research article mathematical model and stability...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 195038, 7 pageshttp://dx.doi.org/10.1155/2013/195038

Research ArticleMathematical Model and Stability Analysis of Inverter-BasedDistributed Generator

Alireza Khadem Abbasi and Mohd Wazir Mustafa

Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia

Correspondence should be addressed to Alireza Khadem Abbasi; alireza [email protected]

Received 23 December 2012; Accepted 18 February 2013

Academic Editor: Vu Phat

Copyright © 2013 A. Khadem Abbasi and M. W. Mustafa. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

This paper presents a mathematical (small-signal) model of an electronically interfaced distributed generator (DG) by consideringthe effect of voltage and frequency variations of the prime source. Dynamic equations are found by linearization about an operatingpoint. In this study, the dynamic of DC part of the interface is included in the model. The stability analysis shows with properselection of system parameters; the system is stable during steady-state and dynamic situations, and oscillatory modes are welldamped. The proposed model is useful to study stability analysis of a standalone DG or a Microgrid.

1. IntroductionDistributed generation (DG) systems have been expectedto be an important electric power supply system for nextgeneration. DGs are able to be installed near the loads,so they can increase the power quality and reliability ofelectricity delivered to sensitive loads. Some of the DGtechnologies require a power electronics interface in orderto convert the energy into the grid compatible AC power.These interface devices make the sources more flexible intheir operation and control compared to the conventionalelectrical machines. However, due to their negligible physicalinertia, they also make the system potentially susceptible tooscillation resulting from network disturbances [1].

The coordinated operation and control of DGs togetherwith loads and storage devices are central to the concept ofmicrogrid [2]. The analysis of the dynamic stability of con-ventional power systems is well established, but formicrogridthere is a need to investigate how circuit and control featuresgive rise to particular oscillatory modes, and which of thesehave poor damping. Finding an exact mathematical model byconsidering DGs and their control is needed to investigatedynamic stability of the microgrid under transient eventssuch as islanding from main grid and small-signal deviationlike slow changing in load.

Reference [3] presented a small-signal model for inverterin stand-alone AC supply system by using only the droop

controller’s variables as state variables of the DG model. Anaveraged current source model has been suggested for theconverter in [4]. In this investigation, the high frequencyconverter current dynamics have been neglected in order tofocus exclusively on the dynamics and control of the islandedmicrogrid. Regardless of the type of the DG, an equiva-lent RLC circuit including output filter and transformersimpedances has been modeled as a power circuit of DG inmost of the literatures [1, 5–8]. In these cases, output currentsand voltages are considered as state variables. References[1, 6, 7] by adding controller equations to the DG systemmade an accurate model for small signal stability analysis ofa microgrid.

In most articles, DC part of the DG (DC filter’s voltageand current) is not considered in dynamic studies, or it isassumed to be a constant value [6], so its effect is neglectedin stability analysis. In the reference [8], the input DC voltagevariation of the inverter has been represented as the externalperturbation in the open-loop model of the DG, but ithas been set as a constant during control loop design andfrequency-domain analysis. The DC voltage has been relatedto the input and output equations of electronically interface ofthe DG by using switching function of rectifier and inverterin [5]. Moreover, two types of models have been presentedin this paper for the prime source of the DG. While theproposedmodels have been used in steady-state and load flow

2 Mathematical Problems in Engineering

Rect

ifier

Inve

rter

Pow

erm

easu

rem

ent

Volta

ge an

dcu

rren

t con

trolle

r

PWM LPF

𝐿𝐶𝐺2 𝐿𝐺2𝑅𝐺2

𝑑𝐺2 𝑄𝑃

𝑣∗𝑞𝑖

𝐸𝐺2𝑉𝐺2

𝐶𝐺2

dc

𝐿𝐺2dc

𝜔

𝑉dc

𝐼𝑑 𝐼dc 𝐼𝑖

Figure 1: Single line equivalent of DG.

analysis, but their linear forms can be used also in dynamicstudies.

The objective of this paper is to find a comprehensivedynamic model of DG, including the prime source, powerelectronically interfaces, output filter, and controller. Theproposed model represents all components of the DG in adq0 reference frame, thus it ensures any application such assteady-state and dynamic analysis that meets requirementsand constraints of both AC and DC parts of the system.

The dynamic stability of the DG is investigated by thesmall signal and step response analysis. In this paper, the DGis connected only to the static load, but it can be extended intothe network or typical microgrid for more applications.

