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Robust Tracking Control of a Three-phase Charger under Unbalanced Grid Condition
10th IFAC Symposium on Control of Power and Energy Systems 10th IFAC Symposium on 4-6, Control Tokyo, Japan, September 2018of Power and Energy Systems 10th IFAC Symposium on 4-6, Control andonline EnergyatSystems Available www.sciencedirect.com Tokyo, Japan, September 2018of Power Tokyo, Japan, September 2018of Power and Energy Systems 10th IFAC Symposium on 4-6, Control Tokyo, Japan, September 4-6, 2018
ScienceDirect
IFAC PapersOnLine 51-28 (2018) 173–178
Robust Tracking Control of a Three-phase Charger Robust Tracking Control of a Three-phase Charger Robust Tracking Control ofGrid a Three-phase under Unbalanced Condition Charger under Unbalanced Grid Condition Robust Tracking Control of a Three-phase under Unbalanced Grid Condition Charger Chivon Choeung*, Meng Leang Kry** and Young Il Lee** under Unbalanced Grid Condition Chivon Choeung*, Meng Leang Kry** and Young Il Lee**
Chivon Choeung*, Meng Leang Kry** and Young Il Lee** * Engineering of Electricity, Polytechnic Institute of Cambodia, Phnom Penh, Cambodia (e-mail: ChivonNational Choeung*, Meng Leang Kry** and Young Il Lee** * Engineering of Electricity, National Polytechnic Institute of Cambodia, Phnom Penh, Cambodia (e-mail: [email protected]). * Engineering of Electricity, National Polytechnic Institute of Cambodia, Phnom Penh, Cambodia (e-mail: [email protected]). * Engineering of Electricity, National Polytechnic Institute of Cambodia, Phnom Penh, Cambodia (e-mail: [email protected]). [email protected]). ** Electrical and Information Engineering, Seoul National University of Science and Technology, Seoul, South Korea ** Electrical and Information Engineering, Seoul National University [email protected]). of Science and Technology, Seoul, South Korea (e-mail: [email protected], ** Electrical and Information Engineering, Seoul National University [email protected]). of Science and Technology, Seoul, South Korea (e-mail: [email protected], ** Electrical and Information Engineering, Seoul National University [email protected]). of Science and Technology, Seoul, South Korea (e-mail: [email protected], (e-mail: [email protected], [email protected]). Abstract: This paper presents a robust control technique for three-phase charger under unbalanced grid Abstract: presents a robust control three-phase chargercontrol under unbalanced grid conditions.This Thepaper control method consists of technique inner-loopfor robust grid-current and outer-loop Abstract: This paper presents a robust control technique for three-phase chargercontrol under unbalanced grid conditions. The control method consists of inner-loop robust grid-current and outer-loop proportional integral control fora constant current (CC) andfor constant voltagecharger (CV) control. A dual-current Abstract: paper presents robust control three-phase under unbalanced grid conditions.This The control method consists of technique inner-loop robust grid-current and outer-loop proportional integral control for constant current (CC) andis constant voltage (CV)control control. A dual-current control for the inner-loop positive and negative employed to eliminate the unbalanced current conditions. The control method consists of sequence inner-loop robust grid-current control and outer-loop proportional integral control for constant current (CC) and constant voltage (CV) control. A dual-current control by forthe the grid inner-loop positive andcurrent negative is employed to eliminate the unbalanced current caused socontrol that positive a constant andsequence voltage can be provided to (CV) the batteries. The inner-loop proportional integral for constant current (CC) and voltage control. A dual-current control forthe the inner-loop andcurrent negative is constant employed to eliminate the unbalanced current caused by grid so thatstate a constant andsequence voltage can be provided to the batteries. The inner-loop robust controllers utilize feedback with integral action in the dq-synchronous frame. A linear matrix control by forthe the grid inner-loop positive andcurrent negative is employed to eliminate the unbalanced current caused so thatstate a constant andsequence voltage can to the batteries. The inner-loop robust controllers utilize feedback with integral action inbe theprovided dq-synchronous frame. A linear matrix inequality-based optimization scheme is used to determine stabilizing gains of the controllers to caused controllers by the gridutilize so thatstate a constant current and voltage caninbe to the batteries. The inner-loop robust feedback with integral theprovided dq-synchronous frame. linear matrix inequality-based optimization scheme isstate used totheaction determine stabilizing gainsThe of uncertainties the Acontrollers to maximize the convergence rate to steady in presence of uncertainties. of the robust controllers optimization utilize state feedback with integral action in thestabilizing dq-synchronous frame. linear matrix inequality-based isstate used determine gainsThe of uncertainties the Acontrollers to maximize the convergence rate scheme to steady in totheof presence of uncertainties. of the system are the described as the potential variation range the inductance and resistance the L-filter. inequality-based optimization scheme isstate used tothedetermine stabilizing gainsThe ofinuncertainties the controllers to maximize convergence rate to steady in presence of uncertainties. of the system are described as the potential variation range of the inductance and resistance in the L-filter. maximize the convergence rate to steady state in the presence of uncertainties. The uncertainties of the system described as the potential variation rangeControl) ofgrid, the inductance resistance in the L-filter. © 2018,are IFAC (International Federation of Automatic Hosting byand Elsevier Ltd.(LMI), All rights reserved. Keywords: Battery charger, fast charging, unbalanced linear matrix inequality robust control. system are described as the potential variation range ofgrid, the inductance andinequality resistance(LMI), in the robust L-filter.control. Keywords: Battery charger, fast charging, unbalanced linear matrix Keywords: Battery charger, fast charging, unbalanced grid, linear matrix inequality (LMI), robust control. Keywords:1.Battery charger, fast charging, unbalanced grid, linear matrix inequality (LMI), robust control. INTRODUCTION In (Gallardo-L., J. 2012), a PI control was used in the AC1. INTRODUCTION In (Gallardo-L., a PIdc-link controlvoltage was used in thewhile ACWith the emerging 1. ofINTRODUCTION enormous amount of plug-in hybrid DC converter sideJ.to2012), maintain constant In (Gallardo-L., 2012), a PIdc-link controlvoltage was used in thewhile ACWith thevehicles emerging ofINTRODUCTION enormous amount of plug-in hybrid DC converter sideJ. toused maintain constant electric (PHEV) and electric vehicles (EV) and the deadbeat control is in the DC-DC converter to regulate 1. In (Gallardo-L., J.to2012), a PIdc-link controlvoltage was used in thewhile ACWith thevehicles emerging of enormous amount of plug-in hybrid DC converter side maintain constant electric (PHEV) and electric vehicles (EV) and the deadbeat control is used in the The DC-DC converter to provides regulate flourishing of the renewable energy, theofdevelopment of the current tomaintain battery. deadbeat control With thevehicles emerging of enormous amount plug-in hybrid DC charging converter sideistoused dc-link voltage constant while electric (PHEV) and electric vehicles (EV) and the deadbeat control in the DC-DC converter to regulate flourishing of has thebecome renewable energy, the development of the charging current to battery.which The deadbeat control provides power an interesting topic. Therefore, the fastcontrol transient respond setting time reaches the electricstoring vehicles (PHEV) and electric (EV) and the deadbeat is used in the The DC-DC converter to provides regulate flourishing of has the renewable energy,vehicles the development of really the charging current to battery. deadbeat control power storing become an interesting topic. Therefore, the really fast transient respond which setting time reaches the technology related to battery charging has drawn a lot of steady state in just a few sampling instants. However, it is flourishing of has thebecome renewable energy, the development of really the charging current to battery.which The deadbeat control provides power storing an interesting topic. Therefore, the fast transient respond setting time reaches the technology related to battery charging has drawn a lot of steady state in just a few sampling instants. However, it is attention from has governments, auto-makers, and researchers. In really sensitive to the parametric uncertainty of the system power storing become an interesting topic. Therefore, the really fast which setting reachesitthe technology related to batteryauto-makers, charging has drawn a lot of steady statetransient in just arespond few sampling instants.time However, is attention from governments, and researchers. In really sensitive to the parametric uncertainty of the system order to answer to the demands of fast and efficient and measurement noise, particularly for high sampling rate. technology related to battery charging has drawn a battery lot of steady state in just a few samplinguncertainty instants. However, it is attention from governments, auto-makers, and researchers. In really sensitive to the parametric of the system order to many answercontrol to thestrategies demandshave of fast and efficient battery and measurement noise, particularly for high sampling rate. charger, been proposed. attention from governments, auto-makers, and researchers. In really sensitive tonoise, the parametric uncertainty of the system order to many answer to thestrategies demands of fast and efficient battery andthis measurement for highofsampling rate. In paper, a robustparticularly