Control System Design for Bi-directional Power Transfer in Single-Phase Back-to-Back Converter Based on the Linear Operating Region Janeth Alcalá, Víctor Cárdenas, Emanuel Rosas and Ciro Núñez Centro de Investigación y Estudios de Posgrado Universidad Autónoma de San Luis Potosí San Luis Potosí, SLP [email protected]
, [email protected]
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Abstract—This paper presents two control strategies for a Single-Phase Back-to-Back (SPBTB) converter for control bidirectional power transfer based on the linear operating region of the topology. The control strategies presented allow an independent control of active/reactive power. The first scheme proposes to cancel the non-linear expressions associated to the linearized model. Therefore, it is obtained a suitable steady state and dynamic response of the system. The second control strategy uses a non-linear method that ensures a better response, since it does not linearize around a specific point, obtaining a global control law for the all operating region. The performance is verified with simulation and experimental results.
It is well kwon that electrical grids restrain active and reactive power flow in the point of common coupling (PCC), which causes over-load and/or sub-utilization problems. Over-load implies higher energy consumption between both interconnected systems. More power transfer though the existing system implicates that some of the lines might operate closer to their capacity limits. Besides, there is a risk that the system operates around to its instability boundaries. Introducing additional interconnections to solve the problem will become the system more complex to operate. Significant benefits can be obtained achieving a load-balancing by introducing topologies that allow a better control of power transfer. In this aspect, voltage source converters (VSC) are a growing technological alternative . Specifically through Back-to-Back (BTB) configuration is possible to accomplish these objectives. BTB converter is formed by two identical VSC connected by a common dc-link. This topology presents several advantages in terms of processing power and allows bidirectional power flow with almost sinusoidal currents at near-unity power factor. Therefore, the power flows can be changed and the thermal limits will not be exceeded. Moreover, losses can be minimized and stability margins are
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not increased. BTB converter is widely used in the power system connected between the line and the load. The load can be active, passive or even another network, Fig. 1(a). The dclink in the middle provides decoupling between both converters; as a result, they could be driven independently. Therefore, it is possible to have a fast and independent control of active and reactive power for both converters and improve the operation of the system. To simultaneously achieve these performances is important to explore control strategies which allow obtaining the desired control objectives. Previous works have presented control strategies were the current controllers are based on PI schemes [2-5]. However, these schemes exhibit coupling restrictions and do not provide an independently control of active/reactive power. In this paper two laws to the control of the BTB converter are discussed based on the dq mathematical model. The first technique presented cancels the coupled terms that appear in the mathematical model of the BTB converter and eliminate locally the nonlinearities [6-7]. The second solution, an input-output linearization technique allows to fully linearize the BTB converter. VSC1 and VSC2 are controlled separately; this can be accomplished because both converters are decoupled through the dc-link capacitor. Nevertheless is not the unique solution to the control of the BTB converter, but significantly simplifies the control of the two supply current components, which allows the SPBTB to transfer bi-directional active power and regulate the reactive power independently. Therefore, this configuration is used to the development of the control strategies. Moreover, the operating capacity of the topology is analyzed to establish the operating active/reactive region of the topology [8-9]. The knowing of the physical restrictions allow identifying what can be done by the control strategy and what cannot be obtained because of the inherent characteristic of the converter. Therefore, to avoid over-modulation conditions and to take advantage of the SPBTB response, an analytical procedure is developed to obtain the operating region.
To the control of the BTB converter outer and inner loops are proposed. The designs of the outer loops are similar in both techniques unlike the inner loops which are different. The inner loops are used to control the current in VSC1 and VSC2. The outer loops in VSC1 control the dc-link voltage and the reactive power and in VSC2 they control active and reactive power. The variables tracking are utilized to prove the local and global tracking capability with suitable steadystate error in both techniques. The simulated and experimental results obtained with both techniques are presented. A SPBTB prototype using IGBT power switches is built and experiments are presented to validate the performance of the proposed controllers. II.