2. System Studied

2.1. Power Circuit. A simplified single-line diagram of thestudied DG is illustrated in Figure 1. DG is included of athree-phase source, a three-phase rectifier, an inverter, and athree-phase static load on the output bus.The DG source canbe representatives of wind turbine or microturbine generatorfor medium and low voltage levels, respectively. Regardlessof the power measurement and control loops, the systemshown in Figure 1 can be studied as a power circuit of DGin microgrid stability analysis.

2.2. Control System. As shown in Figure 1, the control of DGincluded power, voltage, and current controllers. The powercontrol has been achieved by applying droop control forboth active and reactive powers. The droop control for anelectronically interfaced DG has the role of the governor fora synchronously generator. Where, by an increase in load,the reference frequency is decreased. In the similar way, thereactive power is controlled by a droop characteristic in thevoltage magnitude.

In this study, the sin-triangle modulation strategy is usedto prepare applied voltage in the converter. The next parts ofthe control system are the outer voltage and inner currentcontrol loops, which are designed to reject high-frequency

disturbances and provide sufficient damping for the outputfilter by using the proportional integral (PI) compensators.

3. System Equations

3.1. Power Circuit. The dynamic model of AC side of DG inthree-phase abc frame is obtained from

V𝑖𝑎𝑏𝑐

= 𝑅𝐺2𝑖

𝐺2

𝑖𝑎𝑏𝑐

+ 𝐿𝐺2𝜌𝑖

𝐺2

𝑖𝑎𝑏𝑐

+ V𝐺2

𝑎𝑏𝑐, (1)

where 𝑖𝐺2𝑖𝑎𝑏𝑐

, V𝐺2𝑎𝑏𝑐

, and V𝑖𝑎𝑏𝑐

are vectors of the instantaneousamounts of the output currents, bus voltages, and the con-verter output voltages, respectively. Then (1) is transformedto the 𝑑-𝑞 reference frame of DG that rotates at frequency𝜔𝐺2

related to the angular velocity of the voltage space vectorof the bus. The transferred equations are

𝐸

𝐺2

1𝜌𝑖

𝐺2

𝑞𝑑𝑖

= 𝐴

𝐺2

1𝑖

𝐺2

𝑞𝑑𝑖

+ V𝐺2

𝑞𝑑𝑖

− V𝐺2

𝑞𝑑, (2)

where 𝜌 is the operator 𝑑/𝑑𝑡.

3.2. DC Interface Equations. The average voltage and currentof capacitor and inductance of dc filter will change with timeduring transient. For the rectifier under operation withoutphase delay and with commutating inductance, we have [9]

𝑉

𝐺2

𝑑=

3√3

𝜋

𝐸

𝐺2

𝑞−

3

𝜋

𝑋𝐿𝐶𝐺2

𝜔𝑏

𝜔𝐸𝐼

𝐺2

𝑑−

2

𝜔𝑏

𝑋𝐿𝑐𝐺2

𝜌𝐼

𝐺2

𝑑. (3)

We also have

𝑋𝐿𝑑𝑐𝐺2

𝜌𝐼

𝐺2

𝑑= 𝑉

𝐺2

𝑑− 𝑉

𝐺2

𝑑𝑐. (4)

From (3) and (4),

𝑉

𝐺2

𝑑𝑐=

3√3

𝜋

𝐸

𝐺2

𝑞−

3

𝜋

𝑋𝐿𝑐𝐺2

𝜔𝑏

𝜔𝐸𝐼

𝐺2

𝑑

−

1

𝜔𝑏

(𝑋𝐿𝑑𝑐𝐺2

+ 2𝑋𝐿𝑐𝐺2

) 𝜌𝐼

𝐺2

𝑑.

(5)

Equation (5) expresses the relationship between dynamicparts of the dc filter and DG source voltage. The 𝜔

𝐸is the

frequency of the source, and 𝐸𝐺2𝑞

is the network side voltage’s𝑞 component. The voltage is wrote in the form of

𝜌𝑉

𝐺2

𝑑𝑐= 𝜔𝑏𝑋𝑐𝑑𝑐𝐺2

(𝐼

𝐺2

𝑑− 𝐼

𝐺2

𝑑𝑐) . (6)

Apart from the harmonics in the voltage waveform andconsidering sin-triangle modulation for inverter, voltagesbecome

V𝐺2

𝑞𝑖

=

𝑑𝐺2

2

𝑉

𝐺2

𝑑𝑐,

V𝐺2

𝑑𝑖

= 0,

(7)

where 𝑑𝐺2

is duty cycle.We will see that the 𝑞 component of converter voltage

is controlled by controller and V𝐺2𝑑𝑖

is considered to be zero.