tracking control a three-phase charger, control have been proposed. order to answer to the demands of fast and efficient battery and measurement noise, particularly for high sampling rate. charger, control strategies been proposed. In this under paper,unbalanced a robust tracking controlis of a three-phase Classicalmany Proportional Integralhave (PI) control have been charger grid conditions proposed without In this paper, a robust tracking control of a three-phase charger, many control strategies have been proposed. Classical Proportional Integral (PI)(Pinto, control under unbalanced grid is proposed without proposed for single-phase chargers J.G.have 2013 been and charger using converter an conditions interface between three-phase In thisDC-DC paper,unbalanced a robust astracking control a three-phase Classical Proportional Integral (PI)(Pinto, control been charger under is of proposed without proposed for single-phase chargers J.G.have 2013 and using DC-DC converter asgrid an conditions interface between three-phase Zhou, X. 2009). The charging scheme was designed based on AC-DC converter and batteries. The battery is charged with Classical for Proportional Integral (PI)(Pinto, control have been charger under unbalanced grid conditions is proposed without proposed single-phase chargers J.G. 2013 and using DC-DC converter as an interface between three-phase Zhou, X. 2009). The charging scheme was designed based on AC-DC converter and the batteries. The battery isrecommended charged with constant-current (CC) and constant-voltage (CV) charging constant current until voltage reaches the proposed for single-phase chargers J.G. 2013 and DC-DC converter as an interface between three-phase Zhou, X. 2009). The scheme (Pinto, was designed on using AC-DC and the batteries. The battery charged with constant-current (CC)charging and constant-voltage (CV) based charging constant converter current until voltage reaches theisrecommended mode provides shorter chargingwas time compare to that voltage, then the voltage is maintained constant Zhou, which X. 2009). The charging scheme designed based on maximum AC-DC converter and batteries. The battery isrecommended charged with constant-current (CC) and constant-voltage (CV) charging constant current until the voltage reaches the mode which provides shorter method. charging time compare tomain that maximum voltage, then the is maintained constant of fixed voltage charging However, until the current consumed by voltage battery falls tothea residual value. constant-current (CC) and constant-voltage (CV) the charging constant current until thethe voltage reaches recommended mode which provides shorter charging time compare to that maximum voltage, then voltage is maintained constant of fixed of voltage charging method. However, the main until the current consumed byofbattery falls to a residual value. drawback PI controller is gain tuningtime for compare both inner-loop The control method consists inner-loop robust grid-current mode which provides shorter charging that maximum voltage, then the is maintained constant of fixed voltage charging method. However, the tomain until the current consumed byofvoltage battery falls to a residual value. drawback of PI controller is gain tuning for both inner-loop The control method consists inner-loop robust grid-current and fixed outer-loop controller. Furthermore, since the topology of until control and outer-loop proportional integral control for of voltage charging method. However, the main the current consumed byofbattery falls to a residual value. drawback of PI controller is gain tuning for both inner-loop The control method consists inner-loop robust grid-current and outer-loop controller. Furthermore, since the topology of constant control and outer-loop proportional integral control for these proposed methods are based on single-phase converter, current (CC) and constant voltage (CV) control. A drawback of PI controller is gain tuning for both inner-loop The control method consists of inner-loop robust grid-current and outer-loop controller. Furthermore, since the topology of constant control and outer-loop proportional integral control for these proposed methods are based on single-phase converter, current (CC)forand constant voltage (CV) control. A the limit of charging current is smaller compare to those of dual-current control the inner-loop positive and negative and outer-loop controller. Furthermore, since the topology of constant control and outer-loop proportional integral control for these proposed methods are based on single-phase current (CC)forand constant voltage (CV) A the limit of chargers. charging current is smaller compare toconverter, those of sequence dual-current control theeliminate inner-loop positive andcontrol. negative three-phase is employed to the unbalanced current theselimit proposed methods are based on single-phase converter, constant current (CC)forand constant voltage (CV) control. A the of charging current is smaller compare to those of dual-current control the inner-loop positive and negative three-phase chargers. sequence employed to aeliminate unbalanced current by is thecontrol grid sofor that constant the current andand voltage can the limit of chargers. charging current is smaller compare to those of caused dual-current theeliminate inner-loop positive negative three-phase sequence is employed to the unbalanced current Model predictive control have been proposed in (Parvez, M. caused by the grid so that a The constant currentrobust and voltage can provided to the batteries. inner-loop controllers three-phase chargers. sequence employed to aeliminate the unbalanced current Model predictive control have been proposed inThis (Parvez, M. be caused by isthe grid so that constant current and voltage can 2014) for a bidirectional three-phase charger. method be provided to the batteries. The inner-loop robust controllers state feedback integral action the can dqModel predictive control have been proposed inThis (Parvez, M. utilize caused by the grid so thatwith a The constant current and in voltage be provided to the batteries. inner-loop robust controllers 2014) for a bidirectional three-phase charger. method could provide bidirectional power transfer withininstantaneous state feedback with integral action in the dqModel predictive control have been proposed (Parvez, M. utilize synchronous frame. A linear matrix inequality-based 2014) for a bidirectional three-phase charger. This method be provided to the batteries. The inner-loop robust controllers state feedback integral action in the dqcould provide bidirectional power transfer with instantaneous mode capability and fast dynamic response. so, utilize synchronous frame.is used Awith linear matrix inequality-based 2014) charging for a bidirectional three-phase charger. This Even method optimization to determine stabilizing gainsdqof could provide bidirectional power transfer with instantaneous utilize state scheme feedback with integral action in the synchronous frame. A linear matrix inequality-based mode charging capability and fast dynamic response. Even so, the charging fromand power grid to vehicle is based on optimization scheme is used to determine stabilizing gains of could providemethod bidirectional power transfer with instantaneous the controllers to maximize the convergence rate to steady mode charging capability fast dynamic response. Even so, synchronous frame. A linear matrix inequality-based scheme is used to stabilizing of the charging method from power grid to vehicle is based fixed voltage mode therefore the need longer time on to optimization the controllers to maximize thedetermine convergence rate togains steady mode charging capability and fastbatteries dynamic response. Even so, state in the presence of uncertainties. The uncertainties of the the charging method from power grid to vehicle is based on optimization scheme is used to stabilizing of controllers to maximize thedetermine convergence rate togains steady fixed voltage modeMoreover, therefore the batteries needgrid longer time to the be fully charged. in unbalanced conditions state in the presence of uncertainties. The uncertainties of the the charging method from power grid to vehicle is based are presence described asuncertainties. the potential variation range of the fixed voltage modeMoreover, therefore the batteries needgrid longer time on to system the controllers to maximize the convergence rate to steady state in the of The uncertainties of the be fully charged. in unbalanced conditions thisfully method cannot provide constant voltage the batteries are and described as theinpotential variation range of the fixed voltage modeMoreover, thereforea the needtolonger time to system resistance the L-filter. The conventional be charged. in batteries unbalanced conditions state in the ofasuncertainties. The uncertainties of the system are presence described theinpotential variation range of this method cannot providesequence a constant voltagegrid to the batteries inductance the due to the lack of negative compensation. inductance and resistance the L-filter. The conventional be fully charged. Moreover, in unbalanced conditions phase-locked loop (PLL) is in considered in this paper to obtain this method cannot providesequence a constant voltagegrid to the batteries inductance system are and described as the potential variation range of the resistance the L-filter. The conventional due to the lack of negative compensation. phase-locked loop angle. (PLL) is considered in this paper to obtain this to method cannot providesequence a constant voltage to the batteries inductance grid voltageand phase due the lack of negative compensation. resistance the L-filter. Thepaper conventional phase-locked loop (PLL) is in considered in this to obtain grid voltage phase angle. due to the lack of negative sequence compensation. phase-locked loop (PLL) is considered in this paper to obtain grid voltage phase angle. grid voltage phase angle.
