the objective of the genetic algorithm is to determine the values of parameters that ensure, regardless of the operating point, that the system will intercept the sliding part of commutation surface while respecting the following conditions: 8 RCvin λ1 > > < λ2 o minð V ref L Þ Δmin o MaxðΔðλ1 ; λ2 ÞÞ o ΔMax > > : MaxðΔv Þ o Δv o oMax ⋯f or⋯e ¼ eMax
ð29Þ
The power circuit is designed to meet the following specifications:
Fig. 4. Hysteresis band of sliding regime.
output voltage V0 ¼160 V output voltage ripple o2%
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A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643
input voltage VSeff ¼ 115 V, RMS input current ripple ≤5% load resistance R¼ 210 Ω
5. Results
The experimental prototype was built around the dSPACE1104 controller board, which hosts the PowerPC 603e processor, to examine operating characteristic of the proposed method control for PFC. Although the PowerPC and its associated data acquisition circuitry can run up to 1 MHz, computation delay and communication overheads only allowed for the control algorithm to be executed at 20 kHz. The fourth-order Runge–Kutta solver was chosen to discretize the controller for real-time implementation. One Hall-effect CT's LEM (PR30) and isolation amplifier HAMEG (HZ64) were employed to detect the inductor current, input line voltage, and the output DC-bus voltage. Control circuits were built for offset correction and appropriate scaling. To prevent aliasing in the sampling process, second-order low-pass Butterworth filters were used to remove noise and switching frequency ripple in the sensed signals. Step load changes were affected by electronically connecting/disconnecting a parallel load.
A real-time experimental study was performed to capture the performance of the proposed method control for PFC. First, the steady-state performance is evaluated in terms of output voltage regulation, THD, and power factor. Next, the transient performance is evaluated for output voltage response on application of load step changes that are expected in practical applications of this circuit. All the data presented here were captured at 20 kHz using the control desk user interface for the dSPACE1104. 1. Steady-state performance: Fig. 5 shows the corresponding experimental results: the obtained power factor is 0.998% and THD is 2.95%; it is important to note that at nominal line and load condition, the method control has a THD value below 3% even with the limited bandwidth that is allowed by the digital implementation. Line current is very close to sine wave and in phase with the line voltage as shown in Fig. 6; the output voltage error is about less than 2 V. These results show that the proposed PFC control method achieves near unity power factor under steady state, and THD value is much better than the adoption of IEC1000-3-2 as the EN61000-3-2 standard.
[50V/div]
[2A/div]
[100ms/div] Fig. 5. Experimental results for steady state, grid voltage, input current and regulated output DC voltage.
Fig. 8. Experimental results for load disturbances, regulated output DC voltage and input current.
Fig. 6. Experimental measurements, THD, PF and phase.
220 Vref, vo 200
Increasing load 180
Decreasing load 160
140 1.4
1.6
1.8
2
2.2
2.4
Fig. 7. Simulation results for load changes.
2.6
2.8
A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643 240
643
Vref, vo
220
200
180
160
1
1.4
1.8
2.2
2.6
3
3.2
Fig. 9. Simulation results for output voltage reference changes.
[100V/div]
[2A/div] [100ms/div] Fig. 10. Experimental results for output voltage reference variations, regulated output DC voltage and input current.
