energies Article
Large-Signal Stabilization of Three-Phase VSR with Constant Power Load Bo Liu *
ID
, Hongqi Ben and Xiaobing Zhang
School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China;
[email protected] (H.B.);
[email protected] (X.Z.) * Correspondence:
[email protected]; Tel.: +86-156-5762-0553 Received: 18 May 2018; Accepted: 27 June 2018; Published: 1 July 2018
Abstract: As an important interface converter, the three-phase voltage-source rectifier (VSR) connects the grid to DC-input converters. The constant power load (CPL) characteristic of the converter-load often causes large-signal stability problems. In order to solve this problem, the stability boundary of the VSR with CPL is analyzed based on the mixed potential theory, and the stability condition under large-signal disturbance is obtained; thus, the jump range of the load power can be estimated. To improve the stability of the system, a voltage loop control scheme based on ADRC is proposed. A theoretical analysis based on the mixed potential theory shows the proposed scheme expands the power jump range of the VSR with CPL effectively, and improves its disturbance rejection performance. Finally, experimental results prove the effectiveness of the proposed control scheme. Keywords: voltage source rectifier (VSR); constant power load (CPL); large-signal stability; mixed potential theory; active disturbance rejection control (ADRC); power jump range
1. Introduction In order to achieve the goal of multi-functionality, low-cost and high reliability, modularization has become an important trend in power electronics [1–3]. In the process of modularizing power electronic devices, the situation of converters as loads becomes more and more common, and has attracted wide attention [4,5]. As important interface converters, three-phase voltage-source rectifiers (VSRs) connect the grid to DC-input converters, and are widely used in the fields of electric vehicle charging [6], DC micro-grid [7], more electric aircrafts (MEAs) [8], and photovoltaic generation [9]. Many academic papers have focused on stability issues related to three-phase VSRs (or voltage-source inverters, VSIs). In [10], bifurcation and large-signal stability of VSR are analyzed under grid voltage dips, and the stable operating region is identified. Mehrasa M. and Pouresmaeil E. use direct Lyapunov method to discuss a series of stability issues of VSRs applied in different power sectors such as modular multilevel converters (MMCs) [3,11], shunt active power filters [12], and distributed generation [13,14]. A dynamic model in d-q frame with six independent dynamical state variables is obtained, and global asymptotical stability is achieved by the use of direct Lyapunov method [11]. Furthermore, a multi-loop control strategy is proposed to provide stable operation under both MMC’s arm inductance and resistance parameters variations, and also loads changes [3]. However, the above studies do not focus on the effect of load characteristics on the stability of VSR. Figure 1 shows the schematic of a three-phase VSR cascaded with a DC-input converter. In this situation, for VSR, the load is no longer a resistance, but a converter with a constant power load (CPL) characteristic because of its closed-loop control. The CPL causes stability problems easily [15–17]; moreover, the problems are more significant when the DC-bus capacitance is a small-capacity film capacitor to improve the system life and reliability. Some scholars have discussed this problem and put forward some solutions based on small signal stability theory, such as the passive damping Energies 2018, 11, 1706; doi:10.3390/en11071706
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method [18,19], sliding mode control method [17], virtual impedance (active damping) method [20–22], and so on. However, when the cascaded system suffers from a large-signal disturbance such as load power jump or grid voltage drop, the small signal stability theory is no longer applicable, because the theory only studies the stability problems near the operating point. The problems of large-signal stability in converters with CPLs were analyzed in some literature in recent years [16,23,24]. In [23], the large-signal stability region of a DC/DC converter is estimated by a small signal loop. The norm inequalities are used to get the sufficient conditions for converter stability in order to estimate the stable region of the system. Furthermore, the effect of the small signal loop gain on the large-signal stability region is revealed. In [24], the Hamiltonian function