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Distributed generations have been playing an important role in smart power grids to improve grid's performance, reliability and efficiency and greenhouse gas issues. Stability analysis for voltage source inverters is proposed in this paper to

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SRPioneers
© Science and Research Pioneers Institute
4th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE2016), 4-5 February 2016, Dubai, Emirates
Stability Analysis and Dynamic Behavior Improvement of a Stand-alone AC Microgrid
Hossein Karimi, Rahim Rezaei, Mohammad T.H. Beheshti
Electrical and Computer Engineering Tarbiat Modares University Tehran, Iran Hosein.karimi@modares.ac.ir , rahim.rezaei@modares.ac.ir , mbehesht@modares.ac.ir
Abstract
—
Distributed generations have been playing an important role in smart power grids
to improve grid’s
performance, reliability and efficiency and greenhouse gas issues. Stability analysis for voltage source inverters is
proposed in this paper to investigate microgrid’s stability. In
first step, pole placement is implemented in order to find controllers parameters. Particle swarm optimization is applied to find controller parameters. It is shown that although the system is stable for wide area of load variations, the system performance is not that good; therefore, an objective function is proposed to address both stability and performance improvement over wide range of load changes.
Keywords-component; distributed generaation; pole placement; controller parameter; objective function.
I.
I
NTRODUCTION
A collection of Distributed Generations (DG) constitutes microgrids which can either provide power for local loads, or transfer power to the main grid. These small grids can improve grid reliability, integration of renewable energy sources into power grids, and the system efficiency. Fuel cell, microturbine, photovoltaic can be connected to the grid through power electronic inverters [1] - [2].
These devices make DG’s operation more flexible in
comparison to the electrical machine [3]. Microgrid can be operated in two modes which are grid connected and autonomous mode. In autonomous mode, microgrid must regulate frequency and voltage besides providing power for loads. On the other hand, in grid-connected mode, MGs just provide power to the grid, and their frequency and voltage are dictated by the main grid [4] - [6]. Due to the variable nature of sources and uncertainty in load demand, several new challenges arise in this field of study. A sophisticated control strategy is proposed to cope with these challenges. The proposed control strategy includes droop controller, voltage controller, and current controller [7] - [8]. Droop controller is implemented to
mimic conventional generators’ behavior. In the power
grids, generators decrease frequency/voltage as demand increases. There exist two channel, namely P-f and Q-V. Frequency mainly depends on active power, while voltage depends on reactive power. In the other word, frequency can be controlled by P, and voltage by Q [9] - [10]. Generally, VSIs are assumed to be fast and their tracking performance is accurate and perfect. This
assumption means inverter’s bandwidth is above the droop
controller, voltage controller, and frequency controller bandwidth [3]. Small signal representation of a microgrid plays an important role in m
icrogrid’s stability analysis. The small
signal model of a microgrid is presented in [3]. This model consists of three parts, i.e. inverter model, network model, and load model and can be extended to any grid.
Microgrid’s stability is highly affected by lo
ads variations and controller parameters. Careful selection of voltage and current controller parameters is very vital in microgrid stability and performance. Different strategy in choosing these parameters have been proposed in the literature [11] - [13]. In this paper, pole placement is implemented in order to stabilize the system and find controller parameters. The system stability over wide range of load variations is
examined. The only problem is that microgrid’s
performance is not that good in this method. Therefore, another method is proposed to enhance the microgrid performance besides stabilizing it. This method works base on regulation of power, voltage and frequency. The paper is organized as follows; Section II gives the system, inverter and controller models, Section III gives a brief description of PSO. Fitness function is presented in Section IV. Simulation result are shown is Section V, and finally section VI concludes the paper. II.
M
ICROGRID
M
ODEL
A.
