Generation Inverters Connected to the Grid. Juan C. Vásquez, Josep M. Guerrero ... Eduard Gregorio, Pedro RodrÃguez. Department of Electrical Engineering.
Adaptive Droop Control Applied to Distributed Generation Inverters Connected to the Grid Juan C. Vásquez, Josep M. Guerrero
Eduard Gregorio, Pedro Rodríguez
Remus Teodorescu, Frede Blaabjerg
Dept. Automatic Control Systems and Computer Eng. Universitat Politecnica de Catalunya Urgell 187-08036-Barcelona, Spain
Department of Electrical Engineering Universitat Politecnica de Catalunya Colom 1-08222-Terrassa, Spain
Institute of Energy Technology Aalborg University 9220 Aalborg East, Denmark
Abstract-This paper proposes a novel control for voltage source inverters connected to the grid. The control scheme is based on the droop method, and it uses some estimated variables from the grid such as the voltage and the frequency, and the magnitude and angle of the grid impedance. Hence, the inverter is able to inject independently active and reactive power to the grid. The controller provides a proper dynamics decoupled from the grid impedance. Simulation results are provided in order to show the feasibility of the control proposed.
I.
INTRODUCTION
Distributed generation systems and microgrids are taking importance when trying to increase the renewable energy penetration. In this sense, the use of intelligent power interfaces between the sources and the grid is mandatory. Usually, in order to inject energy to the grid current-source inverters (CSI) are used, while in island or autonomous operation voltage-source inverters (VSI) are used [1]. Voltage sources inverters are very interesting since they don’t need any external reference to stay synchronized [2], [3]. In fact, they can operate in parallel with other inverters by using frequency and voltage droops, forming autonomous microgrids [4]. When these inverters are required to operate in grid-connected mode, they often change its behavior from voltage to current sources [5]. To achieve flexible microgrids, which are able to operate in both grid connected and island mode, VSIs are required [6]. The droop method can be used to inject active and reactive power from the VSI to the grid by adjusting the frequency and amplitude of the output voltage [2]-[4]. However, the conventional droop method needs for the knowledge of some parameters of the grid in order to inject independently active and reactive power. In this paper, we propose a control scheme based on the droop method which automatically adjusts their parameters by using an estimation method of the grid impedance based on power variations caused by the VSI at the point of common coupling (PCC). ___________________ This work was supported by the Spanish Ministry of Science and Technology under grants CICYT ENE 2006-15521-C03-01/CON and ENE2007-67878C02-01/ALT.
978-1-4244-1666-0/08/$25.00 '2008 IEEE
II.
VOLTAGE AND CURRENT MONITORING AT THE PCC BY USING A SOGI-FLL
The grid characterization technique used in this paper is based on processing the voltage and current phasors at the point of PCC between the power converter and the grid. To monitoring such voltage and current phasors, the frequency locked loop based on the second order generalized integrator (SOGI-FLL) is used due to its high precision, low computational cost and frequency adaptation capability [7]. A SOGI is a frequency-adjustable resonator which can be implemented by two cascaded integrators working in a closeloop [8]. Since the SOGI acts as an ideal integrator for a sinusoidal input at a particular frequency, the adaptive filter seen in Fig. 1 (SOGI-AF) can be easily implemented. Transfer functions of this filter are given by (1), where ω’ and k set the center frequency and damping factor respectively. D( s ) =
v′ kω ′s ( s) = 2 , v s + kω ′s + ω ′2
(1a)
Q( s ) =
qv′ k ω ′2 ( s) = 2 , v s + kω ′s + ω ′2
(1b)
E ( s) =
v
εv
εv v
( s) =
s 2 + ω ′2 . s + kω ′s + ω ′2
(1c)
2
k
ω′
∫
∫
v′ qv′
εv Fig. 1. SOGI-based adaptive filter (SOGI-AF)
Transfer functions of (1) reveal that the system depicted in of Fig. 1 simultaneously acts as a band-pass, low-pass and notch filter on the input signal v. If v is a sinusoidal signal, v’ and qv’ will be sinusoidal signals as well, being qv’ 90º-lagged respect v’ independently of both the frequency of v and the
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values of ω’ and G k. This interesting characteristic is used later to obtain the V phasor representing the instantaneous voltage v. The error signal εv is zero when the tuning frequency ω’ matches to the frequency ω of the input signal v. Therefore, this error signal is used to render the filter frequency-adaptive by adding an extra frequency-locking loop. Fig. 2 shows the Bode diagram of E(s) and –Q(s). This diagram reveals that εv and -qv’ are in counterphase when ω’< ω and they are in phase when ω’> ω. Therefore, a frequency error signal εf can be achieved by multiplying both signals, this is: ε f = − qv′ ⋅ ε v . (2) The mean value of εf is negative when ω’< ω and vice versa, being only zero when ω’= ω. Since only the dc-value of εf is of interest here, a frequency-locked loop (FLL) can be easily achieved by applying an integral controller with a gain of γ to this frequency error signal as Fig. 3 shows. 10
Magnitude (dB)
0 -10
−Q ( s )
E (s)
-20 -30 -40 -50 180
Phase of − Q ( s )
Phase (deg)
135
εf 0
III.