method is used to discuss the influence of pulse load on the stability of the system, and some details of the transient response are shown. It indicates that pulse load has special effect on the stability of cascaded systems. For example, the system is in an unstable state when the load is connected, however it may attenuate to a stable state during the load disconnection. In general, it was called a metastable state. In [16], the influence of the negative impedance characteristic of CPLs on the stability of the converter is studied based on the mixed potential theory. In addition, a method to improve the large-signal stability in the cascade system is proposed with multistage LC filters. However, this method requires additional passive components, which increases the volume of the device. On the premise of no additional circuits, some meaningful work on control methods has been undertaken to improve the large-signal stability of converters with CPLs [25–28]. The sliding mode control method to improve the stability of VSR is proposed in [25]. In this method, the voltage loop is controlled by sliding mode, and the state observer of the real-time power of the system is designed to observe the power. Furthermore, the parameters of the state observer are derived from the Lyapunov stability theorem. However, for different systems, there is no uniform method to establish the Lyapunov function, so the realization process of the control algorithm is relatively complicated. Moreover, due to the boundary of the current and grid voltage, the range of attraction area for large-signal disturbance is limited, which limits its practicality. In [26], a “stabilizing agent” is implemented on each CPL to reduce the negative impedance characteristic to solve the DC micro-grid stability problem. Thanks to the improved fault tolerance of the solution, the method permits us to consider several fault scenarios, such as the electrical reconfiguration, or the failure of an agent. In [27], multiple linear controllers are designed at different operating points of a three-phase VSR and switched according to certain rules. When the load changes, the drop and overshoot of DC bus voltage are both significantly reduced. The stability of the control method is proved by the general Lyapunov functions. To improve the dynamic performance of three-phase VSR with large-signal disturbance, a direct voltage control method based on feedback linearization is proposed in [28]. The scheme breaks through the conventional double closed-loops control mode, and directly controls input variables by voltage error, so as to stabilize the bus voltage rapidly. However, this scheme relies on an accurate mathematical model of the circuits. Some abnormal operating conditions, such as the imbalance of the grid voltage, would lead to poor control performance. Active disturbance rejection control (ADRC) has been widely studied because of its excellent immunity against disturbance sources [29–31], and has been applied to motion control [30], power electronics, and so on [31]. However, the stability of converters with ADRC has not been adequately analyzed, especially for large-signal stability problems. In this paper, the authors are discussing the large-signal stability of three-phase VSRs with CPL. The main contributions are triple: (1) (2) (3)
Obtaining a large-signal model of three-phase VSR with CPL based on mixed potential theory, whose stable boundary is derived when load power jumps. Proposing a voltage control scheme to improve the large-signal stability based on ADRC, and whose control stability is proven. Deriving the stable boundary of VSR with CPL based on ADRC, which proves that the proposed control scheme expands the load power jump range effectively.
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(2) Proposing a voltage control scheme to improve the large-signal stability based on ADRC, and whose control stability is proven. (3) Deriving the stable boundary of VSR with CPL based on ADRC, which proves that3 ofthe Energies 2018, 11, 1706 14 proposed control scheme expands the load power jump range effectively. The rest of this paper is organized into four sections. Following the introduction, a large-signal The rest of this paper is organized into four sections. Following the introduction, a large-signal stability analysis based on mixed potential theory under PI control is provided in Section 2. The stability analysis based on mixed potential theory under PI control is provided in Section 2. proposed control scheme based on ADRC and the stability analysis of it is presented in Section 3. The proposed control scheme based on ADRC and the stability analysis of it is presented in Section 3. Moreover, the effectiveness of the control scheme is verified by experiments in Section 4. Finally, Moreover, the effectiveness of the control scheme is verified by experiments in Section 4. Finally, some conclusions are drawn in Section 5, and an Appendix A is given after that. some conclusions are drawn in Section 5, and an Appendix A is given after that. S1 ea eb ec