Copyright 2018 IFAC 173Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Copyright 2018 responsibility IFAC 173Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 173 10.1016/j.ifacol.2018.11.697 Copyright © 2018 IFAC 173
IFAC CPES 2018 174 Tokyo, Japan, September 4-6, 2018
Chivon Choeung et al. / IFAC PapersOnLine 51-28 (2018) 173–178
ean 1 n 1 = eb = α e n 3 α 2 c
n e abc
α e a α 2 eb ,
α2 1
(4b)
1 ec
α
2
where α and α are phase-shifting operators and defined as:
2. SYSTEM DESCRIPTION
1 2
α =− + j
3 2
1 2
3 . 2
α =− − j
and
(5)
Substituting (5) in (4) yields:
p e abc
Fig. 1. Three-phase bidirectional charger with L-filter n e abc
A three-phase charger is shown in Fig. 1. The dynamic of the line current is expressed in the abc-axis as follows: dia (t ) L dt + Ria (t ) = ea − v a ,i di (t ) b + Rib (t ) = eb − vb,i L dt L dic (t ) + Ri (t ) = e − v c c c ,i dt
1 1 1 = − 3 2 − 1 2 1 1 1 = − 3 2 − 1 2
1 2
1 −
1 2
−
1 2
1 −
1 2
1 − 2 e a 0 1 − 1 ea 1 1 (6a) − eb + j − 1 0 1 eb 2 2 3 ec 1 − 1 0 ec 1 1 − 2 e a 1 1 − eb − j 2 2 3 ec 1
1/3
(1)
1/6 1/6
ea
All-pass Filter
eb
All-pass Filter
ec
All-pass Filter
where 2u a − ub − uc vo (t ) 6 − u a + 2ub − uc vb,i := vo (t ) . 6 − u a − ub + 2uc vc,i := vo (t ) 6 va ,i :=
1, S x = on; S x = off ; Here, − 1, S x = off ; S x = on;
−
1/3 1/6 1/6
u x ( x = a, b, c) .
1/3 1/6
dt
= Ac idq (t ) + Bc vo (t )u(t ) + d c (t ) .
1/6
(2)
Any unbalanced three-phase voltage can be expressed by the sum of positive, negative and zero sequence (Ma, K. 2015). However, the three-phase charger system has only three wires therefore the zero sequence is does not exist. Hence, the unbalanced current and voltage can be expressed as:
ea
All-pass Filter
eb
All-pass Filter
ec
All-pass Filter 1/3 1/6 1/6
p n e abc = e abc + e abc
(3a)
p n i abc = i abc + i abc
(3b)
The positive and negative sequence of assumed to be: eap 1 α p 1 2 p e abc = eb = α 1 e p 3 α α 2 c
j (eb − ea )
−
+ +
1
+
− −
2 3
−
1
+
2 3
j (ec − eb )
ea+
+ +
eb+ ec+
+
− −
− −
+ +
− −
j (ea − eb )
+ +
1
−
2 3
+
1
−
2 3
j (eb − ec )
ea− − −
+ +
eb− ec−
+
Fig. 3 Negative sequence extraction block diagram
the grid voltage are α 2 e a α eb ,
+
− −
Fig. 2 Positive sequence extraction block diagram
The dynamic in abc-axis (1) can be transformed to dq-axis as follows: didq (t )
0 1 − 1 ea − 1 0 1 e (6b) b 1 − 1 0 ec
An all-pass filter is used to obtain the imaginary part of (6) by shifting 90⁰ from the original phase. The same process is used to extract positive and negative sequence of grid current and voltage. The block diagrams of positive and negative sequence extraction are shown in Fig. 2 and Fig. 3, respectively.
(4a)
1 ec
174
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From equation (3), the current dynamics (2) can be rewritten as a combination of positive and negative sequence. The dynamic of both sequences is almost identical; however, the only difference is the sign of ω due to the inverse direction of the negative sequence vector rotation (Choeung, C. 2016). Then, we have p di dq (t )
p p = Acp i dq − d cp (t ) , (t ) + Bc vo (t )udq dt n (t ) di dq n n (t ) + Bc vo (t )udq = Acn i dq − d cn (t ) dt
The uncertainties of the system can be any kind of variation, but should lies within the range (11). The system uncertain range can be determined as: (13a) L0 / µ ≤ L ≤ µL0 (13b) R0 / µ ≤ R ≤ µR0 where R0 and L0 are nominal value of the filter resistance and inductance, respectively, and µ (>1) can be considered as a tuning parameter. In consideration of compensating the offset error despite the system’s uncertainty model, the control law based on (Lim, J.S. 2014) is employed for (10): p p p p g (k ) = g (k + 1) + ( x ref − x (k − 1)) (14a) p p p p p u K x L g ( k ) = ( k ) + ( k ) n n n n g (k ) = g (k + 1) + ( x ref − x (k − 1)) (14b) n n n n u( k ) = K x ( k ) + L g ( k ) where K pn and Lpn are state feedback and integrator gains, respectively. Because of the integrator in (14), the steadypn state error between the reference state x ref and the grid-
(7a) (7b)
where Acp
R − := L −ω
R
ω − , Acn := L R ω − L
1 − −ω , Bc := 2 L R 0 − L
0 , 1 − 2L
0 0 n p d cp := eq , d cn := eq L L ω is the angular frequency of the AC voltage source. i dq and
current x pn will be compensated provided that the closedloop system is stable.
udq are the grid current and control input in dq-frame,
respectively. The output voltage vo (t ) is governed by the following dynamic: dv (t ) (8) C o = icon (t ) − ibat (t ) dt where 3 T (9) icon (t ) = i dq (t )u(t ) 4 and icon. is the converter current and ibat . is an output current to the battery.
In order to eliminate the unbalanced current caused by negative sequence and provide a constant charging current and voltage to the batteries, the reference state should be given as: p x dref 0 p p x iqref . pn x ref := qref = n 0 x dref n x qref 0 p p where x qref is generated by the outer-loop controller = iqref
Model (7) can be transformed in the following discrete-time system with sampling time h: x p (k + 1) = A p x p (k ) + Bvo (k )u p (k ) − d p (k )
175
depending on its CC and CV control objectives.
(10a)
pn . where A pn = I 4×4 + Acpn , B = Bc h , d pn = d cpn h , x pn = i dq
The same process is used to find optimal gains for positive and negative sequence, so the procedure is unified. A systematic design method was proposed in (Boyd, S.C. 1994) to obtain stabilizing state feedback gain K and integral gain L of (14) using LMI. From relation (10) and (14), we get:
3. ROBUST CONTROLLER DESIGN
z (k + 1) = Aa z (k ) + Ba u(k ) + D(k )
n
n
n
n
n
x (k + 1) = A x (k ) + Bvo (k )u (k ) − d (k )
(10b)
∑
x (k ) z (k ) := g (k )
A Aa := − C d ( k ) B 1 0 ,C = , D (k ) := B a := . 0 1 0 2×2 x ref
In this section, the uncertainties model of the system, offsetfree control and robust optimal gain are discussed. Suppose that the value of L and R in each phase are equal but vary in certain ranges below: (11a) L1 ≤ L ≤ L2 (11b) R1 ≤ R ≤ R2 . Here, we denote the matrices (A, B) corresponding to the four possible combinations of the immoderate value of 1/L and 1/R as (Ai, Bi) (i = 1,2,3,4) and suppose that the matrices (A, B) belongs to the polytopic uncertain set Ψ below: 4 4 (12) Ψ = µ n ( Ai , Bi ) µ n = 1,µ n ≥ 0 . n=1 n =1
where
,
(15) 0 2×2 I 2×2
,
The control input u(k) can be rewritten as:
u(k ) = Fz (k )
F := [K
L] .
(16)
Assume that D(k) = 0 to determine stabilizing gain F, then the closed-loop system can be obtained as follows:
∑
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z (k + 1) = ( Aa + Ba F ) z (k ) .
Here, the outer-loop PI control for constant voltage charging is discussed. Let I r be the output of the outer-loop control, i.e.
(17)
The closed-loop dynamic (17) stable if there exists a positivedefinite matrix W such that T
W − ( Aa + Ba F ) W ( Aa + Ba F ) > 0 .
∫
I r (t ) = K p (voref − vo (t )) + K i (voref − vo (t ))dt
(18)
ref where vo is the constant voltage reference for battery
It is obvious that the condition (18) holds for some W 0 > 0 (W 0 < W )
W0 − ( Aa + Ba F )T W ( Aa + Ba F ) > 0 .