0 -45
0
1
10
2
10
3
10
4
10
10
Frequency (Hz)
Fig. 2. Frequency adaptation principle of the SOGI-FLL
εv
v
k
∫
ω′
∫
IDENTIFICATION OF THE GRID PARAMETERS
The general form of the droop-control method results from a linear interpretation of the grid in which distributed power generators are connected to. Therefore, the grid can be seen from the PCC of a power generator as a simple Thevenin circuit constituted by a grid impedance Zg and a header voltage Vg. Even though the v-i characteristic of the ac grid can not be represented by a simple two-dimensional Cartesian plane, Fig. 4 helps to illustrate further explanations about the impedance detection method used in this work since it depicts the relationship between voltage and current phasors at the PCC for a particular frequency.
Phase of E ( s )
-90 -1 10
reference frame. ThereGfore, a crucial issue in the right detection of the phasor V is that ω’ keeps rigorously equal to the grid frequency ω through different samples of the voltage v. When v is transiently altered by any cause, e.g. by the action of a grid-connected inverter performing droop-control, the detected frequency ω’ will experiment a transitory evolution toward the steady state frequency ω. Such transitory values of ω’ should be discarded by the integration block calculating the angle θ’. In this work, this feature is achieved by setting a sampling-time at the input of the integration block higher than the transitory interval of ω’. The aforementioned SOGI-FLL is also applied to monitoring the current G injected into the PCC i in order to obtain the current phasor I = id + jiq . The fact that the SOGI-FLL acts as a selective filter for detecting two in-quadrature output signals, see (1), is a very interesting feature to attenuate harmonics on the moniGtored voltage and current and accurately detect the G phasors V and I at the fundamental grid frequency. The detected voltage and current in-quadrature signals should be projected on the same d-q rotating reference frame to obtain coherent voltage and current phasors. Therefore, as shown in Fig. 5, the FLL block is only implemented on the monitored voltage v and the angle θ’ is calculated from the integration of the voltage frequency.
v′
qv′
qv′ −1
ω ff γ
∫
ω′ G ΔV pcc ⎧⎨ G⎩ Vg
Fig. 3. Diagram of the SOGI-FLL
G The voltage phasor V = vd + jvq representing the sinusoidal voltage v at the frequency ω is obtained by projecting the two in-quadrature signals [v’; qv’] on a rotating d-q reference frame, that is: ⎡vd ⎤ ⎡ v′ ⎤ ⎡ cos(θ ′) sin(θ ′) ⎤ (3) ⎢ v ⎥ = [Tdq ] ⎢ ′⎥ ; [Tdq ] = ⎢ ⎥, ⎣ qv ⎦ ⎣ − sin(θ ′) cos(θ ′) ⎦ ⎣ q⎦ where θ’ is obtained by integrating the detected frequency ω’. This frequency is the angular speed of the d-q rotating
G I sc
G V pcc G Zg G I pcc
G ΔI pcc
⎧ ⎨ ⎩
εv
εf
Fig. 4. v-i characteristic of the grid for a particular frequency
From the measurement of the voltage and current phasors at the PCC at two different operating points, linearity in the v-i