Mathematical Problems in Engineering 3

𝑃𝜔𝑓

𝑠 + 𝜔𝑓

𝜔𝑓

𝑠 + 𝜔𝑓

��𝜔𝑛 − 𝑘𝑝��

𝜔𝐺2

𝑄�� 𝑣𝑛 − 𝑘𝑣�� 𝑣𝐺2𝑞𝑖

Figure 2: Inverter power controller.

𝑞𝐺

𝑞𝐺2

𝑣𝐺𝑞

𝑑𝐺2

𝑑𝐺

𝛿𝐺2

Figure 3: Common reference frame.

The instantaneous power balance between two sides of theinverter is established by

𝑉

𝐺2

𝑑𝑐𝐼

𝐺2

𝑑𝑐=

3

2

V𝐺2

𝑞𝑖𝑖

𝐺2

𝑞𝑖. (8)

According to (7), we have

𝐼

𝐺2

𝑑𝑐=

3𝑑𝐺2

4

𝑖

𝐺2

𝑞𝑖. (9)

Rewriting (6) with regard to (9),

𝜌𝑉

𝐺2

𝑑𝑐= 𝜔𝑏𝑋𝐶𝑑𝑐𝐺2

(𝐼

𝐺2

𝑑−

3𝑑𝐺2

4

𝑖

𝐺2

𝑞𝑖) . (10)

The dynamic equations of DC part are represented by (5) and(10).

3.3. Control System Equations. Figure 2 shows inverter powercontroller, where 𝑃 and 𝑄 are measured values of inverteractive and reactive output powers. Moreover, 𝑃 and 𝑄 arefiltered values of power after passing through the low-passfilter. As shown in Figure 3, the inverter output voltageV𝐺2𝑖

and inverter output frequency 𝜔𝐺2

are adjusted by thedroop controller characteristics described by (11) and (12),respectively, as follows:

𝜔𝐺2= 𝜔𝑛− 𝑘𝑝𝑃, (11)

V𝐺2∗

𝑞𝑖= V𝑛− 𝑘V

𝑄. (12)

Here, 𝜔𝑛and V𝑛represent the nominal frequency and 𝑞-axis

output voltage set points, respectively. V𝐺2∗𝑞𝑖

is the duty cycle(𝑑𝐺2).The power measurement outputs are given by (13) and

(14), where 𝜔𝑓is the cutoff frequency of the low-pass filter

and 𝑠 is the Laplace operator

𝑃 =

𝜔𝑓

𝑠 + 𝜔𝑓

𝑃, (13)

𝑄 =

𝜔𝑓

𝑠 + 𝜔𝑓

𝑄. (14)

3.4. Common Reference Frame Theory. All equations of DGhave been written in its reference frame. In power systemanalysis, it is necessary to relate all components’ referenceframes to a common frame. This common reference framewould be represented as constant voltages in the synchronousreference frame. In microgrid studies, DG with the biggestnominal power is considered as a common reference frame.

Figure 3 shows the common reference frame of a micro-grid by 𝑞 and 𝑑 axis rotating at the synchronous angularvelocity of𝜔

𝑒.The 𝑞 axis is in orientation of the voltage vector

of the reference DG, V𝐺𝑞. By assuming V𝐺2

𝑞on the 𝑞 axis of

the DG2 reference frame, 𝑞𝐺2, 𝛿𝐺2

is the angle between theDG2 reference frame and common reference frame. In orderto transfer variables in the DG2 reference frame to commonreference frame, the transformation matrix is defined as

𝑓

𝐺= 𝑇𝐺2𝑓

𝐺2, 𝑓

𝐺2= 𝑇

−1

𝐺2𝑓

𝐺, (15)

𝑇𝐺2=[

[

cos 𝛿0𝐺2

sin 𝛿0𝐺2

− sin 𝛿0𝐺2

cos 𝛿0𝐺2

]

]

, (16)

𝑇

−1

𝐺2=[

[

cos 𝛿0𝐺2

− sin 𝛿0𝐺2

sin 𝛿0𝐺2

cos 𝛿0𝐺2

]

]

. (17)

To connect an inverter to the whole system the outputvariables that have the connection with the network need tobe converted to the common reference frame. In this case theoutput variables are the output currents 𝑖𝐺2

𝑞𝑑𝑖and the network

side voltages V𝐺2𝑞𝑑. Because the dynamic of one DG is studied

in this paper, the 𝛿𝐺2

is zero, and transformationmatrix is theunit.