(19)
By applying Schur complement and uncertain set (12) to (19), we get (Boyd, S.C. 1994)
S0 Aai S 0 + Bai H
S 0T
T Aai
+H S
T
T Bai
> 0 , (i = 1,…,4), (20)
where H := FS 0 , S = W −1 , S 0 := S 0−1
A Aai := i − C
0 2×2 , I 2×2
(24)
B Bai := i , (i = 1, 2, 3, 4). 0 2×2
charge and vo (t ) is the output voltage. Suppose that the dynamics of the inner-loop control in previous section is considerably fast so that we can assume (25) icon (t ) ≈ I r (t ) for some current reference I r (t ) . From (8) and (25), we get dv (t ) C o = K p (voref − vo (t )) + K i (voref − vo (t ))dt − ibat (t ) (26) dt or d 2 vo(t) dv (t) (27) C + K p o + K i vo (t ) = K i voref . dt dt The K i and K p gain can be determined by considering the
∫
characteristic polynomial of (27) and can be given as
∆( s ) = s 2 + 2ζω r s + ω r2 or some appropriate value of ζ and
To summarize, closed-loop system (17) is asymptotically stable if there exist symmetric positive definite matrices S and S 0 and a matrix H such that (20) holds and the stabilizing gain is given as (21) F = HS 0−1 .
ω r2 ;
(28)
we get
K i = ω r2
(29)
K p = 2ζω r .
(30)
The control diagram of the constant voltage control of the three-phase charger is shown if Fig. 4. The battery voltage is fed-back to the out-loop controller, which produces a p p . reference current x qref = iqref
Assume that W 0 < αW or equivalent to (22) (0 < α < 1) . S < αS 0 It can be expected that a small α would give a fast convergence of z to the origin. Therefore, to obtain optimal gain F such that the convergence time is minimized, the following optimization problem should be solved. Minimize α subject to (20) and (22) (23)
5.2 Constant current (CC) charging mode In constant current charging stage, the battery pack is charged with fixed current until the voltage reaches the recommended maximum voltage, then switch to constant voltage charging stage. For the control of this constant current charging mode, an outer-loop PI is utilized to generate a reference signal p p for inner-loop robust control with the same x qref = iqref
S , S 0 > 0,
α > 0, H
This optimization scheme is a generalized eigenvalue problem (Boyd, S.C. 1994) which can be solve efficiently by MATLAB Toolbox.
concept as CV charging discussed in previous section. The control structure of the proposed CC charging control is validated as shown in Fig. 4.
4. DUAL-LOOP CHARGING CONTROL In order to perform the battery charging process, most of the battery manufacturers recommend two charging stage i.e. constant current (CC) mode followed by constant voltage (CV) mode. The battery is charged with constant current until the voltage reaches the recommended maximum voltage, then the voltage is maintained constant until the current consumed by battery falls to a residual value. The dual-loop control strategy will be adopted for these two charging states i.e. an p p depending outer-loop controller generates proper x qref = iqref on its CV or CC control objectives and the inner-loop pn controller (14) drives the state of x pn in (10) to follow x ref .
Fig. 4. Control structure of the proposed method
5.1 Constant voltage (CV) charging mode 176
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177
Fig. 6 (a) unbalanced input voltage (b) transient performance of battery current in CC charging mode 300
4. SIMULATION RESULTS
e
a
200
e
b
e
This section presents the results of the simulation to verify the proposed method. The simulation is implemented using PSIM simulation tool and MATLAB LMI toolbox to obtain robust gain for the inner-loop controller. The parameters of the system are shown in Table 1. The control algorithm is conducted using a DLL block from Microsoft Visual Studio and the sampling rate is set to 10 kHz. An equivalent circuit, in Fig. 5, is used for the simulation studies.
c
eabc [V]
100 0 -100 -200 -300 0
0.05
0.1
0.15
0.1
0.15
(a)
500
vo [V]
400 300 200 100 0 0
0.05 (b)
Fig. 5. Battery equivalent circuit Fig. 7 (a) unbalanced input voltage (b) transient performance of battery voltage in CV charging mode
Table 1 Simulation parameters Parameters DC-Link capacitor Filter resistance Filter inductance Sampling rate Constant current reference Constant voltage reference
Value 4700µF 0.1Ω 5mH 10kHz 5A 450V
Let us discuss about the comparison between the controller with and without negative sequence compensator in Fig. 8 and Fig 9, respectively. It can be noted that the controller with negative sequence compensator performs well under unbalanced input grid-voltage. Both the battery current and three-phase grid-current are remarkably acceptable as shown in Fig. 8(b) and (c), respectively. In Fig. 9, the simulation performance of the controller without negative sequence compensator is shown. Its performances, however, are not good compared to those of controller with sequence controller. It can be noted that the battery is has high oscillation and the grid-current does not show an appropriate waveform.
Here, the simulation performances of the proposed charger are discussed. Fig. 6(a) shows an unbalanced input threephase grid-voltage supplied to charger in CC charging mode. In Fig. 6(b), the charger is still able to provide a considerable constant current even under unbalanced grid-voltage thanks to the its negative sequence compensator.
300
The CV charging mode is validated in Fig. 7. We can see that the transient response of the battery voltage in CV mode is fast and constant even the three-phase input is unbalance as shown in Fig 7(b).
e
eabc [V]
200
a
100
eb
0
ec
-100 -200 -300 0.05
0.1
0.15 (a)
0.2
0.25
0.1
0.15 (b)
0.2
0.25
300 e
a
200
eb
100
e
8
ibat [A]
[V]
c
e
abc
0 -100 -200 -300 0
4
2 0.05 0.05
0.1
0.15
15
[A]
5
abc
6
i
(a)
ibat [A]
6
10
ia
5
ib i
0
c
-5
4
-10
3
-15 0.05
0.1
0.15 (c)
0.2
0.25
2
Fig. 8 steady-state performances of (a) input voltage, (b) battery current and (c) grid current; with negative sequence compensator in CC mode
1 0 0
0.05
0.1
0.15
(b)
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Ma, K., Chen, W., Liserre, M., and Blaabjerg, F. (2015). Power Controllability of a Three-Phase Converter With an Unbalanced AC Source, IEEE Trans. Ind. Electron., vol. 30, Issue 3, pp. 1591-1604. Parvez, M., Mekhilef, S., Tan, N.M.L., and Akagi, H. (2014). Model predictive control of a bidirectional AC-DC converter for V2G and G2V applications in electric vehicle battery charger. Transportation Electrification Conference and Expo, Dearborn, MI, USA. Pinto, J.G., Monteiro, V., Goncalves, H., Exposto, B., Pedrosa, D., Couto, C., and Afonso, J.L. (2013). Bidirectional battery charger with Grid-to-Vehicle, Vehicle-to-Grid and Vehicle-to-Home technologies, Industrial Electronics Society, IECON 39th Annual Conference of the IEEE, Vienna, Austria. Zhou, X., Wang, G., Lukic, S., Bhattacharaya, S., Huang, A. (2009). Multi-function bi-directional battery charger for plug-in hybrid electric vehicle application. Energy Conversion Congress and Exposition, San Jose, CA, USA.
300 ea
200
e
eabc [V]
100
b
e
c
0 -100 -200 -300 0.2
0.22
0.24
0.26
0.28
0.3 (a)
0.32
0.34
0.36
0.38
0.4
0.22
0.24
0.26
0.28
0.3 (b)
0.32
0.34
0.36
0.38
0.4
ibat [A]
8
6
4
2 0.2 15
i
i
abc
[A]
10
i
5
a b
ic
0 -5 -10 -15 0.2
0.22
0.24
0.26
0.28
0.3 (c)
0.32
0.34
0.36
0.38
0.4
Fig. 9 steady-state performances of (a) input voltage, (b) battery current and (c) grid current; without negative sequence compensator in CC mode CONCLUSIONS This paper proposes a robust control strategy for a threephase charger under unbalanced grid conditions. The control method consists of inner-loop robust grid-current control and outer-loop proportional integral control for constant current (CC) and constant voltage (CV) control. A paralleled current control for the inner-loop positive and negative sequence is employed to eliminate the unbalanced current caused by the grid so that a constant current and voltage can be provided to the batteries. The simulation results show that the proposed controller can provide a remarkable charging performance to the battery under unbalanced grid. ACKNOWLEDGEMENT . This work was supported by the grant from National Research Foundation of Korea (NRF) (NRF2015R1D1A1A01060451, NRF-2017M3A6A6052595) REFERENCES Boyd, S.C., Ghaoui, L.E., Feron, E., and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, Philadelphia, PA, USA:SIAM. Choeung, C., Park, S.H., Koh, B.K., and Lee, Y.I. (2016). Robust Tracking Control of a Three-Phase DC-AC Inverter for UPS Application under Unbalanced Load Conditions. IFAC Workshop on Control of Transmission and Distribution Smart Grids, Prague, Czech Republic. Gallardo-L., J., Isabel M., M., Guerrero-M., M. A., and Romero-C., E. (2012). Electric vehicle battery charger for smart grids. Electric Power System Research, vol. 90, pp. 18-29. Lim, J.S., Park, C, Han, J., and Lee, Y.I. (2014). Robust Tracking Control of a Three-Phase DC-AC Inverter for UPS Applications, IEEE Trans. on Ind. Electron., vol 61, no. 8, pp. 4142-4151.