Transient performance: To evaluate performances in transient 2. mode, step load changes are effected by disconnecting (or connecting) parallel load. The reference current amplitude is limited to 3.5 A in the control method designs. Figs. 7 and 8 show, respectively, simulation and experimental results of transient response for the proposed method control for PFC for a load resistor step, by 7 33% of the nominal value of the load (212 Ω). After a short transient (about 150 ms), the DC-bus voltage is maintained close to its reference value with a good approximation and stability. The line currents have nearly sinusoidal waveforms. The dynamic behavior of the proposed method under a step change of Vref is presented in Fig. 9 for simulation and Fig. 10 for experimental results. After a short transient (about 100 ms), the DC-bus voltage is maintained close to its new reference (from 180 V to 220 V and vice versa) with good approximation and stability. The line currents have nearly sinusoidal waveforms. 6. Conclusion In this paper, a practical design of sliding mode control for boost power factor controller is established, using a sliding surface which includes all state variables, output regulated voltage and input sinusoidal current. An optimization algorithm was
developed in order to calculate the optimal values of the sliding surface parameters based on a predefined specification. Results show excellent dynamic response of controller and robustness to load and voltage reference with large variations around nominal values. Experimental results show excellent dynamic response, good output regulation, low harmonic distortion, and high power factor can be achieved with the proposed single-stage converter and control scheme based on the proposed sliding mode controller. Finally, to verify the PFC function, the harmonic distortions are measured and compared to the international standards as EN 61000-3-2 and IEEE 519, the power factor is near unity and the THD is less than 3%. References [1]
Mingzhi, H, Jianping, X. Nonlinear PID in digital controlled buck converters. In: Proceedings of the applied power electronics conference APEC 2007, Anahein, CA, USA; 2007. pp. 1461–5. [2] Guo Liping, Hung John Y, Nelms RM. Comparative evaluation of sliding mode fuzzy controller and PID controller for a boost converter. Electric Power Systems Research 2011;81(1):99–106. [3] Arulselvi S, Uma G, ChidambaramM. Design of PID controller for boost converter with RHS zero. In: Proceeedings of the 4th International Power Electronics and Motion Control Conference (IPEMC 2004), China; 2004. [4] Yang Z, Sen PC. A novel technique to achieve unity power factor and fact transient response in ac-to-dc converters. IEEE Transactions on Power Electronics 2001;16(6):764–78. [5] Rashid M. Power electronics handbook. USA: Academic Press; 2001. [6] Utkin VI. Sliding modes and their applications on variable structure system. Moscow Russia: MIR publisher; 1978. [7] Utkin VI, Chang H. Sliding mode control in electromechanical systems. Mathematical Problems in Engineering 2002;8(4–5):451–73. [8] Tan SC, Lai YM, Cheung MKH, Tse CK. On the practical design of a sliding mode voltage controlled buck converters. IEEE Transactions on Power Electronics 2005;20(2):425–37. [9] Ben Saad K, Sahbani A, Benrejeb M. Design procedure and implementation of a robust fuzzy sliding mode controller for buck converters. International Review of Automatic Control 2008;1(3):303–10. [10] Hu ZB, Zhang B, Du GP, Zhong L , Deng WH. Fast transient three-level converters with sliding-mode control. In: Proceedings of the applied power electronics conference and exposition (APEC), vol. 3; 2005. pp. 1436–40. [11] He Zhaolan, Zhang Shuli, Wu Junfeng. Sliding mode control of switched linear systems based on common Lyapunov function. In: Proceedings of the 3rd international symposium on systems and control in aeronautics and astronautics (ISSCAA); 2010. pp. 300–4. [12] Navarro-López Eva M, Cortés Domingo, Castro Christian. Design of practical sliding-mode controllers with constant switching frequency for power converters. Electric Power Systems Research 2009;79(5):796–802. [13] Kessal Abdelhalim, Lazhar Rahmani, Gaubert Jean-Paul, Mohammed Mostefai. Analysis and design of an isolated single-phase power factor corrector with a fast regulation. Electric Power Systems Research 2011;81 (9):1825–31.