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characteristic of Fig. 4 allowsG writing (4), (5) and (6) for G estimating the grid impedanceG Z g , the open-circuit voltage Vg , and the short-circuit current I sc of the grid, respectively. G G G G ΔV pcc V1 − V2 Zg = Zg θg = G = G G (4) I1 − I 2 ΔI pcc GG G G G G G G I1V2 − I 2V1 Vg = Vg φg = V pcc (i ) − Z g I pcc (i ) = G G (5) I1 − I 2 G G G GG G G V pcc (i ) I 2V1 − I1V2 I sc = I sc φsc = I pcc (i ) − G = G G (6) Zg V1 − V2 where Z g and θ g are the magnitude and the angle grid impedance, respectively. Several techniques for detecting the grid impedance are either directly or indirectly based on this basic principle. A well known technique consists on processing the voltage and current variations at the PCC caused by the connection of different loads [9]. Other techniques process the voltage at the PCC after injecting noncharacteristic current harmonics into the grid [10][11]. There are also techniques in which either nonsinusoidal, amplitude modulated or wideband frequency currents are injected to the grid for processing the voltage variation at the PCC [12]-[14]. In this work, the grid parameters are estimated from the active and reactive power variations generated by a grid-connected converter in which a droop-controller is implemented. The cornerstone of this estimation technique is the accuracy in the on-line measurement of voltage and currents phasors at the PCC, which is performed by the system based on the SOGI-FLL described in §II. Fig. 5 shows the whole diagram of the algorithm used in this work to identify the grid G paGrameters. G It is worth to say that the estimated values for Z g , Vg and I sc are transiently wrong after each change in the grid parameters. These transient values can not be sent to the droop-controller of the grid-connected inverter. For this reason, a small buffer of three rows is added at the output of the grid parameters identification block of Fig. 5. The value of any grid parameter used by the droop-control only will be updated when the differences between all the three values stored in the three-row buffer are inside a 5% tolerance band.
i pcc
i
ω′ v pcc
v
ω′
ω′
i′ qi′
⎡⎣Tdq ⎤⎦
εi
θ′
v′ qv′
⎡⎣Tdq ⎤⎦
εv
θ′
id iq
G Zg
G Vg vd vq
G I sc
∫
qv′
ωss′
εv
ω′ Fig. 5. Block diagram of the grid parameters identification algorithm
IV.
ADAPTIVE DROOP CONTROL
In this section, based on the estimation of the grid parameters provided by the identification algorithm, an adaptive droop controller able to inject active and reactive power into the grid with high accuracy is proposed.
A. Power flow analysis From Fig. 4 we can calculate the active P and reactive Q powers injected to the grid by the VSI [15], [16] 1 ⎡( EV cos φ − V 2 ) cos θ + EV sin φ sin θ g ⎤ P= (7a) ⎦ Zg ⎣ Q=
1 ⎡( EV cos φ − V 2 ) sin θ − EV sin φ cos θ g ⎤ ⎦ Zg ⎣
(7b)
where E and φ are the magnitude and phase of the VSI, V is the grid voltage respectively. Notice that both expressions depend highly on the grid impedance (Z∠θ). Consequently, we propose to transform P and Q to novel variables, which are independent from the magnitude and phase of the grid impedance: (8a) Pc = Z g ( P sin θ − Q cos θ ) Qc = Z g ( P cos θ + Q sin θ )
(8b)
being Pc(s) and Qc(s) the linear compensators of the phase and the amplitude. By substituting (7) into (8), it yields the following expressions: Pc = EV sin φ (9a)
Qc = EV cos φ − V 2
(9b) Note that Pc is mainly dependent on the phase φ, while Qc is depends on the voltage difference between the VSI and the grid (E – V). These control variables (Pc and Qc) are independent from the grid impedance, so we can use them into the droop control method to inject active and reactive power.