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IFAC PapersOnLine 51-28 (2018) 173–178
Robust Tracking Control of a Three-phase Charger Robust Tracking Control of a Three-phase Charger Robust Tracking Control ofGrid a Three-phase under Unbalanced Condition Charger under Unbalanced Grid Condition Robust Tracking Control of a Three-phase under Unbalanced Grid Condition Charger Chivon Choeung*, Meng Leang Kry** and Young Il Lee** under Unbalanced Grid Condition Chivon Choeung*, Meng Leang Kry** and Young Il Lee**
Chivon Choeung*, Meng Leang Kry** and Young Il Lee** * Engineering of Electricity, Polytechnic Institute of Cambodia, Phnom Penh, Cambodia (e-mail: ChivonNational Choeung*, Meng Leang Kry** and Young Il Lee** * Engineering of Electricity, National Polytechnic Institute of Cambodia, Phnom Penh, Cambodia (e-mail: [email protected]). * Engineering of Electricity, National Polytechnic Institute of Cambodia, Phnom Penh, Cambodia (e-mail: [email protected]). * Engineering of Electricity, National Polytechnic Institute of Cambodia, Phnom Penh, Cambodia (e-mail: [email protected]). [email protected]). ** Electrical and Information Engineering, Seoul National University of Science and Technology, Seoul, South Korea ** Electrical and Information Engineering, Seoul National University [email protected]). of Science and Technology, Seoul, South Korea (e-mail: [email protected], ** Electrical and Information Engineering, Seoul National University [email protected]). of Science and Technology, Seoul, South Korea (e-mail: [email protected], ** Electrical and Information Engineering, Seoul National University [email protected]). of Science and Technology, Seoul, South Korea (e-mail: [email protected], (e-mail: [email protected], [email protected]). Abstract: This paper presents a robust control technique for three-phase charger under unbalanced grid Abstract: presents a robust control three-phase chargercontrol under unbalanced grid conditions.This Thepaper control method consists of technique inner-loopfor robust grid-current and outer-loop Abstract: This paper presents a robust control technique for three-phase chargercontrol under unbalanced grid conditions. The control method consists of inner-loop robust grid-current and outer-loop proportional integral control fora constant current (CC) andfor constant voltagecharger (CV) control. A dual-current Abstract: paper presents robust control three-phase under unbalanced grid conditions.This The control method consists of technique inner-loop robust grid-current and outer-loop proportional integral control for constant current (CC) andis constant voltage (CV)control control. A dual-current control for the inner-loop positive and negative employed to eliminate the unbalanced current conditions. The control method consists of sequence inner-loop robust grid-current control and outer-loop proportional integral control for constant current (CC) and constant voltage (CV) control. A dual-current control by forthe the grid inner-loop positive andcurrent negative is employed to eliminate the unbalanced current caused socontrol that positive a constant andsequence voltage can be provided to (CV) the batteries. The inner-loop proportional integral for constant current (CC) and voltage control. A dual-current control forthe the inner-loop andcurrent negative is constant employed to eliminate the unbalanced current caused by grid so thatstate a constant andsequence voltage can be provided to the batteries. The inner-loop robust controllers utilize feedback with integral action in the dq-synchronous frame. A linear matrix control by forthe the grid inner-loop positive andcurrent negative is employed to eliminate the unbalanced current caused so thatstate a constant andsequence voltage can to the batteries. The inner-loop robust controllers utilize feedback with integral action inbe theprovided dq-synchronous frame. A linear matrix inequality-based optimization scheme is used to determine stabilizing gains of the controllers to caused controllers by the gridutilize so thatstate a constant current and voltage caninbe to the batteries. The inner-loop robust feedback with integral theprovided dq-synchronous frame. linear matrix inequality-based optimization scheme isstate used totheaction determine stabilizing gainsThe of uncertainties the Acontrollers to maximize the convergence rate to steady in presence of uncertainties. of the robust controllers optimization utilize state feedback with integral action in thestabilizing dq-synchronous frame. linear matrix inequality-based isstate used determine gainsThe of uncertainties the Acontrollers to maximize the convergence rate scheme to steady in totheof presence of uncertainties. of the system are the described as the potential variation range the inductance and resistance the L-filter. inequality-based optimization scheme isstate used tothedetermine stabilizing gainsThe ofinuncertainties the controllers to maximize convergence rate to steady in presence of uncertainties. of the system are described as the potential variation range of the inductance and resistance in the L-filter. maximize the convergence rate to steady state in the presence of uncertainties. The uncertainties of the system described as the potential variation rangeControl) ofgrid, the inductance resistance in the L-filter. © 2018,are IFAC (International Federation of Automatic Hosting byand Elsevier Ltd.(LMI), All rights reserved. Keywords: Battery charger, fast charging, unbalanced linear matrix inequality robust control. system are described as the potential variation range ofgrid, the inductance andinequality resistance(LMI), in the robust L-filter.control. Keywords: Battery charger, fast charging, unbalanced linear matrix Keywords: Battery charger, fast charging, unbalanced grid, linear matrix inequality (LMI), robust control. Keywords:1.Battery charger, fast charging, unbalanced grid, linear matrix inequality (LMI), robust control. INTRODUCTION In (Gallardo-L., J. 2012), a PI control was used in the AC1. INTRODUCTION In (Gallardo-L., a PIdc-link controlvoltage was used in thewhile ACWith the emerging 1. ofINTRODUCTION enormous amount of plug-in hybrid DC converter sideJ.to2012), maintain constant In (Gallardo-L., 2012), a PIdc-link controlvoltage was used in thewhile ACWith thevehicles emerging ofINTRODUCTION enormous amount of plug-in hybrid DC converter sideJ. toused maintain constant electric (PHEV) and electric vehicles (EV) and the deadbeat control is in the DC-DC converter to regulate 1. In (Gallardo-L., J.to2012), a PIdc-link controlvoltage was used in thewhile ACWith thevehicles emerging of enormous amount of plug-in hybrid DC converter side maintain constant electric (PHEV) and electric vehicles (EV) and the deadbeat control is used in the The DC-DC converter to provides regulate flourishing of the renewable energy, theofdevelopment of the current tomaintain battery. deadbeat control With thevehicles emerging of enormous amount plug-in hybrid DC charging converter sideistoused dc-link voltage constant while electric (PHEV) and electric vehicles (EV) and the deadbeat control in the DC-DC converter to regulate flourishing of has thebecome renewable energy, the development of the charging current to battery.which The deadbeat control provides power an interesting topic. Therefore, the fastcontrol transient respond setting time reaches the electricstoring vehicles (PHEV) and electric (EV) and the deadbeat is used in the The DC-DC converter to provides regulate flourishing of has the renewable energy,vehicles the development of really the charging current to battery. deadbeat control power storing become an interesting topic. Therefore, the really fast transient respond which setting time reaches the technology related to battery charging has drawn a lot of steady state in just a few sampling instants. However, it is flourishing of has thebecome renewable energy, the development of really the charging current to battery.which The deadbeat control provides power storing an interesting topic. Therefore, the fast transient respond setting time reaches the technology related to battery charging has drawn a lot of steady state in just a few sampling instants. However, it is attention from has governments, auto-makers, and researchers. In really sensitive to the parametric uncertainty of the system power storing become an interesting topic. Therefore, the really fast which setting reachesitthe technology related to batteryauto-makers, charging has drawn a lot of steady statetransient in just arespond few sampling instants.time However, is attention from governments, and researchers. In really sensitive to the parametric uncertainty of the system order to answer to the demands of fast and efficient and measurement noise, particularly for high sampling rate. technology related to battery charging has drawn a battery lot of steady state in just a few samplinguncertainty instants. However, it is attention from governments, auto-makers, and researchers. In really sensitive to the parametric of the system order to many answercontrol to thestrategies demandshave of fast and efficient battery and measurement noise, particularly for high sampling rate. charger, been proposed. attention from governments, auto-makers, and researchers. In really sensitive tonoise, the parametric uncertainty of the system order to many answer to thestrategies demands of fast and efficient battery andthis measurement for highofsampling rate. In paper, a robustparticularly tracking control a three-phase charger, control have been proposed. order to answer to the demands of fast and efficient battery and measurement noise, particularly for high sampling rate. charger, control strategies been proposed. In this under