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Analysis and design of sliding mode controller gains for boost power factor corrector Abdelhalim Kessal a,n, Lazhar Rahmani b a b
Mohammed el bachir el ibrahimi University, Bordj Bou Arréridj, Algeria Automatic Laboratory of Sétif (LAS), Ferhat Abbas University, Algeria
art ic l e i nf o
a b s t r a c t
Article history: Received 26 March 2013 Accepted 3 May 2013 Available online 2 June 2013 This paper was recommended for publication by Jeff Pieper
This paper presents a systematic procedure to compute the gains of sliding mode controller based on an optimization scheme. This controller is oriented to drive an AC–DC converter operating in continuous mode with power factor near unity, and in order to improve static and dynamic performances with large variations of reference voltage and load. This study shows the great influence of the controller gains on the global performances of the system. Hence, a methodology for choosing the gains is detailed. The sliding surface used in this study contains two state variables, input current and output voltage; the advantage of this surface is getting reactions against various disturbances—at the power source, the reference of the output, or the value of the load. The controller is experimentally confirmed for steadystate performance and transient response. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: PFC Power factor Sliding mode control Switching frequency
1. Introduction Usually, traditional PID controllers are used for the control of power converters [1–3]. Simple models of converters are generally obtained from signals averaging and linearization techniques; these models may then be used for control design [4,5]. On the other hand, PID controllers failed to satisfactorily perform constrained specifications under large parameter variations and load disturbances [2]. Another choice for controlling power converter is to use the sliding control techniques. Sliding mode control (SMC) of variable structure systems such as power converters is particularly interesting because of its natural robustness, its capability of system order reduction, and suitability for the nonlinearity aspect of power converters [5–7]. However, despite being a popular research subject, SMC is still rarely applied in practical AC–DC converters. It is mainly due to the fact that no systematic procedure is available for the design of SMC in practical applications [8]. For example, the influence of the controller gains on the closed loop system performances for a given application is not properly clarified, and most of the previous works are limited to the study of the influence of these parameters only on the existence and stability of sliding mode [9,10]. In other cases an empirical approach is adopted for selecting these gains of SMC; computer simulation and experiments were performed to study the effect of the various control gains on the response of the output voltage [10]. Therefore in this paper, analysis and design of
n
Corresponding author. Tel.: +213 66404 8090; fax: +213 35 674543. E-mail address: [email protected] (A. Kessal).
SMC for power factor corrector (PFC) are studied. After studying and analyzing different existing solutions for sliding mode control of PFC, a control mode that allows a direct control of the voltage of boost converter is proposed. The performances of the controller in terms of robustness and dynamic response will be improved. Most literature works are concerned with the study of hitting, existence and stability conditions of the SMC. The contribution of this paper goes beyond this direction by involving the study of the influence of control parameters on system performances. In this context, an optimization algorithm is developed in order to choose the controller parameters based on a predefined specification for a given real application. Accordingly, this paper is oriented in the application of the sliding modes for control of the bench of the power factor corrector (PFC). Principle of control by sliding modes is described briefly. Thereafter, the application of this principle for the control of the bench of PFC will be evoked. Based on the choice of the sliding surface, various modes of control will be studied. Then a mode of control based on a sliding surface utilizing all the variables of state are studied; this is in order to improve the performances of the closed loop. The important concepts associated to this type of control such as the convergence conditions, existence, or stability of the sliding mode, are considered carefully. This paper proposes a systematic analysis, design and digital implementation of the proposed controller, composed by linear controller in the DC voltage loop and sliding mode controller in the current loop. This controller is verified by detailed MATLAB/ Simulink based on simulations through the use of a continuous time plant model and a discrete time controller. Design is comprehensive in the sense that it accounts for sampling effects,
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.05.002
A. Kessal, L. Rahmani / ISA Transactions 52 (2013) 638–643
639
Fig. 1. Boost converter circuit.
computation delays, hardware filtering for antialiasing, and software filtering for measurement noise reduction, where necessary. Real-time implementation is done on an experimental prototype using the dSPACE DS1104 controller board. This controller is experimentally compared for steady-state performance and transient response over the entire range of input and load conditions for which the system is designed. The paper is organized as follows. In Sections 2 and 3, a description of converter, and a design and analysis of controllers are given. The experimental setup is detailed in Section 4. Section 5 presents the obtained results with discussions.
2. Mathematical model of boost converter The basic circuit diagram of the DC–DC converter with front end solid state input power factor conditioner used in the proposed scheme is shown in Fig. 1. The power circuit is that of an elementary step-up converter. When the boost switch Sw is turned on (u ¼1), the inductor current builds up, and energy is stored in the magnetic field of the inductor, whereas the boost diode D is reverse biased, and the capacitor supplies power to the load. This is the first mode operation. As soon as the boost switch is turned off (u ¼0), the power circuit changes mode, and the stored energy in the inductor, together with the energy coming from the input AC source, is pumped to the output circuitry (capacitor–load combination). This is mode 2 of the circuit. Then the state space model for the boost PFC in continuous current mode can be found by the circuit analysis. The output voltage and inductor current dynamics are governed by the variable structure real switched system. 8 < C dvo ¼ ð1−uÞiL −io dt ð1Þ : L didtL ¼ vin −ð1−uÞvo In order to obtain a sinusoidal input current in phase with the input voltage, the control unit should act in such a way that vin sees a resistive load equal to the ratio of vin and iL. This has been done by comparing the actual current passing through the inductor with a current reference, which is derived from vin and has an amplitude determined by the output voltage controller.