constant value. The dq model is given by (1) – (5), where m1dq and m2dq are the modulating signals, which must fulfill the constraints1
The configuration of the proposed system is showed in Fig. 1(a) and the state equivalent model circuit in Fig. 1(b). Load1
Cdc Vdc (a) i1 +
v1 ( t )
R2 + −
v2 ( t )
q 1, 2
. ω is the angular
d d 1 1 i1 = ωi1q + v1d − vdc m1d dt L1 uT L1
d q 1 i1 = −ωi1d − vdc m1q dt uT L1
d d 1 1 i2 = ωi2q − v2d + vdc m2d dt L2 uT L2
d q 1 i2 = −ωi2d − vdc m2q dt uT L2
d 1 ⎡ d d vdc = m1 i1 + m1qi1q − m2d i2d + m2qi2q ⎤ ⎦ dt 2uT ⎣
B. Linear Operating Region The operation region of the SPBTB converter is the graphic representation of all the equilibrium points, which are defined by the constraints imposed to the modulation indexes and by the topology characteristic . The linear operating region guarantees power transfer between both converters in the SPBTB without over-modulation and a lower THDV (Total Harmonic Voltage Distortion). To define the linear operating region, an analytical procedure is studied. The system input restrictions and the constraint imposed by the voltage are:
idc 2 idc Cdc
A. Operation requirements The SPBTB converter must keep the dc-link voltage level constant and the active power transmitted through the dc-link should be adjustable. As no reactive power is transmitted through the dc-link, the reactive power compensation is attributed to the VSC corresponding to each side control. One VSC must control the voltage level in the dc-link and the other the active power flow. Both converters can perform any of the two tasks; therefore the decision is subjective. In the SPBTB it was decided VSC1 will be in charge of controlling the voltage level in the link and VSC2 must control the active power flow. Based on the linear operating region of the SPBTB this task is guaranteed because the active/reactive power boundary of the converter will not be overcome.
d 1, 2
frequency of ac mains; uT is the peak amplitude of the triangular carrier.
SYSTEM MODELING AND PRINCIPLES OF OPERATION
( m ) + ( m ) < (u )
(b) Fig. 1. SPBTB. (a) Power topology. Fig. (b). Equivalent circuit.
The mathematical model of the SPBTB converter is a nonlinear-type because the state variables are multiplied by the control inputs. Theoretical results report that it is difficult to obtain global stability or a suitable tracking for sinusoidal waveforms. Therefore, the single-phase dq transformation  is used and the sinusoidal voltages and currents are expressed in rotating coordinates by two constant dq variables which provide phase and amplitude, respectively. As a result, through the dq model the control objectives are simplified and the sinusoidal tracking problem becomes a regulation problem. Hence, the control objectives can be achieved by regulating the dq variables to the desired
(m ) + (m ) 2 d 1, 2
2 q 1, 2
Vdc > 2V1
Solving (1)-(5) at the equilibrium point, yields to the following2: I1d = − I 2d = − 1 2
M 1qVdc ω L1
M 2qVdc − V2q ω L2
M 1d Vdc − V1d ω L1
I 2q =
M 2d Vdc − V2d ω L2
Sub-indices 1, 2 are referred to each one of the converters respectively. Capital letters are used to represent the steady state and peak values.
General expressions for active and reactive powers transmitted for both converters are defined by:
Q1, 2 =
d d 1, 2 1, 2
q d 1, 2 1, 2
+ V1,q2 I1q, 2
d q 1, 2 1, 2
M 1q = −0.132
V1d, 2 are taken as references for VSC1,2. Therefore, V1q, 2 = 0 .