paper,unbalanced a robust tracking controlis of a three-phase Classicalmany Proportional Integralhave (PI) control have been charger grid conditions proposed without In this paper, a robust tracking control of a three-phase charger, many control strategies have been proposed. Classical Proportional Integral (PI)(Pinto, control under unbalanced grid is proposed without proposed for single-phase chargers J.G.have 2013 been and charger using converter an conditions interface between three-phase In thisDC-DC paper,unbalanced a robust astracking control a three-phase Classical Proportional Integral (PI)(Pinto, control been charger under is of proposed without proposed for single-phase chargers J.G.have 2013 and using DC-DC converter asgrid an conditions interface between three-phase Zhou, X. 2009). The charging scheme was designed based on AC-DC converter and batteries. The battery is charged with Classical for Proportional Integral (PI)(Pinto, control have been charger under unbalanced grid conditions is proposed without proposed single-phase chargers J.G. 2013 and using DC-DC converter as an interface between three-phase Zhou, X. 2009). The charging scheme was designed based on AC-DC converter and the batteries. The battery isrecommended charged with constant-current (CC) and constant-voltage (CV) charging constant current until voltage reaches the proposed for single-phase chargers J.G. 2013 and DC-DC converter as an interface between three-phase Zhou, X. 2009). The scheme (Pinto, was designed on using AC-DC and the batteries. The battery charged with constant-current (CC)charging and constant-voltage (CV) based charging constant converter current until voltage reaches theisrecommended mode provides shorter chargingwas time compare to that voltage, then the voltage is maintained constant Zhou, which X. 2009). The charging scheme designed based on maximum AC-DC converter and batteries. The battery isrecommended charged with constant-current (CC) and constant-voltage (CV) charging constant current until the voltage reaches the mode which provides shorter method. charging time compare tomain that maximum voltage, then the is maintained constant of fixed voltage charging However, until the current consumed by voltage battery falls tothea residual value. constant-current (CC) and constant-voltage (CV) the charging constant current until thethe voltage reaches recommended mode which provides shorter charging time compare to that maximum voltage, then voltage is maintained constant of fixed of voltage charging method. However, the main until the current consumed byofbattery falls to a residual value. drawback PI controller is gain tuningtime for compare both inner-loop The control method consists inner-loop robust grid-current mode which provides shorter charging that maximum voltage, then the is maintained constant of fixed voltage charging method. However, the tomain until the current consumed byofvoltage battery falls to a residual value. drawback of PI controller is gain tuning for both inner-loop The control method consists inner-loop robust grid-current and fixed outer-loop controller. Furthermore, since the topology of until control and outer-loop proportional integral control for of voltage charging method. However, the main the current consumed byofbattery falls to a residual value. drawback of PI controller is gain tuning for both inner-loop The control method consists inner-loop robust grid-current and outer-loop controller. Furthermore, since the topology of constant control and outer-loop proportional integral control for these proposed methods are based on single-phase converter, current (CC) and constant voltage (CV) control. A drawback of PI controller is gain tuning for both inner-loop The control method consists of inner-loop robust grid-current and outer-loop controller. Furthermore, since the topology of constant control and outer-loop proportional integral control for these proposed methods are based on single-phase converter, current (CC)forand constant voltage (CV) control. A the limit of charging current is smaller compare to those of dual-current control the inner-loop positive and negative and outer-loop controller. Furthermore, since the topology of constant control and outer-loop proportional integral control for these proposed methods are based on single-phase current (CC)forand constant voltage (CV) A the limit of chargers. charging current is smaller compare toconverter, those of sequence dual-current control theeliminate inner-loop positive andcontrol. negative three-phase is employed to the unbalanced current theselimit proposed methods are based on single-phase converter, constant current (CC)forand constant voltage (CV) control. A the of charging current is smaller compare to those of dual-current control the inner-loop positive and negative three-phase chargers. sequence employed to aeliminate unbalanced current by is thecontrol grid sofor that constant the current andand voltage can the limit of chargers. charging current is smaller compare to those of caused dual-current theeliminate inner-loop positive negative three-phase sequence is employed to the unbalanced current Model predictive control have been proposed in (Parvez, M. caused by the grid so that a The constant currentrobust and voltage can provided to the batteries. inner-loop controllers three-phase chargers. sequence employed to aeliminate the unbalanced current Model predictive control have been proposed inThis (Parvez, M. be caused by isthe grid so that constant current and voltage can 2014) for a bidirectional three-phase charger. method be provided to the batteries. The inner-loop robust controllers state feedback integral action the can dqModel predictive control have been proposed inThis (Parvez, M. utilize caused by the grid so thatwith a The constant current and in voltage be provided to the batteries. inner-loop robust controllers 2014) for a bidirectional three-phase charger. method could provide bidirectional power transfer withininstantaneous state feedback with integral action in the dqModel predictive control have been proposed (Parvez, M. utilize synchronous frame. A linear matrix inequality-based 2014) for a bidirectional three-phase charger. This method be provided to the batteries. The inner-loop robust controllers state feedback integral action in the dqcould provide bidirectional power transfer with instantaneous mode capability and fast dynamic response. so, utilize synchronous frame.is used Awith linear matrix inequality-based 2014) charging for a bidirectional three-phase charger. This Even method optimization to determine stabilizing gainsdqof could provide bidirectional power transfer with instantaneous utilize state scheme feedback with integral action in the synchronous frame. A linear matrix inequality-based mode charging capability and fast dynamic response. Even so, the charging fromand power grid to vehicle is based on optimization scheme is used to determine stabilizing gains of could providemethod bidirectional power transfer with instantaneous the controllers to maximize the convergence rate to steady mode charging capability fast dynamic response. Even so, synchronous frame. A linear matrix inequality-based scheme is used to stabilizing of the charging method from power grid to vehicle is based fixed voltage mode therefore the need longer time on to optimization the controllers to maximize thedetermine convergence rate togains steady mode charging capability and fastbatteries dynamic response. Even so, state in the presence of uncertainties. The uncertainties of the the charging method from power grid to vehicle is based on optimization scheme is used to stabilizing of controllers to maximize thedetermine convergence rate togains steady fixed voltage modeMoreover, therefore the batteries needgrid longer time to the be fully charged. in unbalanced conditions state in the presence of uncertainties. The uncertainties of the the charging method from power grid to vehicle is based are presence described asuncertainties. the potential variation range of the fixed voltage modeMoreover, therefore the batteries needgrid longer time on to system the controllers to maximize the convergence rate to steady state in the of The uncertainties of the be fully charged. in unbalanced conditions thisfully method cannot provide constant voltage the batteries are and described as theinpotential variation range of the fixed voltage modeMoreover, thereforea the needtolonger time to system resistance the L-filter. The conventional be charged. in batteries unbalanced conditions state in the ofasuncertainties. The uncertainties of the system are presence described theinpotential variation range of this method cannot providesequence a constant voltagegrid to the batteries inductance the due to the lack of negative compensation. inductance and resistance the L-filter. The conventional be fully charged. Moreover, in unbalanced conditions phase-locked loop (PLL) is in considered in this paper to obtain this method cannot providesequence a constant voltagegrid to the batteries inductance system are and described as the potential variation range of the resistance the L-filter. The conventional due to the lack of negative compensation. phase-locked loop angle. (PLL) is considered in this paper to obtain this to method cannot providesequence a constant voltage to the batteries inductance grid voltageand phase due the lack of negative compensation. resistance the L-filter. Thepaper conventional phase-locked loop (PLL) is in considered in this to obtain grid voltage phase angle. due to the lack of negative sequence compensation. phase-locked loop (PLL) is considered in this paper to obtain grid voltage phase angle. grid voltage phase angle.