Fig. 2. Boost converter circuit governed by sliding mode controller.
inductor current via sliding mode that replace classical hysteresis current control (Fig. 2). This control of the output voltage of AC–DC converter meets the criteria of stability and existence of sliding mode. However, it is difficult to determine the gains of the voltage loop since sliding mode is a highly nonlinear method [2]. Furthermore, since SMC is only applied to current regulation, the voltage loop will be more sensitive to high frequencies phenomena and to uncertainties in the reference current. In order to improve the performances of the controller, a control mode based on a sliding surface which involves output voltage will be treated. Let (Vequ, Iequ) be the desired equilibrium point, where Vequ is the output voltage, and Iequ is the inductor current peak at equilibrium point. The input current peak IL can be expressed as [13] IL ¼
π ð1−αÞ Vo 2 R
S ¼ λ1 ðvo −V ref Þ þ λ2 ðiL −iref Þ
The control objectives of the PFC are twofold: regulate the output voltage vo to a reference voltage Vref and give the input current iL a rectified sine waveform in phase with the rectified voltage vin. The design of sliding mode controller for PFC starts with the choice of sliding surface. As it is shown in [11], it is clear that direct surface vo−Vref can tend to zero only if the current increases continuously. Usually, a cascade control structure is used, which leads to solve the control problem using two control loops [12]: an outer voltage loop which generates the reference current from voltage error and an inner current loop which controls the
ð3Þ
where λ1 and λ2∈R+. The control by current imposes the average power passed to the load with the ideal PFC pre-regulators [13]. P¼
V SM I ref ¼ vo io 2
ð4Þ
The reference current peak depends on the operating point; it can be taken as V SM I ref io vo ¼ ¼ 2V ref ð1−cos 2ωtÞ Rð1−cos 2ωtÞ
ð5Þ
2vo V ref io ¼ ð1−cos 2ωtÞ V SM Rð1−cos 2ωtÞ
ð6Þ
I ref ¼ 3. Design of sliding mode controller
ð2Þ
equilibrium point becomes ðV equ ; I equ ¼ π=2 V equ So, the ð1−αÞ=R , and the sliding surface shall be given according to the expression
Sliding surface coefficients (λ1, λ2) should be chosen such that the sliding mode exists at least around the desired equilibrium point, and the dynamics of the system will reach the surface and lead toward the equilibrium point. 3.1. Existence condition The existence condition of sliding mode implies that both S_ and _S will tend to zero (when t-∞), which means that the system dynamics remains on the sliding surface. The existence condition of the sliding mode is SS_ o0 (when S-0); achieving this
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inequality guarantees the existence of the sliding mode around the switching surface. The model of PFC can be written in a state space where the equilibrium point is the origin, whether x1 ¼ vo −V ref and x2 ¼ iL −iref where iref ¼ I ref jsin ωtj, so
the surface is given by vin =L − λ10 ðx1 þ V ref Þ=λ2 RC ueq ¼ 1− x1 þ V ref =L − λ10 ðx2 þ iref Þ=λ2 C
8 < C dx1 ¼ ð1−uÞðx2 þ iref Þ−ðx1 þV ref Þ R dt
Replacing the equivalent control (12) in the state space model (7) and from S¼ 0, the dynamic of x1 at the sliding regime is given as 2 vin iref − x1 λ10 =λ2 − x1 þ V ref =R dx1 ð13Þ ¼ dt C x1 þ V ref −L iref − λ10 x1 =λ2 λ10 =λ2
ð7Þ
: L dx2 ¼ vin −ð1−uÞðx1 þ V ref Þ dt