The (11-14) expressions are obtained by evaluating (7) and (8) at the maximum and minimum values of the control inputs. −
V1d Vdc V dV < P1 < 1 dc 2ω L1 2ω L1
(V1d − Vdc ) d (V d + Vdc ) d V1 < Q1 < 1 V1 2ω L1 2ω L1 −
V2d Vdc V dV < P2 < 2 dc 2ω L2 2ω L2
(V2d − Vdc ) d (V d + Vdc ) d V2 < Q2 < 2 V2 2ω L2 2ω L2
M 1d = 0.386 P1 = 200W
Active Power (kW)
P1, 2 =
Active Power Operating Region
(13) (14) M 1d = 0 . 386 M 1q = − 0 .132
By mapping the operating region of the dq active and reactive current components into P and Q domain it is possible to determine the power transfer limits for a given dclink voltage. As example, Fig. 2 shows the power transfer limits for VSC1. The system parameters used to plot the linear operating region are shown in Table I. The dc-link capacitor was selected to transfer the maximum active power obtained from the graphics of the active operating regions.
Q1 = 0VAR
The colors gradient gives the value of the transmitted power from V1 to VSC1 according to the dq coordinates for the modulating indexes md × mq . As example, from Fig. 2, VSC1 and VSC2 should have M 1d = 0.386 , M 1q = −0.132 to transmit 200W from V1 to V2 at unitary power factor in both converters.
Fig. 2(c) shows the modulating indexes for VSC1 depending on the power factor. The blue – green surface and the red – orange surface represents the lagged and leaded power factor for the VSC1 respectively. It is observed that both converters are able to operate in a wider range in the inductive power factor region than in the capacitive one. The operation of the system closer to the periphery is equivalent to work with over-modulation conditions and all the problems that this involves. Therefore, to take advantage of a SPBTB response it must operate under its boundary region. For the system proposed the maximum active power that the converter is able to transfer from the mains to the load is around 1.5kW as can be note in Figs. 2(a) and 2(b).
m1d Power Factor
M 1d = 0.386 M 1q = −0.132
PF1 = 1
m1d (c) Fig. 2. VSC1 Operating Region. (a) Active power transfer. (b) Reactive power transfer. (c) SPBTB Power Factor.
TABLE I SYSTEM PARAMETERS Parameter Value V1d, V2d ω L1, L2 R1, R2 Vdc
Therefore, to cancel the non-linear expressions the following control law is proposed :
30 Vrms 377r/s 4.1 mH 284mΩ 110V
m1d = k p ( i1dref − i1d ) + ki ∫ ( i1dref − i1d )dt +
uT v1d u L + ωi1q T 1 (15) vdc vdc
m1q = k p ( i1qref − i1q ) + ki ∫ ( i1qref − i1q )dt − ω i1d
The control objectives assigned to the SPBTB converter are: dc-link voltage vdc regulated and active/reactive power flow control. The proposed control block is shown in Fig. 3; the outer loops connected in cascade with the inner loops provide the dq current references. Two control strategies are proposed for the inner current loops. The first one is based on a scheme that allows decoupling active/reactive components of the input current. The second one proposes use the inputoutput linearization method.
uT L1 vdc
From (15) and (16) is obtained the closed loop transfer function of the active/reactive current, this yields the system to a linear decoupled system around the operation point. i1d ( s ) d 1ref
i1q ( s ) q 1ref
−k p vdc s − ki vdc uT L1 s 2 − k p vdc s − ki vdc
Two outer loops based on PI Controllers are added for VSC1. One controls the reactive power by manipulating the reactive current component and the second one controls the dc-link voltage. These control loops will be regulated to the desired level indirectly once the line current tracking is achieved. The transfer function for vdc and q1 is given by (18) and (19). Besides, another two outer loops to control active/reactive power in VSC2 are used. The p2 and q2 transfer function are given by (20) and (21). To guarantee the decoupled between the inner current loop and outer loops the bandwidth of the outer loop must be a decade lower than the inner loop. The control system parameters are shown in Table II. The parameters are chosen in order to obtain a satisfactory dynamic response.