B. Droop control technique In order to inject the desired active and reactive powers (defined as P* and Q*), the following droop method which uses the transformation (8) is proposed φ = −G p ( s ) Z g ⎡⎣( P − P* ) sin θ g − ( Q − Q* ) cos θ g ⎤⎦ (10a) E = E * − Gq ( s ) Z g ⎡⎣( P − P* ) cos θ g + ( Q − Q* ) sin θ g ⎤⎦ (10b) Fig. 6 shows the block diagram of the implementation of the droop controller to inject the desired active P* and reactive power Q*. Gp(s) and Gq(s) are the compensator transfer functions of Pc and Qc, respectively mi + m p s + md s 2 G p ( s) = (11a) s ni + n p s Gq ( s ) = (11b) s
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Note that in practice the derivative term in Gq(s) is avoided since it barely affects to the system dynamics. Adaptive Droop Controller
P* P
−
+
Pc
G p (s)
+
Q
+
Q* −
(Eq. 8)
Zg
E*
Qc
Gq ( s )
φ
(18)
a4 = T 2 + 2Tmd VEcosΦ
a3 = 4T + 4md n pV 2 E + 2VcosΦ ( 2md E + Tn p + Tm p E )
a2 = 2VcosΦ (Tmi E + Tni + 2n p + 2m p ) + 4V 2 E ( m p n p + md ni ) + 4
1 − e −Ts 1 Qi ≈ Qi . Ts 1 + (T 2 ) s
(14b)
By substituting (10) into (14), and doing a small signal approximation in order to linearize the equations, it yields 1 pˆ c ( s ) = V sin Φ eˆ( s ) + VE cos Φ φˆ( s ) (15a) 1 + (T 2 ) s
(
)
(
)
1 V cos Φ eˆ( s ) − VE sin Φ φˆ( s ) 1 + (T 2 ) s
1 + (T 2 ) s
Being
Vc * = E sin(ω t − φ )
C. System dynamics A small signal analysis is provided, in order to show the system stability and the transient response, allowing the designer to adjust the control parameters [17-18]. Taking into account that P and Q are the average values of instantaneous active and reactive power p(t) and q(t) 1 t −T P = vd id + vq iq = ∫ p (t )dt (12a) T t 1 t −T Q = vq id − vd iq = ∫ q (t )dt (12b) T t being T the period of the grid frequency. By using the first order Padé approximation, 2 − Ts e −Ts ≈ (13) 2 + Ts the average value P and Q can be expressed as follows 1 − e −Ts 1 (14a) P= Pi ≈ Pi s 1 + (T 2 ) s
qˆc ( s ) =
⎟ ⎠
a4 s 4 + a3 s 3 + a2 s 2 + a1 s + a0 = 0
Vref *
Fig. 6. Block diagram of the adaptive droop control.
Q=
s
By combining (17a) and (17b), it can be obtained the following fourth order characteristic equation
*
φ
Droop functions
θg
E ω
⎞ V sin Φ eˆ( s ) + VE cos Φ φˆ( s ) (17a) ⎟⎟ 1 + (T 2 ) s ⎠ ⎛ ni + n p s ⎞ V cos Φ eˆ( s ) − VE sin Φ φˆ( s ) . (17b)
eˆ( s) = − ⎜ ⎝
−
P/Q Decoupling Transformation
⎛ mi + m p s + md s 2 ⎜ s ⎝
φˆ( s) = − ⎜
(15b)
where the low-case variables with the symbol ^ indicate small signal values, and uppercase variables are the steady-state values. By using (10), (11) and (15), it can be obtained ⎛ mi + m p s + md s 2 ⎞ = − ( ) φ s (16a) ⎜⎜ ⎟⎟ pˆ c ( s ) s ⎝ ⎠ ⎛ ni + n p s ⎞ eˆ( s ) = − ⎜ (16b) ⎟ qˆc ( s ) . s ⎝ ⎠ From (16), it can be derived the following expressions
a1 = 4VcosΦ ( ni + mi E ) + 4V 2 E ( mi n p + m p ni ) a0 = 4ni mi EV 2
Where the steady-state values of the active and reactive power are P = P* and Q = Q*, and, from (7), the steady-state phase and amplitudes can be calculated as follows ⎛ ⎞ P* sinθ g − Q* cosθ g ⎟ , (19a) Φ = tan −1 ⎜ * ⎜ P cosθ g + Q* sinθ g + V 2 Z g ⎟ ⎝ ⎠ 2 V cosθ g + PZ g E= (19b) V cosθ g cosΦ + sinθ g sinΦ
(
(
V.