Replacing iref by its value from (6) in the expression of the commutation surface (3), the surface becomes
2vo V ref jsin ωtj S ¼ λ1 vo −V ref þ λ2 iL − V SM Rð1−cos 2ωt Þ
2vo V ref jsin ωtj V SM Rð1−cos 2ωtÞ 2λ2 V ref jsin ωtj vo −λ2 iL −λ1 V ref ¼ λ1 − V SM Rð1−cos 2ωtÞ ¼ λ1 ðvo −V ref Þ þ λ2 iL −λ2
ð8Þ
Introducing the Lyapunov function V ¼ 1=2x21 , its derivative is V_ ¼ x1 x_ 1 , so vin ðλ1 =λ2 Þ þ ðv0 þ V ref Þ=R 2 _ ð14Þ V ¼ −x1 Cðx1 þ V ref Þ−Lðiref − λ10 x1 =λ2 Þðλ10 =λ2 Þ The condition for V_ to be negative is 0 LV 2ref =Rvin ðλ10 =λ2 Þ−CV ref λ10 x1 λ1 4 0⇒x1 4 CðV ref þ x1 Þ−L iref − λ2 λ2 C þ Lðλ10 =λ2 Þ2 ð15Þ
This equation can be written in the coordinate system (x1, x2): 2λ2 V ref jsin ωtj 2λ2 V ref jsin ωtj x1 þ λ1 − V S ¼ λ1 − V SM Rð1−cos 2ωtÞ V SM Rð1−cos 2ωtÞ ref þλ2 x2 þ λ2 iref −λ1 V ref ¼ λ01 x1 þ λ2 x2
ð9Þ
where λ01 ¼ λ1 −ð2λ2 V ref jsin ωtj=V SM Rð1−cos 2ωtÞÞ The state of the switch (u∈{0,1}) imposes two signs of the derivative of the sliding surface; replacing u in the state system (7), the boundaries of the sliding area are deduced by 8 < −x1 −λ10 RC
: x1 ð−
λ01
RC
V ref RC
þ λ2 vLin 4 0
− λL2 Þ þ x2
λ01 C
ðS_ 4 0Þ
þ λ2 ðvLin −
V ref 0 vref L Þ−λ1 ð RC
−
iref C Þo0
ðS_ o 0Þ
ð10Þ
To ensure that the sliding mode exists at least around the equilibrium point (x1 ¼x2 ¼0), the following condition must be satisfied: λ′1 RCvin o λ2 V ref L
ð11Þ
3.2. Stability condition To ensure stability, the system dynamics during sliding mode is directed to the desired equilibrium point. The goal is to determine the dynamic of x1 and x2 when the sliding regime is achieved. Taking the state space model in (7) and the commutation surface in (8), from S_ ¼ 0, the equivalent average control that must be applied to the system in order that the system state slides along
ð12Þ
Based on the sliding region defined by (10) and the existence condition (11), the condition given in (15) is always satisfied along the sliding region of the commutation surface. According to the theorem of Lyapunov stability, the system is globally asymptotically stable. 3.3. Controller parameters and system performance Inequality (11) provides only general information concerning the existence of sliding mode. On the other hand, performances of the closed loop system are influenced by the choice of parameters of the controller, especially when the system presents large variations around nominal. To choose these parameters, the size of the sliding part in the switching surface must be taken into consideration. Really, sliding condition is only satisfied on a subpart of the surface and not on the entire surface as shown in (10). Therefore, the controller parameters must be carefully chosen to ensure that the system dynamic will intercept the commutation surface in the sliding part. For this, precautions against unwanted behaviors that cause an overshoot response must be taken (Fig. 3). The two points A(x1A,x2A) and B(x1B,x2B) are assumed which are respectively the crossing of system dynamic together with the commutation surface, and the boundary of the sliding parcel (Fig. 3). Supposing at t¼ 0, the output voltage is Vini (initial voltage value) and the input current is null, since the surface S will be negative, then u ¼1 such that the space vector becomes 8 < C dx1 ¼ −ðx1 þV ref Þ R dt ð16Þ : L dx2 ¼ vin dt
Fig. 3. Sliding mode with and without overshoot.