Fig. 3. Proposed Control Scheme for the SPBTB.
vdc ( s )
vdcref ( s )
A. Active and reactive current decoupled control strategy From equations (1)-(5) can be seen that the active/reactive line currents components are coupled and do not allow an independently control of the active and reactive power. The control block diagram for VSC1 is shown in Fig. 4.
k p v1d s + ki v1d
2Cdc vdcref s 2 + k p v1d s + ki v1d
v1d (k p s + ki ) q1 ( s ) =− q1ref ( s ) (2 − v1d k p ) s − v1d ki
v2d (k p s + ki ) p2 ( s ) = p2 ref ( s ) (2 + v2d k p ) s + v2d ki
v2d (k p s + ki ) q2 ( s ) =− q2 ref ( s ) (2 − v2d k p ) s − v2d ki
B. Non-linear control strategy
SPBTB model in steady - state can be represented as: ⎡ di1d ⎤ ⎡ d ⎤ ⎡ vdc ⎢ ⎥ ⎢ωi q − R1i1d + v1 ⎥ ⎢ − ⎢ dt ⎥ ⎢ 1 L1 ⎥ ⎢ uT L1 ⎢ ⎢ di q ⎥ ⎢ d q ⎥ ⎢ 1 ⎥ = ⎢ −ωi1 − R1i1 ⎥ + ⎢ 0 ⎢ ⎢ dt ⎥ ⎢ ⎥ ⎢ −idc 2 ⎢ dvdc ⎥ ⎢ ⎥ ⎢ i1d ⎢ ⎥ ⎢ Cdc ⎥ ⎦ ⎢ 2C ⎢⎣ dt ⎥⎦ ⎣ dc ⎣
Fig. 4. Control Block Diagram for VSC1.
⎤ ⎥ ⎥ ⎡ i1d ⎤ ⎢ ⎥ ⎡ m d ⎤ (22) v ⎥ ⎡u ⎤ − dc ⎥ ⎢ 1 ⎥ , x = ⎢ i1q ⎥ , u = ⎢ 1 ⎥ uT L1 ⎥ ⎣u2 ⎦ ⎢ ⎥ ⎢⎣ m1q ⎥⎦ ⎥ ⎢ vdc ⎦⎥ q ⎣ i1 ⎥ 2Cdc ⎥⎦ 0
According to the theory of input–output linearization , an input-output system is input-output linearize while exist a control law, such as: ⎡ Lρf −1h1 ( x ) ⎤ ⎡ v1 ⎤ ⎢ ⎥ ⎢ ⎥ u = − E −1 ( x ) ⎢ ... ⎥ + E −1 ( x ) ⎢ ... ⎥ ⎢ ρm ⎥ ⎢⎣ vm ⎥⎦ ⎢⎣ L f hm ( x ) ⎥⎦
⎡ ⎤ 0 ⎥ ⎢ − x3 ⎡u ⎤ ⎥ 1 ⎢ − x3 ⎥ ⎢ 1 ⎥ ⎥ +u ⎢ 0 ⎥ u T ⎥ x2 ⎥ ⎣ 2 ⎦ ⎢ x1 ⎦ 2⎦ ⎣ 2
R1x1 + v1d ⎤
y = [ x1
i1d ( s) i1q ( s ) −k1 = = υ1 ( s ) υ2 ( s ) s − k1
Where E(x) is called decoupling matrix of the MIMO system, ρ is the relative degree of the system and v denote the auxiliary control variable. Therefore, calculating the Lie Derivates (24) is obtained. This control law exists for all vdc≠0. ⎡ L1x1 ⎤ ⎡ω L1x2 − ⎢ L x ⎥ = ⎢ ⎢ 1 2 ⎥ ⎢ −ω L1x1 − R1x2 ⎢⎣Cdc x3 ⎥⎦ ⎢⎣ idc 2
The parameter of the controller k1 = –1535 is chosen to have an acceptable dynamic response. The closed loop transfer function (28) is a first – order system.
As a result, it is possible to control independently the active and reactive components of the line current and according to the linear relations between the power quantities: vd id vd iq vd iq p1 = 1 1 , q1 = 1 1 , q2 = 2 2 (29) 2 2 2 Hence, the controls of the system powers correspond to the control of the corresponding currents, because v1d and v2d are constants. Moreover, the SPBTB is able to supply active/reactive power without modify the transfer of reactive/active power in both steady state and transient conditions. The outer control loops are designed as (18) – (21).