Copyright 2018 IFAC 173Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Copyright 2018 responsibility IFAC 173Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 173 10.1016/j.ifacol.2018.11.697 Copyright © 2018 IFAC 173
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ean 1 n 1 = eb = α e n 3 α 2 c
n e abc
α e a α 2 eb ,
α2 1
(4b)
1 ec
α
2
where α and α are phase-shifting operators and defined as:
2. SYSTEM DESCRIPTION
1 2
α =− + j
3 2
1 2
3 . 2
α =− − j
and
(5)
Substituting (5) in (4) yields:
p e abc
Fig. 1. Three-phase bidirectional charger with L-filter n e abc
A three-phase charger is shown in Fig. 1. The dynamic of the line current is expressed in the abc-axis as follows: dia (t ) L dt + Ria (t ) = ea − v a ,i di (t ) b + Rib (t ) = eb − vb,i L dt L dic (t ) + Ri (t ) = e − v c c c ,i dt
1 1 1 = − 3 2 − 1 2 1 1 1 = − 3 2 − 1 2
1 2
1 −
1 2
−
1 2
1 −
1 2
1 − 2 e a 0 1 − 1 ea 1 1 (6a) − eb + j − 1 0 1 eb 2 2 3 ec 1 − 1 0 ec 1 1 − 2 e a 1 1 − eb − j 2 2 3 ec 1
1/3
(1)
1/6 1/6
ea
All-pass Filter
eb
All-pass Filter
ec
All-pass Filter
where 2u a − ub − uc vo (t ) 6 − u a + 2ub − uc vb,i := vo (t ) . 6 − u a − ub + 2uc vc,i := vo (t ) 6 va ,i :=
1, S x = on; S x = off ; Here, − 1, S x = off ; S x = on;
−
1/3 1/6 1/6
u x ( x = a, b, c) .
1/3 1/6
dt
= Ac idq (t ) + Bc vo (t )u(t ) + d c (t ) .
1/6
(2)
Any unbalanced three-phase voltage can be expressed by the sum of positive, negative and zero sequence (Ma, K. 2015). However, the three-phase charger system has only three wires therefore the zero sequence is does not exist. Hence, the unbalanced current and voltage can be expressed as:
ea
All-pass Filter
eb
All-pass Filter
ec
All-pass Filter 1/3 1/6 1/6
p n e abc = e abc + e abc
(3a)
p n i abc = i abc + i abc
(3b)
The positive and negative sequence of assumed to be: eap 1 α p 1 2 p e abc = eb = α 1 e p 3 α α 2 c
j (eb − ea )
−
+ +
1
+
− −
2 3
−
1
+
2 3
j (ec − eb )
ea+
+ +
eb+ ec+
+
− −
− −
+ +
− −
j (ea − eb )
+ +
1
−
2 3
+
1
−
2 3
j (eb − ec )
ea− − −
+ +
eb− ec−
+
Fig. 3 Negative sequence extraction block diagram
the grid voltage are α 2 e a α eb ,
+
− −
Fig. 2 Positive sequence extraction block diagram
The dynamic in abc-axis (1) can be transformed to dq-axis as follows: didq (t )
0 1 − 1 ea − 1 0 1 e (6b) b 1 − 1 0 ec
An all-pass filter is used to obtain the imaginary part of (6) by shifting 90⁰ from the original phase. The same process is used to extract positive and negative sequence of grid current and voltage. The block diagrams of positive and negative sequence extraction are shown in Fig. 2 and Fig. 3, respectively.
(4a)
1 ec
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From equation (3), the current dynamics (2) can be rewritten as a combination of positive and negative sequence. The dynamic of both sequences is almost identical; however, the only difference is the sign of ω due to the inverse direction of the negative sequence vector rotation (Choeung, C. 2016). Then, we have p di dq (t )
p p = Acp i dq − d cp (t ) , (t ) + Bc vo (t )udq dt n (t ) di dq n n (t ) + Bc vo (t )udq = Acn i dq − d cn (t ) dt
The uncertainties of the system can be any kind of variation, but should lies within the range (11). The system uncertain range can be determined as: (13a) L0 / µ ≤ L ≤ µL0 (13b) R0 / µ ≤ R ≤ µR0 where R0 and L0 are nominal value of the filter resistance and inductance, respectively, and µ (>1) can be considered as a tuning parameter. In consideration of compensating the offset error despite the system’s uncertainty model, the control law based on (Lim, J.S. 2014) is employed for (10): p p p p g (k ) = g (k + 1) + ( x ref − x (k − 1)) (14a) p p p p p u K x L g ( k ) = ( k ) + ( k ) n n n n g (k ) = g (k + 1) + ( x ref − x (k − 1)) (14b) n n n n u( k ) = K x ( k ) + L g ( k ) where K pn and Lpn are state feedback and integrator gains, respectively. Because of the integrator in (14), the steadypn state error between the reference state x ref and the grid-
(7a) (7b)
where Acp
R − := L −ω
R
ω − , Acn := L R ω − L
1 − −ω , Bc := 2 L R 0 − L
0 , 1 − 2L
0 0 n p d cp := eq , d cn := eq L L ω is the angular frequency of the AC voltage source. i dq and
current x pn will be compensated provided that the closedloop system is stable.
udq are the grid current and control input in dq-frame,
respectively. The output voltage vo (t ) is governed by the following dynamic: dv (t ) (8) C o = icon (t ) − ibat (t ) dt where 3 T (9) icon (t ) = i dq (t )u(t ) 4 and icon. is the converter current and ibat . is an output current to the battery.
In order to eliminate the unbalanced current caused by negative sequence and provide a constant charging current and voltage to the batteries, the reference state should be given as: p x dref 0 p p x iqref . pn x ref := qref = n 0 x dref n x qref 0 p p where x qref is generated by the outer-loop controller = iqref
Model (7) can be transformed in the following discrete-time system with sampling time h: x p (k + 1) = A p x p (k ) + Bvo (k )u p (k ) − d p (k )
175
depending on its CC and CV control objectives.
(10a)
pn . where A pn = I 4×4 + Acpn , B = Bc h , d pn = d cpn h , x pn = i dq
The same process is used to find optimal gains for positive and negative sequence, so the procedure is unified. A systematic design method was proposed in (Boyd, S.C. 1994) to obtain stabilizing state feedback gain K and integral gain L of (14) using LMI. From relation (10) and (14), we get:
3. ROBUST CONTROLLER DESIGN
z (k + 1) = Aa z (k ) + Ba u(k ) + D(k )
n
n
n
n
n
x (k + 1) = A x (k ) + Bvo (k )u (k ) − d (k )
(10b)
∑
x (k ) z (k ) := g (k )
A Aa := − C d ( k ) B 1 0 ,C = , D (k ) := B a := . 0 1 0 2×2 x ref
In this section, the uncertainties model of the system, offsetfree control and robust optimal gain are discussed. Suppose that the value of L and R in each phase are equal but vary in certain ranges below: (11a) L1 ≤ L ≤ L2 (11b) R1 ≤ R ≤ R2 . Here, we denote the matrices (A, B) corresponding to the four possible combinations of the immoderate value of 1/L and 1/R as (Ai, Bi) (i = 1,2,3,4) and suppose that the matrices (A, B) belongs to the polytopic uncertain set Ψ below: 4 4 (12) Ψ = µ n ( Ai , Bi ) µ n = 1,µ n ≥ 0 . n=1 n =1
where
,
(15) 0 2×2 I 2×2
,
The control input u(k) can be rewritten as:
u(k ) = Fz (k )
F := [K
L] .
(16)
Assume that D(k) = 0 to determine stabilizing gain F, then the closed-loop system can be obtained as follows:
∑
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z (k + 1) = ( Aa + Ba F ) z (k ) .
Here, the outer-loop PI control for constant voltage charging is discussed. Let I r be the output of the outer-loop control, i.e.
(17)
The closed-loop dynamic (17) stable if there exists a positivedefinite matrix W such that T
W − ( Aa + Ba F ) W ( Aa + Ba F ) > 0 .
∫
I r (t ) = K p (voref − vo (t )) + K i (voref − vo (t ))dt
(18)
ref where vo is the constant voltage reference for battery
It is obvious that the condition (18) holds for some W 0 > 0 (W 0 < W )
W0 − ( Aa + Ba F )T W ( Aa + Ba F ) > 0 .