R
L
S3
S5
iL
S2
Load: DC-Input Converter (CPL)
ia ib ic
C
S4
S6
vdc
VSR Figure 1. 1. Schematic Schematicof ofthree-phase three-phase VSR VSR cascaded cascaded with with aa DC-input DC-input converter. converter. Figure
2. 2. Large-Signal Large-Signal Stability Stability Analysis Analysis Based Based on on Mixed Mixed Potential Potential Theory Theory under under PI PI Control Control 2.1. Introduction Introduction to to the the Mixed Mixed Potential Potential Theory Theory 2.1. The mixed theory is a is method to study stability nonlinear by constructing The mixedpotential potential theory a method to the study the of stability of circuits nonlinear circuits by Lyapunov functions, especially forespecially circuits with [32,33]. The mixed functions are constructing Lyapunov functions, for CPLs circuits with CPLs [32,33].potential The mixed potential functions related to energy, which composed current potential functions and voltage potential functions are functions related to are energy, whichofare composed of current potential functions and functions. The core content The of the theory is toofestablish mixed to satisfy certain voltage potential functions. core content the theory is topotential establishfunctions mixed potential functions stability and finally, to obtain rangestoofobtain the system parameters ensure parameters stability. to to satisfytheorems, certain stability theorems, andthe finally, the ranges of thetosystem The mixed potential function P consists of resistance, capacitance, and inductance in a nonlinear ensure stability. network. i1 , . . . , ir represents and vr+1 ,capacitance, . . . , vr+s are capacitor voltages.inP is The Assume mixed potential function Pinductor consistscurrents, of resistance, and inductance a defined as: nonlinear network. Assume i1, …, ir represents inductor currents, and vr+1, …, vr+s are capacitor Z r +s voltages. P is defined as: P= vµ diµ + iσ vσ (1)
∑
µ >r + s
∫∑
∑
σr=+rs+1
∑
P right is vthe iσ vσ (1) In Equation (1), the first = item on the potential of all the non-storage elements; µ dicurrent µ + µ >r + s σ = r +1 the second is the sum of energy in capacitors. If vµ can not be represented by iµ , the current potential can be by voltage In written Equation (1), the potential first itemand onEquation the right(2):is the current potential of all the non-storage Z elements; the second is the sum of Zenergy in capacitors. If vμ can not be represented by iμ, the vµ dipotential − Equation iµ dvµ (2) µ = vµ iµ and current potential can be written by voltage (2):
∫
∫
vµ= diµ state vµ iµequation − iµ dvµ can be described as: The relationship between P and the system
(2)
( distate equation The relationship between P and the system can be described as: ∂P L dtρ = ∂i ρ
(3) ∂P σ −Cdidv ρ =∂P dt ∂vσ = L dt ∂iρ where L and C are inductance and capacitance (3) respectively in the circuit. P dvσthe ∂validity Equation (3) gives the criterion to verify of the mixed potential functions. −C dt = ∂vσ Generally speaking, the unified form of P is:
where L and C are inductance and capacitance respectively in the circuit. P(i, v) = − A(i ) + B(v) + (i, γv − α) (4) Equation (3) gives the criterion to verify the validity of the mixed potential functions. Generally the unifiedthe form of P is: where A(i)speaking, and B(v) represents current potential and voltage potential of the non-storage elements in the circuit respectively, (i, γv − α) represents the energy of capacitors and parts of non-storage
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P(i , v) = − A(i ) + B( v) + (i , γ v − α )
(4)
Energies 2018, 11, 1706 4 of 14 where A(i) and B(v) represents the current potential and voltage potential of the non-storage elements in the circuit respectively, (i, γv − α) represents the energy of capacitors and parts of non-storage elements the circuits, where γmatrix is a constant matrix not necessarily with elements in the circuits,inwhere γ is a constant not necessarily associated with associated circuit topology circuit topology with elements ±1, 0, and α is constant vector. with elements ±1, 0, and α is constant vector.