3.5. Small-Signal Model. By liberalization of nonlinear differ-ential equations around the operating steady-state point, thesmall-signal model of the system has been created [9].

For inverter equations, first output variables must betransferred to the common reference frame and then belinearized. By using (15) inverter output currents and voltagesin the common reference frame can be achieved as

𝑖

𝐺2

𝑞𝑑𝑖= 𝑇

−1

𝐺2𝑖

𝐺

𝑞𝑑𝑖,

V𝐺2

𝑞𝑑= 𝑇

−1

𝐺2V𝐺

𝑞𝑑.

(18)


Then with linearization technique, we have

Δ𝑖

𝐺2

𝑞𝑑𝑖= 𝑇

−1

𝐺2Δ𝑖

𝐺

𝑞𝑑𝑖+ 𝑖

0

𝛿𝐺2

Δ𝛿𝐺2,

ΔV𝐺2

𝑞𝑑= 𝑇

−1

𝐺2ΔV𝐺

𝑞𝑑+ V0

𝛿𝐺2

Δ𝛿𝐺2,

(19)

where

𝑖

0

𝛿𝐺2

= [−𝑖

𝐺2

𝑑𝑖0𝑖

𝐺2

𝑞𝑖0]

𝑇

,

V0

𝛿𝐺2

= [−V𝐺2

𝑑0V𝐺2𝑞0]

𝑇

.

(20)

The subscript 0 denotes steady-state quantities. The Δ𝛿𝐺2

could be rewritten as a function of output voltages in commonreference frame as follows

𝛿𝐺2= tan−1(

V𝐺𝑑

V𝐺𝑞

) . (21)

With linearization of (21),

Δ𝛿𝐺2= 𝑚𝑞𝛿𝐺2

ΔV𝐺

𝑞+ 𝑚𝑑𝛿𝐺2

ΔV𝐺

𝑑, (22)

where

𝑚𝑞𝛿𝐺2

= −

V𝐺𝑑0

V𝐺𝑞0

2+ V𝐺𝑑0

2,

𝑚𝑑𝛿𝐺2

=

V𝐺𝑞0

V𝐺𝑞0

2+ V𝐺𝑑0

2.

(23)

For more information, see [3, 7]. Furthermore, the linearequation of the inverter output voltage could be written as

ΔV𝐺2

𝑞𝑖

=

𝑑𝐺20

2

Δ𝑉

𝐺2

𝑑𝑐+

𝑉

𝐺2

𝑑𝑐0

2

Δ𝑑𝐺2.

(24)

By replacing (7) and (18) into the inverter equations andlinearization with considering (19), (22), and (24) the linearequations of the DG have been arranged in the basic style ofthe linear differential equations as

𝜌Δ𝑋

DG= 𝐴

DGΔ𝑋

DG+ 𝐵

DGΔ𝑈

DG, (25)

Δ𝑋

DG= [Δ𝑋

𝐺2Δ𝑋

𝐺2

𝐶] ,

Δ𝑈

DG= [Δ𝑈

𝐺2Δ𝑈

𝐺2

𝐶] ,

(26)

where Δ𝑈DG and Δ𝑋

DG are control and state variablesrespectively. The state matrix of the system is expressed by𝐴

DG that is used in dynamic analysis of the DG.The output vector of the system is defined as a linear

combination of control and state variables in the form of

Δ𝑌

DG= 𝐶Δ𝑋

DG+ 𝐷Δ𝑈

DG. (27)

In the power system small signal stability analysis, the currentinjection into the network from the device is considered as

the output signal [10]. For the DG, the output current is oneof the state variables, so we have

Δ𝑖

𝐺

𝑞𝑑𝑖= 𝐶

DGΔ𝑋

DG. (28)

All matrices are available in Appendix A in detail.Finally, the transfer function is defined by

Δ𝑌

DG

Δ𝑈

DG = 𝐶

DG(𝑆𝐼 − 𝐴

DG)

−1

𝐵

DG+ 𝐷

DG.

(29)

3.6. Eigenvalue Analysis. In power systems small-signal sta-bility analysis is aimed to determine the properties ofoperation parameter variations that are independent fromdisturbance intensity. Eigenvalue analysis is used to show theinformation of different stability modes for power systemsmall-signal stability problems. A system is stable when allof its modes are stable. Furthermore, it is required that alloscillations are well and quickly damped. This technique isdeclared in [11] in details.