Where x = ⎡⎣i1d i1q vdc ⎤⎦ , u = ⎡⎣u1d u1q ⎤⎦ .
IV. SIMULATION AND EXPERIMENTAL RESULTS The proposed control strategies have been simulated using Matlab/Simulink under the following conditions: V1,2 = 30Vrms , f = 60 Hz , L1,2 = 4.1mH , R1,2 = 284mH ,
The control law is obtained using (23)
⎡ ⎤ R1 v1d ⎢ω x2 − L1 x1 + L1 − v1 ⎥ ⎥ x3 ⎢⎢ R −ω x1 − 1 x2 − v2 ⎥ L1 ⎣⎢ ⎦⎥
The original nonlinear system (22) is transformed into the (input-output) linearized system such that: ⎡ y1 ⎤ ⎡ v1 ⎤ ⎢ y ⎥ = ⎢v ⎥ ⎣ 2⎦ ⎣ 2⎦
The plant in the Laplace domain is seen through the control law as (27), which is a first – order plant. Therefore it is possible to achieve a zero steady-state error in the tracking toward its references using a simple proportionally controller. Fig. 5 shows the proposed control scheme. Y1 ( s ) v1 ( s )
Y2 ( s ) v2 ( s )
It is then possible to choose suitable constants such that the error dynamics are stable and achieve the tracking of the constant reference by usual linear pole placement technique. y ref
Fig. 5. Control scheme for the linearized plant.
UT = 5V , Vdc = 110V , Cdc = 1050 μ F , P1,2 = 200W , PF1,2 = 1 to
verify their feasibility. Figs. 6 and 7 show the relevant waveforms for a transferred active power of 200W with a 100% load reversed transient at t=0.4s. The results show that the dq current components present a good tracking for both control strategies. Nevertheless, there are differences in the transient response, which are mainly due to the second order dynamic that presents the decoupled active/reactive current control strategy. These differences can be observed in the oscillations of the currents. The input-output linearization strategy reduces the system to a first order type. Therefore, the control response converges exponentially and do not present the oscillation that appears in the first control strategy. Besides, the settling time is lower for the second proposed control strategy (t=10ms). Nevertheless, the difference is not significant (less than 5ms). The first control strategy have a similar performance the second one for the simulation parameters, but it is important to emphasize that this strategies decoupled locally around a set-point; this disadvantage is not presented by the non-linear control strategies, because the input-output linearization provides global stability. It means that the system is decoupled for all the operating range. In order to validate the operating region presented in section II, simulations were done. From Fig. 2(a) is observed that the active power boundary for the SPBTB converter is around 1.5kW. Therefore, a power ramp was applied to the converter using the input – output linearization technique to guarantee global stability.
0.4 Time (s)
0 -50 0.37
0 -20 0.37 50
vdc (V )
0.4 Time (s)
100 50 1.15 3000
2000 1000 1.15 200
0 -5 1.15
100 0 1.15 5
ma (V )
current decoupled control strategy. (a) vdc . (b)
i2dq currents tracking. (d) Supply voltage v1 and current i1.
i1dq currents tracking.
P1 (W )
V vdcdc((V) V) dq (A) ii11dq ( A) dq ii22dq ((A) A)
(V/A) v1 v/1i/i11 (V / A)
-20 0.37 20
m1d / m1q (V ) i1d / i1dref ( A)
Fig. 6. Simulated results for a 100% load reversed. Active/reactive (c)
Fig. 7. Simulated results for a 100% load reversed. Input-output
-20 0.37 50
current decoupled control strategy. (a) vdc . (b)
-20 0.37 20
dq ii2dq A) 2 ((A)
From the simulated results can be conclude that even if the control dynamic is very good it is not possible to achieved the control objectives because of the inherent characteristic of the SPBTB converter. Therefore, the operating region provides essential information that can be used to evaluate the performance of a controller and with the information discarded if the desired goals are not obtained because of the control technique used or if the problem is due to the own behavior of the topology.
v1v/1/ii11 ((V/A) V / A)
The objective of the simulation was to overcome the physical power constraints of the converter given by the active power operating region (Fig. 2(a)). Fig. 8 shows the dc-link voltage, the active power, the d current component and its reference, the dq components of the control signals and finally the control signal in the abc frame for VSC1. It can be observed that close to the 1.35ms (when the converter reaches 1.5kW) the control signals are saturated and consequently the d current is unable to follow its reference. In turn, the system becomes unstable.
i1dq currents tracking.
i2dq currents tracking. (d) Supply voltage v1 and current i1.