Inverter model
SRPioneers
© Science and Research Pioneers Institute
4th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE2016), 4-5 February 2016, Dubai, Emirates
Voltage Source Inverters (VSI) are commonly used to
connect DGs to the grid [3]. VSI’s performance is similar
to electric machines witch regulate voltage and frequency simultaneously [14]. DG structure is shown in Fig. 1. The system model is obtained from [3] and [15]. The system is divided in two parts. The first part is output LC filter and coupling inductance to reject high frequency disturbance. The second part consists of droop controller, voltage controller, and current controller. To share active and reactive power among inverters, a droop is introduced in the voltage and frequency magnitude. In (1),
n
is nominal frequency and V
n
is nominal voltage.
p
m
and
q
n
are droop for voltage and frequency multiplied by active and reactive power. The instantaneous active and reactive power are passed through a low-pass filter to obtain P and Q, as shown in (2) and (3).
p(t)
( )
cod od oq oqc
P i i s
(2)
( )
( )
cod oq oq od cq t
Q i i s
(3) By linearizing the above formula, small signal power controller model can be written in state space form as shown in (4).
p
A
And
p
B
are given in (5).
ldqQ Q p p odq P P odq
i A B ii
(4)
000 00 0
c pcc oq c od c oq c od pc od c oq c od c oq
A I I V V B I I V V
(5) The algebraic equations for voltage, frequency, and current controller are as follows:
Figure 1. Distributed generation structure
Output of voltage controller is obtained by a PI controller and as follows:
* *
,
qd q pqd d ld q lq
d d V n f mdt dt d d i i i I dt dt
(7) Therefore, for voltage, frequency and current controller:
1 2
0
V dq dq p f
E E
(8)
*1 2
0
ldqdq dq c dq c odqodq
i B i B V i
(9) Where
1 2
0 1 0,0 0 1
q p
n E E m
1 2
1 0 1 0 0 0 0 0,0 1 0 1 0 0 0 0
c c
B B
Equations for LC filter and coupling inductance are given in (10). State space combination of droop controller, voltage and current controller, and LCL filter gives the microgrid state space model. This model is obtained from (1) to (10), and shown in (11), (12). A
Inv
, B
Inv
and B
u
are shown in (13) and (14).
B.
Network model
The test network has three buses and two lines, and is shown in Fig. 2. Line equations and its state space form are given in (15) and (16), respectively. Matrices A
NET
, B
1NET
, and B
2NET
are shown in (17).
n pn q
m P V V n Q
(1)
*** * ** * *
( ) ( )( ) ( )( ) ( )( ) ( )
d pv q iv qq pv p iv pd pc d ld ic d ld n f lq od q pc q lq ic q lq n f ld oq
i K V n q K V n q dt i K V m p K V m p dt V K i i K i i dt L i V V K i i K i i dt L i V
(6)
z
L
f
C
f
R
f
LcRc
l o a d
inverter
n
w
v
od
v
oq
i
ld
i
lq
V
d*
V
q*
V
f
-m
p
-n
q
pq
Droop
controlVF controller Current controller
PQ
dq to abc
L
i
o
i
o
v
SRPioneers
© Science and Research Pioneers Institute
4th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE2016), 4-5 February 2016, Dubai, Emirates
C.
Load model
Though there are several types of load in the grid, general RL type of load is considered in this study. Loads equations are given is (17) and (18).
D.
Microgrid model
Combining inverter model, network model, and load model gives the whole model for the microgrid. State space equation for the microgrid is given in (19). A virtual resistor is used to define buses voltage. It is assumed to be large enough such that its introduction would have minimum effect on system dynamic. More detail about the virtual resistor and the microgrid can be found in [3].
1 11 11 11 11 11 1
f ld ld lq id od f f f lq f lq ld iq oq f f f od oq ld od f f oqod lq oq f f od cod oq od bd c c coq coq od oq bqc c c
Rdii i v vdt L L Ldi Ri i v vdt L L Ldvv i idt C C dvv i idt C C di Ri i v vdt L L Ldi Ri i v vdt L L L
(10)
V inv invi invi invi bDQi f
x A x B v Bu
(11)
i i i dqi dqildqi odqi odqi
Xinv Q pi v i
(12)
1 11 1
lineDi lineilineDi lineQi bDj bDk linei linei lineilineQi lineilineQi lineDi bQj bQk linei linei linei
di r i i v vdt L L Ldi r i i v vdt L L L
(15)
.12
lineDQ NET lineDQ NET bDQ NET
i A i B B
(16) III.