In this paper, the participation factors and eigenvalues ofthe statematrix are computed and analyzed. Eigenvalues withnegative real parts show that the system is stable. the statevariable with the highest normalized participation factor isthe best choice for feedback signal if it may be a measurablephysical variable.

4. Case Study and Results

To study small-signal stability of the model, a 10 kVA DGconnected to a 9 kW static load is considered as the casestudy. All DG information is taken from [1]. The outputimpedance of DG is 0.1 + 𝑗0.4239Ω. The cutoff frequency 𝜔

𝑓

is considered 31.41 Rad/s.The rectifier’s input inductance 𝑋

𝐿𝑐

is used to amend theinput current. The inductor with low value may not restrictthe peak current, and its high amount may reduce the inputvoltage [10]. Hence, a 1.06 (mH) inductor is chosen as anoptimized value.

The DC filter is involved in a small series inductance,𝑋𝐿𝑑𝑐

, (here 0.03mH) and a shunt capacitance,𝑋𝑐𝑑𝑐

, to smooththe diode rectifier output. With regard to [10], a 165 𝜇F/kWcapacitance is selected in this study. For DG, we have

Power = 10 kW. (30)

Hence

𝐶 = 165 ∗ 100 = 1650 𝜇F. (31)

The steady-state values of the system are calculated by loadflow program.

In order to realize the sensitivity and dynamic behaviour,eigenvalues trajectory plots are drawn as a function ofsome system parameters. Figure 4 shows the locations ofeigenvalues as a function of variation of 𝑘

𝑝from 0.00000004

to 0.000094 with the rate of 0.000001. In this case, 𝑘V is equalto 0.0013.

Figure 5 shows the loci of eigenvalues corresponding tovariation of 𝑘V from 0.00003 to 0.1 with the rate of 0.001. 𝑘

𝑝is

considered to be 0.000015.


0 50

0200400600800

54

2

3

1

6

−400 −350 −300 −250 −200 −150 −100 −50−800

−600

−400

−200

Figure 4: Loci of eigenvalues according to kp changing.

0 20

0200400600800

5

1

24

3

6

−160 −140 −120 −100 −80 −60 −40 −20−800

−600

−400

−200

Figure 5: Loci of eigenvalues according to kv changing.

0

0200400600800

1

2

56

3

4

−180 −160 −140 −120 −100 −80 −60 −40 −20−800

−600

−400

−200

Figure 6: Loci of eigenvalues according to wf changing.

Trajectory plots show instability in modes of 1, 2, 3, and 4if the 𝑘

𝑝and 𝑘V increase to big values.

With regard to 𝑘𝑝and 𝑘V as constant values of 0.000015

and 0.0013, Figure 6 shows the root locus plot for the modelas a function of the filter cutoff frequency 𝜔

𝑓from 37 to 25

with the rate of −0.5. It can be seen that eigenvalues 5 and 6move to right side of the plot.

Results are used to adjust the gains of the power con-troller. With optimal values of controller gains and appropri-ate selection of the DG parameters, all oscillatory modes arewell damped, and the proposed model is stable.

By using QR-decomposition method in the MATLABsoftware environment, the eigenvectors and eigenvalues ofthe state matrix have been computed in operating point.The DG system along with its controller is described by sixeigenvalues: 𝜆

1,2= −44.90 ± 698.37𝑖, 𝜆

3,4= −31.14 ±

244.66𝑖, 𝜆5= −156.58, and 𝜆

6= −32.62.

Table 1: Participation factors.

𝜆1

𝜆2

𝜆3

𝜆4

𝜆5

𝜆6

Δ𝑖

𝐺2

𝑞𝑖0.2428 0.2428 0.2001 0.2001 0.0781 0.0

Δ𝑖

𝐺2

𝑑𝑖0.0497 0.0497 0.3646 0.3646 0.0829 0.0

ΔV𝐺2𝑑𝑐

0.4458 0.4458 0.0528 0.0528 0.0 0.0Δ𝐼

𝐺2

𝑑0.2522 0.2522 0.2025 0.2025 0.0463 0.0

Δ𝜔𝐺2

0.0094 0.0094 0.1782 0.1782 0.7923 0.0005Δ𝑑𝐺2

0.0001 0.0001 0.0019 0.0019 0.0004 0.9995

Linear simulation results

Time (s)A

mpl

itude

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1

1.5

2

2.5

𝑖𝑞𝑖𝑑

𝑋: 1.001𝑌: 0.3656𝑋: 1.003𝑌: 0.0831

Figure 7: Step response of current to the source voltage.