0 -5 1.15
Fig. 8. VSC1. (a) Dc-link voltage. (b) Active power ramp. (c) d current component and its reference. (d) dq control inputs components. (e) Control input component in the abc frame.
The control strategies were implemented and tested in an experimental SPBTB prototype under the same simulated conditions. In Figs. 9(a) and 10(a), CH1-CH4 show v1 vs . i1 (top) and v2 vs . i2 (bottom) when the SPBTB converter is transferring 200W from v1 to v2 at unitary power factor. It can be observed that the power transferred is sustained. A power step was applied to the converter, in order to evaluate the dynamic response. Figs. 9(b) and 10(b) show the dc-link voltage response to the power step when the dc-link. The power variation in the experiment was from 150W − 200W . It ca n be notes that the dc-link voltage does not fluctuate significantly with both control schemes. In fact, the response of the decoupled active/reactive control strategy has a good performance considering that this control law only provides locally stability. The SPBTB converter was tested under regeneration, when the power is negative and flows into the dc-link, making dq current components negative. The line current change from 5.67Arms to -5.67Arms, experimental results are shown in Figs. 9(c) and 10(c); it is observed that the reversal response in the current is fast. Therefore, it can be conclude that both controllers present a good dynamical performance under reversal of load current, ensuring bidirectional power flow.
v1 i1 v2 i2
VI. CONCLUSIONS This paper has presented and compared two control strategies of the SPBTB converter for providing an independent active/reactive power control. Moreover, in order to find the linear operating region of the SPBTB an analysis was derived, it was found that depending on dc-link voltage and the systems parameters (inductor link and line voltage), there is a limited range of active/reactive power that can be managed by the topology. These region boundaries allow identifying the physical restrictions of the system. Therefore, it is possible to assure from the control response if the desired results are not been reached because of the control strategy or by the restrictions imposed by the system. Based on the results two control strategies were proposed: a decoupled active/reactive current control strategy and an input-output linearization control strategy. The decoupling control has a good stability response. However, the PI parameters must be designed to a specific operating point. Nevertheless, experimental result show that the system recovers rapidly from power steps and reversed transient. The input – output linearizing current control algorithm, developed, linearizes the SPBTB converter to first – order ones and removes the cross – coupling between d and q current components. Therefore, it is possible to control independently active and reactive power, which improves the operation of the converter. Theoretical and experimental results show that both control strategies have a good dynamic of controlled variables. Moreover, by the experimental results it was shown that the control strategies allow bi-directional power flow and ensures power balancing control.
(c) Fig. 9. Experimental results for the active and reactive decoupled control strategy: (a) Voltage and current supply in VSC 1 (top) and VSC2 (bottom) when active power flows from V1 to V2. (b) Dc-link voltage and supply current response in VSC1 for a power step (150W-200W). (c) Voltage and current supply in VSC1 for a 100% (200W to -200W) reversed power transient (under regeneration) (top) and d component and its reference (bottom).
i1 v1 i2
TABLE II. CONTROL PARAMETERS Gain
kp ki τ
1 100 10(ms)
-0.001 -15.153 66(us)
0.001 15.153 66(us)
-0.001 -15.153 66(us)
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(c) Fig. 10. Experimental results for the input-output linearization control strategy: (a) Voltage and current supply in VSC1 (top) and VSC2 (bottom) when active power flows from V1 to V2. (b) Dc-link voltage and supply current response in VSC1 for a power step (150W-200W). (c) Voltage and current supply in VSC1 for a 150% (200 W to -300W) reversed power transient (under regeneration) (top) and d component and its reference (bottom).