P
ARTICLE
S
WARM
O
PTIMIZATION
Particle Swarm Optimization (PSO) algorithm was presented by kennedy and Eberhart in 1995[16]. This algorithm is inspired by a group of fish/birds in finding food. PSO is an evolutionary iterative algorithm which search the space to find optimal value. Each particle movement incorporates three terms. First term is random movement, second is movement toward its best experience, and finally, movement toward the best global experience.
0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 00 0 0 0 0 00
c c oq c od c oq c od c c od c oq c od c oqq pq pv iv p pv ivq pv pc pc iv pc f ic p lq n f f f f p pv pc p ld inv
I I V V I I V V nmn K K m K K n K K K K K R K m i L L L Lm K K m i A L
0 0 0 0 0 01 10 0 0 0 0 0 0 01 10 0 0 0 0 0 0 010 0 0 0 0 0 0 010 0 0 0 0 0 0 0
pc iv pc f icn f f f f p oq f f p od f f c p oqc cc p od c c
K K K R K L L Lm V C C m V C C Rm i L L Rm i L L
(13)
12 2
10 0 0 00 0 1 0 0 0 0 0 0 0 0,1 0 0 0 1 00 0 0 0
T T pvcinv u pv lq ld oq od oq od c
K L B B K i i V V i i L
(14)
1 2 1 11 2 1 146 46
inv inv DG DGc inv NET inv load MG NET DG DGc NET pw NET NET NET NET load load DG DGc load pw load NET load load load
A B M C B M B M A B M C B C A B M B M B M C B C B M A B M
1 2 3 1 2 336 36 36,22 121 2 36 361 36
0 0 1 0; ; ,0 0 0 10 0 0 0
inv inv inv inv MG u u u DGc pw p DGc DGc DGc DGc
A diag A A A B B B B C C m C diag C C C
(20)
SRPioneers
© Science and Research Pioneers Institute
4th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE2016), 4-5 February 2016, Dubai, Emirates
1011012022021 11 11 11 11 1 12 21 12 21
0 00 00 00 00 0 0 00 0 0 00 0 0 00 0 0 02
r line Lliner line Lline NET r line Lliner line Lline L Lline line L Lline line NET L Lline line L Lline linelineQ
A B NET
I B
122
lineDlineQlineD
I I I
(17)
.12
loadDQ load loadDQ load bDQload
i A i B B
(18)
10,10
loadiloadi loadiloadi loadiloadiloadiloadi
R L L A B R L L
(19) In this paper, this algorithm is chosen to find optimum value of fitness function since it is easy to implement. It
doesn’t use complex function and its answer is very
accurate. Also, previous experience on microgrids [17] - [18] and microgrid feature inspired the authors to apply this algorithm. More detail about the way PSO works are presented in [18]. Briefly, PSO is implemented as follows to find optimum values: 1. Specify PSO parameters; number of particles,
iteration and inertia constant ω and particle u
pper and lower limits. 2. Initialization; PSO initialized itself randomly. 3. a. For the first objective function: PSO presents PI parameters and runs MATLAB/Mfile to calculates A
mg
and its Eigen values. b. For the Second objective function PSO calls Simulink and provides it with PI coefficients and runs it; system states are given to PSO to calculate the objective function 4. Iteration increases and particles move to new locations. 5. PSO calls Mfile/Simulink and provides Mfile/Simulink with new PI coefficients. 6. If this iteration result in a better answer, save the answer as the best answer 7. If iteration does not reach its maximum value go to 4. 8. Exit and print the objective function as well as the best PI coefficients. IV.
O
BJECTIVE
F
UNCTION
A system is stable if its Eigen values places on the left hand side of the rectangular coordinate. Therefore, first Objective Function (OF) is to define controller parameters
such that the system’s Eigen values place on the left hand
side. It is formulated as follows:
1
( ( ( )))
mg
OF max real eigen A
(20) This OF
1
should be minimized. If it gets negative, it means the system is stable. The second OF is to minimize frequency and voltage fluctuation, and is as follows:
2 ,01
100*
DG
n j reference j j reference j
OF V V f f dt
(21) V
reference
and f
reference
are voltage and frequency reference which are obtained from (1). PSO should calculate controller parameters so that frequency and voltage deviation from their reference values decrease. V.