)
)
SIMULATION RESULTS
The control proposed was tested through proper simulations of a single-phase VSI connected to the grid. Table I shows the main control and system parameters used. Fig. 7 shows the transient response of P and Q for changes step in P* (from 1500 to 2000 W at t=1s) and Q* (from 500 to 250 VAr at t=2s). Note the good decoupling of P and Q injection. Fig. 8 shows the good resemblance between the system phase dynamics and the model (18). The model has been proved for wide system parameters, showing its validity. By using that model, we can extract the root locus family that can be seen in Figs. 9, 10, and 11, by changing mp, md, and nd. In order to guarantee the stability condition (input/output behavior) of the closed-loop system dynamics, a poles study of the fourth order identified model is employed. The performance of this kind of systems is often viewed in terms of a pole dominance set of it can be seen. In one hand, the A0 coefficient of the characteristic equation depends basically of mi and ni parameters that influence directly over the fast response of the system making it more damped. In some practical cases is possible to adjust this parameters for finetuning purposes. The fourth order system can be simplified to a third, second, or even first-order system.
2423
Fig. 12 shows an example of P dynamics, using the control without and with the Z estimation loop, for grid impedance variations (R = 1, 2, and 3 Ω). As it can be seen, this loop decouples to a large extend the dynamics from the grid impedance.
-3
x 10
10
8
Symbol V
ω∗ R L Z
θ S mi mp md ni np
TABLE I. SYSTEM PARAMETERS Parameter Value Grid voltage amplitude 311 Grid frequency 50 Grid resistance 2 Grid inductance 3 Grid impedance module 2.3 Grid impedance angle 28.8 Nominal apparent power 4 Integral droop Pc coefficient 0.0018 Proportional droop Pc coefficient 0.00005 Derivative droop Pc coefficient 4·10–7 Integral droop Qc coefficient 0.05 Proportional droop Qc coefficient 0.0004
Units V Hz Ω mH Ω º kVA Ws/rd W/rd W/rd·s VAr·s/V VAr/V
Phase [rd]
6
4
2 Model Real
0
-2
0
0.2
0.4
0.6
0.8
1
Time [s]
Fig. 8. Dynamic response of the system and the model (18). Roo t Loc u s
30
20
Im aginary Ax is
λ3
2500
P
P [W] & Q [VAr]
2000
1500
10 λ1
λ2
0
-10
λ4
-20
-30 -100
1000
-80
-60
-40
-20
0
R e a l A x is
Fig. 9. Root locus for 1·10– 6 < md < 4·10– 5.
500
Q 0 0
0.5
1
1.5 time [s]
2
2.5
Root Loc us
3
1 0.8 0.6
Fig. 7. Transient response of P and Q injected to the grid by the VSI.
Im aginary Ax is
0.4 0.2
λ1
λ2
0
λ3
λ4
- 0.2 - 0.4 - 0.6 - 0.8 -1 - 350
- 300
- 250
-200
-150
Real A x is
Fig. 10. Root locus for 0.0004 < np < 0.01.
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-100
- 50
0
Root Locus
reactive power to the grid. The proposed droop control uses such parameters to close the loop, achieving a tight P and Q regulation. Thanks to the feedback variables of the estimator, the system dynamics is well decoupled from the grid parameters. The results point out the applicability of the proposed control scheme to distributed generation VSIs for microgrid applications.
1 0.8 0.6
Imaginary Axis
0.4 0.2
λ3
λ2
λ1
0
λ4
REFERENCES
-0.2 -0.4
[1]
-0.6
[2]
-0.8 -1 -100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
[3]
Real Axis
Fig. 11. Root locus for 0.00005 < mp < 0.0001.
[4]
1600
[5]
1400 1200
[6]
1000 800 600
[7]
400 200
R=1 R=2 R=3
0 -200
0
0.5
1
1.5
[8] 2
(a)
[9]
1600 1400
[10]
1200 1000 800
[11]
600 400 200 0 -200
[12]
R=1 R=2 R=3 0
0.5
1
1.5
[13]
2
(b) Fig. 12. Start up of P for different line impedances, (a) without and (b) with the estimation of Z.
VI.
CONCLUSION
[14]
[15]
In this paper has been presented a novel control for VSI connected to the grid that is able to inject active and reactive power. The control has two main structures. The first one is the grid parameters estimation, which calculates the amplitude and frequency of the grid, as well as the magnitude and phase of the grid impedance. The second one is a droop control scheme, which uses these parameters to inject independently active and
[16] [17] [18]
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