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Taking into consideration the initial values, the resolution of this system gives Lðx2 þiref Þ
−
x1 ¼ V ini e
RC:vin
−V ref
ð17Þ
This solution represents the dynamic of the system before intercepting the surface at point A. From S¼0 and from (16) we deduce the coordinate of point A: Lðx2A þiref Þ λ10 − x2A ¼ V ini e RC:vin þ V ref ð18Þ λ2 From the equation of the commutation surface (9) and the limit of sliding zone defined by (10), the coordinate of point B is 0 ! λ1 V ref =RC − iref =C =λ2 − vin −V ref =L λ10 ð19Þ x2B ¼ 2 λ2 λ10 =λ2 þ λ10 =λ2 RC þ 1=C By deduction, the system intercepts the commutation surface in the right part if the controller parameters are selected in such a way that ‖OA‖ o ‖OB‖
ð20Þ
The theory of sliding mode assumes that the hysteresis band shall be null, so frequency approaches infinity. It is obvious that this assumption could not be made owing to the frequency limitation caused by the feature of circuit components and losses. Generally, a hysteresis window is added around the surface to maintain the operating frequency; from Fig. 4, the rise time ton and fall time toff can be expressed as 2Δ t on ¼ þ S
t of f
2Δ ¼ S−
1 f¼ t on þ t of f
ð21Þ
From (9) and (21), the hysteresis band expression is deduced, so
Δ¼
1 2f
terms of steady state error. So the choice of controller parameters must be taken into account to make the closed loop system less sensitive to an error in the current part of the surface. It is supposed that the measured reference current (IrefMEAS) can be expressed as the sum of the expression of reference current given by (4) and an error term (eIref), where e is defined as the error percentage, so that I ref Meas ¼ I ref −eI ref
ð23Þ
Replacing Irefmeas in the expression of sliding surface (3) λ2 eV 2ref λ2 eV ref ðvo −V ref Þ þ λ2 ðiL −I ref Þ þ S ¼ λ10 þ vin R vin R
ð24Þ
The average value of the term λ2 eV 2ref =vin R is constant, so λ2 eV ref ðvo −ðV ref þ Δvo ÞÞ þ λ2 ðiL −ðI ref þ ΔiL ÞÞ ð25Þ S ¼ λ10 þ vin R where Δvo, and ΔiL, are the voltage and the current steady state errors respectively. Error part in the sliding surface is given by 2
ðλ10 þ
λ2 eV ref λ2 eV ref ÞðΔvo Þ þ λ2 ðΔiL Þ ¼ − vin R vin R
ð26Þ
The control u will tend S to zero, so vo -V ref þ Δvo and iL -I ref þ ΔiL ; from the balance of input/output power ðV ref þ Δvo Þio ¼ vin ðI ref þ ΔiL ÞÞ
1 þ 1= λ10 V ref =RC 1=vin −1 þ λ2 =L vin −V ref 1= λ2 =L vin − λ10 V ref =RC
Consequently, from the expression of Δ(λ1,λ2), a limitation on the choice of the controller parameters is maintained. In fact, values of these parameters must ensure that the hysteresis window is greater than the perturbation generated by the converter, and in the same time, the band value should be limited to guarantee the robustness. In addition, the main goal is to regulate the output voltage, but the commutation surface (3) depends on the current errors, so an important optimization study of controller gains should be the analysis of the sensitivity of the controller in front of a measurement or a current reference estimation error. Actually, the current reference is unknown; it can be extracted from the load current (6). However, this latter can be measured or observed through an extended Luenberger observer. In both cases, an error can occur which affects the response of the system in
641
ð27Þ
! ð22Þ
Then, expression of Δvo will be vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 u 2 4eV 2ref Rvin @−λ1 V ref eV ref u λ1 V ref eV ref A Δvo ¼ − − þt þ − − 2 λ2 Rvin Rvin λ2 Rvin Rvin v2 R2 in
ð28Þ
4. Experimental setup In this study, the gains λ1 and λ2 of sliding controller are adjusted employing the off-line iterative genetic algorithm (GA). Thus, GA determines the controller gains which are the most compatible and provide optimum performance. The criteria are based on practical specifications. Thus the objective of the genetic algorithm is to determine the values of parameters that ensure, regardless of the operating point, that the system will intercept the sliding part of commutation surface while respecting the following conditions: 8 RCvin λ1 > > < λ2 o minð V ref L Þ Δmin o MaxðΔðλ1 ; λ2 ÞÞ o ΔMax > > : MaxðΔv Þ o Δv o oMax ⋯f or⋯e ¼ eMax
ð29Þ
The power circuit is designed to meet the following specifications:
Fig. 4. Hysteresis band of sliding regime.