2.2. Stability Problems of the Three-Phase VSR with CPL Because VSRs are often used as grid interface converters, converters, their their loads loads are are usually usually not not resistors, resistors, but power electronic converters. Power Power electronic electronic converters converters are are usually usually controlled controlled by by closed closed loops; loops; therefore, they are regarded as CPLs. Taking the DC/DC converter as an example of the load, when therefore, they are regarded as CPLs. Taking the DC/DC converter as an example of the load, when the the input current suddenly the input voltage will decrease to maintain theconstant. output input current suddenly rises,rises, the input voltage will decrease in orderintoorder maintain the output constant. The characteristic of maintaining constant power is called the CPL This characteristic. This The characteristic of maintaining constant power is called the CPL characteristic. characteristic is characteristic is bad for the will stability, which will be explained below. bad for the stability, which be explained below. Take the the voltage voltage source sourcein inseries serieswith withCPL CPLasasan anexample, example,asasshown shownininFigure Figure When the CPL 2. 2. When the CPL is is disturbed to cause the input current i CPL to rise, in order to maintain its output constant, the input disturbed to cause the input current iCPL to rise, in order to maintain its output constant, voltage vCPL of the the CPL CPL will will decrease. decrease. However, However,this this will will lead lead to to an an increase increase in in the the voltage on the CPL of resistance Rs again. This response, similar to positive feedback, takes the Rs to to make makethe thecurrent currentiCPL iCPLrise rise again. This response, similar to positive feedback, takes circuit away from the operating point, and causes it to lose stability. The larger the disturbance, the the circuit away from the operating point, and causes it to lose stability. The larger the disturbance, more significant stability problems caused by CPL willwill be. be. the more significant stability problems caused by CPL iCPL
Rs vCPL
vs
CPL
Figure CPL. Figure 2. 2. The The circuit circuit of of voltage voltage source source in in series series with with CPL.
2.3. Analysis of Power Jump Range of Three-Phase VSR with CPL 2.3. Analysis of Power Jump Range of Three-Phase VSR with CPL The average model equivalent circuit of three-phase VSR with CPL in dq coordinates system is The average model equivalent circuit of three-phase VSR with CPL in dq coordinates system is shown in Figure 3. Neglecting the power loss, Equation (5) can be obtained based on the power shown in Figure 3. Neglecting the power loss, Equation (5) can be obtained based on the power balance. balance. vd id + vq iq io = vd id + vq iq (5) io = vdc (5) vdc where vd,q are the control voltage in the dq coordinates respectively, id,q are the grid current in the where vd,q are the control voltage in the dq coordinates respectively, are thesource grid current dq dq coordinates respectively, io is the output current of the equivalentid,qcurrent on the in DCthe side, coordinates respectively, i o is the output current of the equivalent current source on the DC side, and and vdc is the output voltage of the VSR. vdc is According the outputto voltage of the VSR. theory, the current potential function of non-storage elements in the mixed potential According to the mixed potential theory, the current potential function of non-storage the circuit is: elements in the circuit is: Ri R R id R id R id R iq ∑ vµ diµ =id 0 d ed did i− id d + 0 3ωLiiqq di d + 0 eq di q 0 Rid diidd − 0 vd di d µ>r +s v di = (6) ed did − Rid did − vRd did + 3ω Liq di μ μ R di + 0 eq diq R i i 0R i 0 Ri 0 0 μ >r + s − 0 q Riq diq − 0 q vq diq − 0 q 3ωLid diq + 0 o vdc dio − 0 L vdc di L (6)
iq
iq
iq
io
iL
− Riq diq − vq diq − 3ω Lid diq + vdc dio − vdc diL where vd,q are the control voltage in the dq0 coordinates respectively, 0 0 0 R is the 0equivalent series resistance of the grid side, iL is the load current of the VSR. where vd,q are the control voltage in the dq coordinates respectively, R is the equivalent series The potential energy stored in the capacitor in the circuit is: resistance of the grid side, iL is the load current of the VSR. The potential energy stored in ther+ capacitor in the circuit is: s (7) ∑ iδ vδ = −io vdc + PCPL δ =r =1
where PCPL is the power of the CPL.