R
ESULTS
A.
Case Study
The case study is a three bus microgrid including two loads, a network and three inverters, and is shown in Fig. 2. Microgrid parameters are given in table I. Inverters are equally rated and each 10 KVA. Nominal frequency and voltage (phase) is 50 Hz and 220 V RMS. m
p
and n
q
are chosen so that frequency and three phase voltage deviation to be at most 0.5 Hz and 20 V, respectively. In this study, first controller (Droop) is just implemented and secondary controller, which eliminate error caused by Droop controller, is not applied. Moreover, in design of inverter LC filter, capacitor and inductor is chosen to remove high frequency harmonics. Also, to decrease inductor voltage drop in inverter output, it is chosen smaller in comparison to the capacitor.
TABLE I: System Parameters
Parameter Value Parameter Value
m
p
2.09e-4 r
f
0.1
Ω
n
q
1.3e-3 L
c
0.35mH C
f
50
μ
F r
lc
0.03
Ω
L
f
1.35mH F 0.75
SRPioneers
© Science and Research Pioneers Institute
4th International Conference on Electrical, Computer, Mechanical and Mechatronics Engineering (ICE2016), 4-5 February 2016, Dubai, Emirates Figure 2. Distributed Generation Structure Figure 3. First Objective Function Value Figure 4. Controller Parameters
B.
Pole Placement
State Space model of the microgrid is implemented in MATLAB/Mfile. Each time PSO gives some coefficient for controller and runs the state space model to find A
mg
and its Eigen values. In each iteration, PSO moves to find optimum value of the OF. In this study, PSO population and iteration are chosen to be 50 and 100, respectively. At first, loads are chosen to be 3 kW and 1.5 kVar (total 6kW and 3 kVar). First OF value is depicted in Fig. 3. At first runs,
max(real (eign(A
mg
)))
was positive, which means the system was unstable at first guess of the PSO. After several running, the system becomes stable. The controller parameters are shown in Fig.4. At the end, the maximum real value of the Eigen values is -4.37. PI parameters for K
pv
, K
iv
, K
pc
, and K
ic
are 1.98, 34.26, 27.56, and 470.36,
Figure 5. Eigen Values for the Given Loads Figure 6. Frequency Fluctuations Using Pole Placement Technique
respectively. Eigen value for the first load is depicted in Fig. 5. In this figure, those Eigen values which are close to the srcin are only depicted. Other Eigen values are far from srcin and have no effect on the system. At times 0.3S and 0.6S loads are changed to 2kW and 1kVar (total 4kW and 2kVar) and 6kW and 3kVar (total 12kW and 6kVar), respectively. Again, Eigen values for the second and third load are depicted on one figure (Fig. 5) to give a better comparison. Fig. 5 shows the system is still stable for the second and third load. Frequency variations are shown in Fig. 6. Although the system is stable for these loads, the frequency fluctuations is very high. It can be improved if the regulation method changes.
C.
Proposed method
Another objective function is suggested to improve microgrid performance. In this method, PSO gives controller parameters, and runs MATLAB/Simulink. After each simulation, PSO obtains frequency and voltage fluctuations to calculate objective function. In this scenario, PSO is run 40 times and moves its particles to find better performance. The controller parameter, for K
pv
, K
iv
, K
pc
, and K
ic
are 0.36, 1.30, 18.97, and 42.36, respectively. Fig. 7 shows frequency deviation using second method. To have a better comparison, frequency deviation for the first method is also depicted on this figure. Fig. 7 suggests that second strategy give better performance for the microgrid in comparison to the first method besides the system is stable. Load variations is depicted in Fig. 8 as well.
DG 1DG 2DG 2LOAD 2LOAD 2
r=0.23
Ω
x=0.1
Ω
r=0.35
Ω
x=0.58
Ω
i
o1
i
o2
i
o3
V
b1
V
b2
V
b3

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