output voltage V0 ¼160 V output voltage ripple o2%
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input voltage VSeff ¼ 115 V, RMS input current ripple ≤5% load resistance R¼ 210 Ω
5. Results
The experimental prototype was built around the dSPACE1104 controller board, which hosts the PowerPC 603e processor, to examine operating characteristic of the proposed method control for PFC. Although the PowerPC and its associated data acquisition circuitry can run up to 1 MHz, computation delay and communication overheads only allowed for the control algorithm to be executed at 20 kHz. The fourth-order Runge–Kutta solver was chosen to discretize the controller for real-time implementation. One Hall-effect CT's LEM (PR30) and isolation amplifier HAMEG (HZ64) were employed to detect the inductor current, input line voltage, and the output DC-bus voltage. Control circuits were built for offset correction and appropriate scaling. To prevent aliasing in the sampling process, second-order low-pass Butterworth filters were used to remove noise and switching frequency ripple in the sensed signals. Step load changes were affected by electronically connecting/disconnecting a parallel load.
A real-time experimental study was performed to capture the performance of the proposed method control for PFC. First, the steady-state performance is evaluated in terms of output voltage regulation, THD, and power factor. Next, the transient performance is evaluated for output voltage response on application of load step changes that are expected in practical applications of this circuit. All the data presented here were captured at 20 kHz using the control desk user interface for the dSPACE1104. 1. Steady-state performance: Fig. 5 shows the corresponding experimental results: the obtained power factor is 0.998% and THD is 2.95%; it is important to note that at nominal line and load condition, the method control has a THD value below 3% even with the limited bandwidth that is allowed by the digital implementation. Line current is very close to sine wave and in phase with the line voltage as shown in Fig. 6; the output voltage error is about less than 2 V. These results show that the proposed PFC control method achieves near unity power factor under steady state, and THD value is much better than the adoption of IEC1000-3-2 as the EN61000-3-2 standard.
[50V/div]
[2A/div]
[100ms/div] Fig. 5. Experimental results for steady state, grid voltage, input current and regulated output DC voltage.
Fig. 8. Experimental results for load disturbances, regulated output DC voltage and input current.
Fig. 6. Experimental measurements, THD, PF and phase.
220 Vref, vo 200
Increasing load 180
Decreasing load 160
140 1.4
1.6
1.8
2
2.2
2.4
Fig. 7. Simulation results for load changes.
2.6
2.8
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643
Vref, vo
220
200
180
160
1
1.4
1.8
2.2
2.6
3
3.2
Fig. 9. Simulation results for output voltage reference changes.
[100V/div]
[2A/div] [100ms/div] Fig. 10. Experimental results for output voltage reference variations, regulated output DC voltage and input current.
Transient performance: To evaluate performances in transient 2. mode, step load changes are effected by disconnecting (or connecting) parallel load. The reference current amplitude is limited to 3.5 A in the control method designs. Figs. 7 and 8 show, respectively, simulation and experimental results of transient response for the proposed method control for PFC for a load resistor step, by 7 33% of the nominal value of the load (212 Ω). After a short transient (about 150 ms), the DC-bus voltage is maintained close to its reference value with a good approximation and stability. The line currents have nearly sinusoidal waveforms. The dynamic behavior of the proposed method under a step change of Vref is presented in Fig. 9 for simulation and Fig. 10 for experimental results. After a short transient (about 100 ms), the DC-bus voltage is maintained close to its new reference (from 180 V to 220 V and vice versa) with good approximation and stability. The line currents have nearly sinusoidal waveforms. 6. Conclusion In this paper, a practical design of sliding mode control for boost power factor controller is established, using a sliding surface which includes all state variables, output regulated voltage and input sinusoidal current. An optimization algorithm was
developed in order to calculate the optimal values of the sliding surface parameters based on a predefined specification. Results show excellent dynamic response of controller and robustness to load and voltage reference with large variations around nominal values. Experimental results show excellent dynamic response, good output regulation, low harmonic distortion, and high power factor can be achieved with the proposed single-stage converter and control scheme based on the proposed sliding mode controller. Finally, to verify the PFC function, the harmonic distortions are measured and compared to the international standards as EN 61000-3-2 and IEEE 519, the power factor is near unity and the THD is less than 3%. References [1]
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