δ = r =1 iδ vδ δ = r =1
= −io vdc + PCPL
(7)
where PCPL is the power of the CPL. where PCPLEquations is the power the(7), CPL. From (6) of and the mixed potential function of the system can be obtained as: From Equations (6) and (7), the mixed potential function of the system can be obtained as: r+s id iq 1 1 Energies 2018, 5 of 14 P (11, v , i1706 vμ diμ + r + s iδ vδ = ed id + eq iq − 1 Rid22 − 1 Riq22 − id vd did − iq vq diq )= 0 0 P ( v , i ) = μ >r + s vμ diμ + δ =r =1 iδ vδ = ed id + eq iq − 2 Rid − 2 Riq − vd did − vq diq 0 0 2 2 (8) μ >r + s δ = r =1 id iq vdc vdc P (8) CPL − vdc iocan dvdcbe+ obtained dv id 3ω Liq di vdc P From Equations (6) and (7), the mixed+potential function ofLithe as: d − iq 3ω d diqsystem dc CPL + 0 3ω Liq did − 0 3ω Lid diq − 0 io dvdc + 0 vdc dvdc 0 0 0 0 vdc r +s R R id R iq 1 2 vδ = ed id + eqbetween iq − 21 Ri2d P P(v, i ) = to Equation − and Ri − v di − v di ∑ vµ diµ +(3),∑theiδ relationship q q According the state equations of the circuit can d d q 0 2 0 µ> r +s δ =r = 1 relationship between P and the state equations of the circuit can (8) According to Equation (3), the R R R R iq be used to verify the correctness of the function: id vdc vdc PCPL dv + 3ωLi di − 3ωLi di − i dv + q q o dc d d dc 0 vdc 0 0 0 be used to verify the correctness of the function: ∂P did According to Equation (3), the relationship P and the state equations of the circuit can be ∂P = ed − Ribetween d − vd + 3ω Liq = L did ∂ i d = ed − Rid − vd + 3ω Liq = L dt used to verify the correctness of the function: dt ∂id diq ∂ P Riq − vq − 3ω Lid = L didiqd (9) ∂P ∂=P e= e−q − Rid − vd + 3ωLiq = Ldt ∂i (9) d ∂iq =d eq − Ri q − vq − 3ω Lid = L dt di dtq ∂P ∂iq = e − Ri − v − 3ωLi = L (9) q q q d Pcpl dt dvdc ∂iq ∂P PP −C dv ∂P dvdcdc ∂= P −=i−i+ o + cplcpl==− C vdc = −C dtdt ∂vdc∂vdc = o−io +vdc ∂v vdc dt dc
id id
3L 3L
Rs Rs
3ωLiq 3ωLiq vd vd
ed ed iq iq
3L 3L
Rs Rs
C C
3ωLid 3ωLid
eq eq
io io
PCPL PCPL
vdc vdc
vq vq
Figure 3. The equivalent circuit of three-phase VSR with CPL in dq coordinates. Figure Figure 3. 3. The The equivalent equivalent circuit circuit of of three-phase three-phase VSR VSR with with CPL CPL in in dq dq coordinates. coordinates.
This indicates that Equation (9) satisfies the form Equation (3), which proves that the mixed indicatesPthat Equation (9) satisfies the the form form Equation Equation (3), (3), which which proves proves that that the the mixed mixed This function indicates Equation potential is correct. potential function PP is is correct. correct. potential The function proportional integral (PI) regulators are commonly used to control voltage and current of The proportional (PI) regulators are commonly used to control voltage current of The proportional integral regulators are commonly to control andand current of the the three-phase VSR, asintegral shown(PI) in Figure 4. The control loopused equations are: voltage the three-phase as shown in Figure 4. The control equations three-phase VSR,VSR, as shown in Figure 4. The control looploop equations are: are:
Iqref
+ Iqref − + −
iq iq
id id
− −
Vdcref Vdcref+
PI + − vdc PI − vdc
PI PI
− −
vq vq
− 3ωLi−d
3ωLid PI
PI + +idref idref
vd
− vd − + 3ωLiq + 3ωLiq
Figure 4. Control block diagram of the three-phase VSR with conventional PI control. Figure 4. Control block diagram of the three-phase three-phase VSR VSR with with conventional conventional PI PI control. control.