Am

plitu

de 0

0.02

0.04 Linear simulation results

Time (s)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

𝑖𝑞𝑖𝑑

−0.08

−0.06

−0.04

−0.02

𝑋: 1.003𝑌: −0.002899𝑋: 1.003𝑌: −0.01276

Figure 8: Step response of current to the source angular velocity.

The normalized participation factors of the dynamicmodel are represented in Table 1. It is a useful indication ofthe role of a state on the mode. As seen in oscillatory modesof 1, 2, 3, and 4, the state variables of DC part have the mainrole.

Figures 7 and 8 show step responses of the output currentaccording to the source voltage and frequency. The smallsteady-state error in 𝑖

𝑞𝑖

is caused by the effects of them onDC voltage and DC current. The variation in 𝑖

𝑑𝑖

is due tocoupling between 𝑞 and 𝑑 components of current. By controlof rectifier and making a constant DC voltage this effects willbe removed. Error specifications also could be adjusted byadding current and voltage control loops.


5. Conclusion

The mathematical model of the inverter-based DG has beenproposed in this paper. The proposed model has includedpower and control circuits of the DG. The dynamics of theDC filter and the effects of the prime source have been alsoconsidered in the DG model. By using eigenvalue analysis,the stability of proposed model has been studied in detail.This study presented that all oscillatory modes are welldamped with suitable selection of system parameters. Sincethe proposed model is stable, it can be used in the dynamicand steady-state study of a microgrid.

Appendices

Linear equations of the DG.

A. Power Circuit

A.1. AC Part. By substituting (18) into the (2) and lineariza-tion

𝐸

𝐺2

1Δ𝑖

𝐺

𝑞𝑑𝑖

= 𝐴

𝐺2

1Δ𝑖

𝐺2

𝑞𝑑𝑖

+ 𝐵

𝐺2

1ΔV𝐺2

𝑞𝑑𝑖

+ 𝐵

𝐺2

2ΔV𝐺

𝑞𝑑+ 𝐵

𝐺2

3Δ𝜔𝐺2,

(A.1)

where

𝐸

𝐺2

1=

[

[

[

[

𝑋𝑙𝐺2

𝜔𝑏

0

0

𝑋𝑙𝐺2

𝜔𝑏

]

]

]

]

, 𝐴

𝐺2

1=

[

[

[

[

−𝑅𝐺2

−

𝜔𝐺20

𝑋𝑙𝐺2

𝜔𝑏

𝜔𝐺20

𝑋𝑙𝐺2

𝜔𝑏

−𝑅𝐺2

]

]

]

]

,

𝐵

𝐺2

1= 𝑇𝐺2[

1

0

] ,

𝐵

𝐺2

2= [(𝑇

𝐺2𝐴

𝐺2

1𝑖

0

𝛿𝐺2

− 𝑇𝐺2V0

𝛿𝐺2

)𝑀

𝐺2

𝑞𝑑− 𝐼] ,

𝐵

𝐺2

3= −𝑇𝐺2𝐸

𝐺2

1𝑖

0

𝛿𝐺2

, 𝑀

𝐺2

𝑞𝑑= [𝑚𝑞

𝛿𝐺2

𝑚𝑑𝛿𝐺2

] ,

Δ𝜔𝐺2= Δ

𝛿𝐺2.

(A.2)

By (23) and ΔV𝐺2𝑑𝑖

= 0

𝐸

𝐺2

1Δ𝑖

𝐺

𝑞𝑑𝑖

= 𝐴

𝐺2

1Δ𝑖

𝐺

𝑞𝑑𝑖

+ 𝐵

𝐺2

1

𝑑𝐺20

2

Δ𝑉

𝐺2

𝑑𝑐+ 𝐵

𝐺2

1

𝑉

𝐺2

𝑑𝑐0

2

Δ𝑑𝐺2

+ 𝐵

𝐺2

2ΔV𝐺

𝑞𝑑+ 𝐵

𝐺2

3Δ𝜔𝐺2,

𝐸

𝐺2

1Δ𝑖

𝐺

𝑞𝑑𝑖

= 𝐴

𝐺2

1Δ𝑖

𝐺

𝑞𝑑𝑖

+ 𝐹

𝐺2

1[

[

Δ𝑉

𝐺2

𝑑𝑐

Δ𝐼

𝐺2

𝑑

]

]

+ 𝐵

𝐺2

7Δ𝑑𝐺2

+ 𝐵

𝐺2

2ΔV𝐺

𝑞𝑑+ 𝐵

𝐺2

3Δ𝜔𝐺2,

𝐹

𝐺2

1= 𝐵

𝐺2

1

𝑑𝐺20

2

[1 0] , 𝐵

𝐺2

7= 𝐵

𝐺2

1

𝑉

𝐺2

𝑑𝑐0

2

.