(19)
By applying Schur complement and uncertain set (12) to (19), we get (Boyd, S.C. 1994)
S0 Aai S 0 + Bai H
S 0T
T Aai
+H S
T
T Bai
> 0 , (i = 1,…,4), (20)
where H := FS 0 , S = W −1 , S 0 := S 0−1
A Aai := i − C
0 2×2 , I 2×2
(24)
B Bai := i , (i = 1, 2, 3, 4). 0 2×2
charge and vo (t ) is the output voltage. Suppose that the dynamics of the inner-loop control in previous section is considerably fast so that we can assume (25) icon (t ) ≈ I r (t ) for some current reference I r (t ) . From (8) and (25), we get dv (t ) C o = K p (voref − vo (t )) + K i (voref − vo (t ))dt − ibat (t ) (26) dt or d 2 vo(t) dv (t) (27) C + K p o + K i vo (t ) = K i voref . dt dt The K i and K p gain can be determined by considering the
∫
characteristic polynomial of (27) and can be given as
∆( s ) = s 2 + 2ζω r s + ω r2 or some appropriate value of ζ and
To summarize, closed-loop system (17) is asymptotically stable if there exist symmetric positive definite matrices S and S 0 and a matrix H such that (20) holds and the stabilizing gain is given as (21) F = HS 0−1 .
ω r2 ;
(28)
we get
K i = ω r2
(29)
K p = 2ζω r .
(30)
The control diagram of the constant voltage control of the three-phase charger is shown if Fig. 4. The battery voltage is fed-back to the out-loop controller, which produces a p p . reference current x qref = iqref
Assume that W 0 < αW or equivalent to (22) (0 < α < 1) . S < αS 0 It can be expected that a small α would give a fast convergence of z to the origin. Therefore, to obtain optimal gain F such that the convergence time is minimized, the following optimization problem should be solved. Minimize α subject to (20) and (22) (23)
5.2 Constant current (CC) charging mode In constant current charging stage, the battery pack is charged with fixed current until the voltage reaches the recommended maximum voltage, then switch to constant voltage charging stage. For the control of this constant current charging mode, an outer-loop PI is utilized to generate a reference signal p p for inner-loop robust control with the same x qref = iqref
S , S 0 > 0,
α > 0, H
This optimization scheme is a generalized eigenvalue problem (Boyd, S.C. 1994) which can be solve efficiently by MATLAB Toolbox.
concept as CV charging discussed in previous section. The control structure of the proposed CC charging control is validated as shown in Fig. 4.
4. DUAL-LOOP CHARGING CONTROL In order to perform the battery charging process, most of the battery manufacturers recommend two charging stage i.e. constant current (CC) mode followed by constant voltage (CV) mode. The battery is charged with constant current until the voltage reaches the recommended maximum voltage, then the voltage is maintained constant until the current consumed by battery falls to a residual value. The dual-loop control strategy will be adopted for these two charging states i.e. an p p depending outer-loop controller generates proper x qref = iqref on its CV or CC control objectives and the inner-loop pn controller (14) drives the state of x pn in (10) to follow x ref .
Fig. 4. Control structure of the proposed method
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Fig. 6 (a) unbalanced input voltage (b) transient performance of battery current in CC charging mode 300
4. SIMULATION RESULTS
e
a
200
e
b
e
This section presents the results of the simulation to verify the proposed method. The simulation is implemented using PSIM simulation tool and MATLAB LMI toolbox to obtain robust gain for the inner-loop controller. The parameters of the system are shown in Table 1. The control algorithm is conducted using a DLL block from Microsoft Visual Studio and the sampling rate is set to 10 kHz. An equivalent circuit, in Fig. 5, is used for the simulation studies.
c
eabc [V]
100 0 -100 -200 -300 0
0.05
0.1
0.15
0.1
0.15
(a)
500
vo [V]
400 300 200 100 0 0
0.05 (b)
Fig. 5. Battery equivalent circuit Fig. 7 (a) unbalanced input voltage (b) transient performance of battery voltage in CV charging mode
Table 1 Simulation parameters Parameters DC-Link capacitor Filter resistance Filter inductance Sampling rate Constant current reference Constant voltage reference
Value 4700µF 0.1Ω 5mH 10kHz 5A 450V
Let us discuss about the comparison between the controller with and without negative sequence compensator in Fig. 8 and Fig 9, respectively. It can be noted that the controller with negative sequence compensator performs well under unbalanced input grid-voltage. Both the battery current and three-phase grid-current are remarkably acceptable as shown in Fig. 8(b) and (c), respectively. In Fig. 9, the simulation performance of the controller without negative sequence compensator is shown. Its performances, however, are not good compared to those of controller with sequence controller. It can be noted that the battery is has high oscillation and the grid-current does not show an appropriate waveform.
Here, the simulation performances of the proposed charger are discussed. Fig. 6(a) shows an unbalanced input threephase grid-voltage supplied to charger in CC charging mode. In Fig. 6(b), the charger is still able to provide a considerable constant current even under unbalanced grid-voltage thanks to the its negative sequence compensator.
300
The CV charging mode is validated in Fig. 7. We can see that the transient response of the battery voltage in CV mode is fast and constant even the three-phase input is unbalance as shown in Fig 7(b).
e
eabc [V]
200
a
100
eb
0
ec
-100 -200 -300 0.05
0.1
0.15 (a)
0.2
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0.15 (b)
0.2
0.25
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a
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eb
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e
8
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[V]
c
e
abc
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4
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[A]
5
abc
6
i
(a)
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6
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ia
5
ib i
0
c
-5
4
-10
3
-15 0.05
0.1
0.15 (c)
0.2
0.25
2
Fig. 8 steady-state performances of (a) input voltage, (b) battery current and (c) grid current; with negative sequence compensator in CC mode
1 0 0
0.05
0.1
0.15
(b)
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Ma, K., Chen, W., Liserre, M., and Blaabjerg, F. (2015). Power Controllability of a Three-Phase Converter With an Unbalanced AC Source, IEEE Trans. Ind. Electron., vol. 30, Issue 3, pp. 1591-1604. Parvez, M., Mekhilef, S., Tan, N.M.L., and Akagi, H. (2014). Model predictive control of a bidirectional AC-DC converter for V2G and G2V applications in electric vehicle battery charger. Transportation Electrification Conference and Expo, Dearborn, MI, USA. Pinto, J.G., Monteiro, V., Goncalves, H., Exposto, B., Pedrosa, D., Couto, C., and Afonso, J.L. (2013). Bidirectional battery charger with Grid-to-Vehicle, Vehicle-to-Grid and Vehicle-to-Home technologies, Industrial Electronics Society, IECON 39th Annual Conference of the IEEE, Vienna, Austria. Zhou, X., Wang, G., Lukic, S., Bhattacharaya, S., Huang, A. (2009). Multi-function bi-directional battery charger for plug-in hybrid electric vehicle application. Energy Conversion Congress and Exposition, San Jose, CA, USA.
300 ea
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i
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ic
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Fig. 9 steady-state performances of (a) input voltage, (b) battery current and (c) grid current; without negative sequence compensator in CC mode CONCLUSIONS This paper proposes a robust control strategy for a threephase charger under unbalanced grid conditions. The control method consists of inner-loop robust grid-current control and outer-loop proportional integral control for constant current (CC) and constant voltage (CV) control. A paralleled current control for the inner-loop positive and negative sequence is employed to eliminate the unbalanced current caused by the grid so that a constant current and voltage can be provided to the batteries. The simulation results show that the proposed controller can provide a remarkable charging performance to the battery under unbalanced grid. ACKNOWLEDGEMENT . This work was supported by the grant from National Research Foundation of Korea (NRF) (NRF2015R1D1A1A01060451, NRF-2017M3A6A6052595) REFERENCES Boyd, S.C., Ghaoui, L.E., Feron, E., and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, Philadelphia, PA, USA:SIAM. Choeung, C., Park, S.H., Koh, B.K., and Lee, Y.I. (2016). Robust Tracking Control of a Three-Phase DC-AC Inverter for UPS Application under Unbalanced Load Conditions. IFAC Workshop on Control of Transmission and Distribution Smart Grids, Prague, Czech Republic. Gallardo-L., J., Isabel M., M., Guerrero-M., M. A., and Romero-C., E. (2012). Electric vehicle battery charger for smart grids. Electric Power System Research, vol. 90, pp. 18-29. Lim, J.S., Park, C, Han, J., and Lee, Y.I. (2014). Robust Tracking Control of a Three-Phase DC-AC Inverter for UPS Applications, IEEE Trans. on Ind. Electron., vol 61, no. 8, pp. 4142-4151.
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