R v = − K i − i + K i − i ii d dre f d dt + 3ωLiq d ip dre f R vq = − Kip Iqre f − iq + Kii Iqre f − iq dt − 3ωLid R idre f = Kvp Vdcre f − vdc + Kvi Vdcre f − vdc dt
(10a) (10b) (10c)
where Vdcref is the output voltage reference of the VSR, Iqref is the current reference of the q axis, ω is the frequency of the grid, Kip and Kii are the proportion and integral coefficients of PI regulator of d
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axis current and q axis current respectively, Kvp , Kvi are the proportion and integral coefficients of PI regulator in voltage loop, respectively. According to Appendix A, the current potential function of the circuit is a second order matrix: " A( i ) =
−ed id + 12 Ri2d +
R id 0
vd did +
R id 0
3ωLiq did
−eq iq + 12 Ri2q +
0
#
0
R iq 0
vq diq −
R iq 0
3ωLid diq
(11)
and the voltage potential function is a first order matrix: B( v ) = −
Z v dc 0
io dvdc +
Z v dc PCPL 0
vdc
dvdc
(12)
In order to analyze the stability, the second order partial derivatives of A(i) and B(v) are solved as: ∂2 A( i ) = Aii_PI (i ) = ∂i2
Bvv_PI (v) =
∂2 B( v ) ∂v2
=
"
R + Kip 0
Kvi Kip id C Vdcre f − vdc ed id − PCPL
0 R + Kip
−
# (13)
Kvp Kip id v i −P + d d 2 CPL vdc vdc
(14)
According to the mixed potential stability theorem in the Appendix A, the sufficient condition for the stability of the three-phase VSR with CPL under large-signal perturbation is: PCPL_PI
0, the three-phase VSR based on ADRC is stable. 3.3. Large Sigal Stability Analysis of Three-Phase VSR with CPL Based on ADRC The power jump range is still analyzed by mixed potential theory in this section to discuss the stability of three-phase VSR with CPL based on ADRC. Assume z1 = v2dc and b0 = b, then . z2 = f y, y, u = −2PCPL /C
(32)
The control function of voltage loop is given by: idre f =
2 2 CK p (Vdcre u0 − z2 f − vdc ) + 2PCPL = b0 2ed
(33)
The control loop equations of a three-phase VSR based on ADRC are constructed by Equations (10a), (10b) and (33). The current potential function and the voltage potential function are still Equations (11) and (12) respectively. In the same way as in Section 2.3, the stability is analyzed by the theorem in the Appendix A. According to the control loop Equations (10a), (10b) and (33), the second order partial derivatives of A(i) and B(v) are solved as: " # R + Kip 0 Aii_ADRC (i ) = (34) 0 R + Kip Bvv_ADRC (v) = −
K p Kip Cid v i −P + d d 2 CPL ed vdc
(35)
According to the mixed potential stability theorem in the Appendix A, the sufficient condition for the stability of the three-phase VSR with CPL based on ADRC under large-signal perturbation is: PCPL_ADRC
0 (A1) and when |v| + |i | → ∞ , there is P(i, v) =
1 µ1 − µ2 1 P(i, v) + PiT L−1 Pi + PvT C −1 Pv → ∞ 2 2 2
(A2)
Then when t → ∞ , all variables of the circuit will approach the operating point, that is, no matter how the voltage and current change, the system will eventually become stable. References 1.
2. 3.
4.
5. 6.
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