(A.3)

A.2. DC Part. By rearranging (5) and (10) in the frame lineardifferential equation

𝐸

𝐺2

2𝜌[

[

Δ𝑉

𝐺2

𝑑𝑐

Δ𝐼

𝐺2

𝑑

]

]

= 𝐴

𝐺2

2[

[

Δ𝑉

𝐺2

𝑑𝑐

Δ𝐼

𝐺2

𝑑

]

]

+ 𝐴

𝐺2

3Δ𝑖

𝐺2

𝑞𝑑𝑖

+ 𝐵

𝐺2

4Δ𝑑𝐺2+ 𝐵

𝐺2

5[

[

Δ𝐸

𝐺2

Δ𝜔𝐸

]

]

,

𝐸

𝐺2

2= [

1 0

0 1

] ,

𝐴

𝐺2

2=

[

[

[

[

[

[

3𝑑𝐺20

𝑖

𝐺2

𝑞𝑖0𝑋𝐶𝑑𝑐𝐺2

𝜔𝑏

4𝑉

𝐺2

𝑑𝑐0

𝑋𝐶𝑑𝑐𝐺2

𝜔𝑏

−

1

𝑍2

−

3𝑋𝐿𝑐𝐺2

𝜔𝐸0

𝜋𝑍2𝜔𝑏

]

]

]

]

]

]

,

𝐴

𝐺2

3=

[

[

[

−

3𝑑𝐺20

𝑋𝐶𝑑𝑐𝐺2

𝜔𝑏

4

0

0 0

]

]

]

,

𝐵

𝐺2

4=

[

[

[

[

−

3𝑖

𝐺2

𝑞𝑖0𝑋𝐶𝑑𝑐𝐺2

𝜔𝑏

2𝑉

𝐺2

𝑑𝑐0

0

]

]

]

]

,

𝐵

𝐺2

5=

[

[

[

[

0 0

3√3

𝜋𝑍2

−

3𝐼

𝐺2

𝑑0

𝑋𝐿𝑐𝐺2

𝜋𝑍2𝜔𝑏

]

]

]

]

,

𝑍2=

1

𝜔𝑏

(𝑋𝐿𝑑𝑐

+ 2𝑋𝐿𝑐

) .

(A.4)

By replacing (19) and (22) into the (A.4),

𝐸

𝐺2

2𝜌[

[

Δ𝑉

𝐺2

𝑑𝑐

Δ𝐼

𝐺2

𝑑

]

]

= 𝐴

𝐺2

2[

[

Δ𝑉

𝐺2

𝑑𝑐

Δ𝐼

𝐺2

𝑑

]

]

+ 𝐴

𝐺2

4Δ𝑖

𝐺

𝑞𝑑𝑖

+ 𝐵

𝐺2

4Δ𝑑𝐺2

+ 𝐵

𝐺2

5[

[

Δ𝐸

𝐺2

Δ𝜔𝐸

]

]

+ 𝐵

𝐺2

6ΔV𝐺

𝑞𝑑,

𝐴

𝐺2

4= 𝐴

𝐺2

3𝑇𝐺2

−1, 𝐵

𝐺2

6= 𝐴

𝐺2

3𝑖

0

𝛿𝐺2

𝑀

𝐺2

𝑞𝑑.

(A.5)


A.3. Differential Equation of Power Circuit

𝐸

𝐺2Δ𝑋

𝐺2

= 𝐴

𝐺2Δ𝑋

𝐺2+ 𝐵

𝐺2Δ𝑈

𝐺2+ 𝐶

𝐺2Δ𝑋

𝐺2

𝐶,

Δ𝑋

𝐺2= [Δ𝑖

𝐺2

𝑞𝑖Δ𝑖

𝐺2

𝑑𝑖ΔV𝐺2𝑑𝑐

Δ𝐼

𝐺2

𝑑] ,

𝐸

𝐺2=[

[

𝐸

𝐺2

10

0 𝐸

𝐺2

2

]

]

, 𝐵

𝐺2=[

[

𝐵

𝐺2

20

𝐵

𝐺2

6𝐵

𝐺2

5

]

]

,

𝐶

𝐺2=[

[

𝐵

𝐺2

3𝐵

𝐺2

7

0 𝐵

𝐺2

4

]

]

, 𝐴

𝐺2=[

[

𝐴

𝐺2

1𝐹

𝐺2

1

𝐴

𝐺2

4𝐴

𝐺2

2

]

]

.

(A.6)

B. Control Circuit

B.1. Active Power Droop Controller. Linear form of (11) and(13),

Δ𝜔𝐺2= −𝑘𝑝Δ𝑃, (B.1)

Δ𝑃 =

𝜔𝑓

𝑠 + 𝜔𝑓

Δ𝑃. (B.2)

By combining (B.1) and (B.2),𝑠Δ𝜔𝐺2= −𝜔𝑓Δ𝜔𝐺2− 𝑘𝑝𝜔𝑓Δ𝑃. (B.3)

B.2. Reactive Power Droop Controller. In the same way of theactive power control, from (12) and (14) we have

𝑠Δ𝑑𝐺2= −𝜔𝑓Δ𝑑𝐺2− 𝑘V𝜔𝑓Δ𝑄. (B.4)

B.3. Differential Equation of Control Circuit

Δ𝑋

𝐺2

𝐶= 𝐴

𝐺2

𝐶Δ𝑋

𝐺2

𝐶+ 𝐵

𝐺2

𝐶0Δ𝑈

𝐺2

𝐶,

Δ𝑋

𝐺2

𝐶= [Δ𝜔𝐺2

Δ𝑑𝐺2] ,

Δ𝑈

𝐺2

𝐶= [Δ𝑃 Δ𝑄] , 𝐴

𝐺2

𝐶= [

−𝜔𝑓

0

0 −𝜔𝑓

] .

(B.5)

With writing the active and reactive power in terms of voltageand current

Δ𝑋

𝐺2

𝐶= 𝐴

𝐺2

𝐶Δ𝑋

𝐺2

𝐶+ 𝐶

𝐺2

𝐶Δ𝑋

𝐺2+ 𝐵

𝐺2

𝐶6ΔV𝐺

𝑞𝑑,

𝐶

𝐺2

𝐶= [𝐵

𝐺2

𝐶50] , 𝐵

𝐺2

𝐶5= 𝐵

𝐺2

𝐶1𝐵

𝐺2

𝐶2𝑇

−1

𝐺2,

𝐵

𝐺2

𝐶6= 𝐵

𝐺2

𝐶1𝐵

𝐺2

𝐶3𝑇

−1

𝐺2+ 𝐵

𝐺2

𝐶4𝑀

𝐺2

𝑞𝑑,

𝐵

𝐺2

𝐶4= 𝐵

𝐺2

𝐶1𝐵

𝐺2

𝐶2𝑖

0

𝛿𝐺2

+ 𝐵

𝐺2

𝐶1𝐵

𝐺2

𝐶3V0

𝛿𝐺2

,

𝐵

𝐺2

𝐶1=

[

[

[

−

3

2

𝑘𝑝𝜔𝑓

0

0 −

3

2

𝑘V𝜔𝑓

]

]

]

,

𝐵

𝐺2

𝐶2=[

[

−V𝐺2𝑑0

V𝐺2𝑞0

V𝐺2𝑞0

V𝐺2𝑑0

]

]

, 𝐵

𝐺2

𝐶3=[

[

𝑖

𝐺2

𝑑𝑖0−𝑖

𝐺2

𝑞𝑖0

𝑖

𝐺2

𝑞𝑖0𝑖

𝐺2

𝑑𝑖0

]

]

.

(B.6)

C. Linear Differential Equations of DG (25)

𝜌Δ𝑋

DG= 𝐴

DGΔ𝑋

DG+ 𝐵

DGΔ𝑈

DG,

𝐴

DG=[

[

𝐸

𝐺2−1

𝐴

𝐺2𝐸

𝐺2−1

𝐶

𝐺2

𝐶

𝐺2

𝐶𝐴

𝐺2

𝐶

]

]

,

𝐵

DG=[

[

𝐸

𝐺2−1

𝐵

𝐺2

𝐵

𝐺2

𝐶7

]

]

, 𝐵

𝐺2

𝐶7= [𝐵

𝐺2

𝐶60] .

(C.1)

References

[1] N. Pogaku,M. Prodanovic, andT. C.Green, “Modeling, analysisand testing of autonomous operation of an inverter-basedmicrogrid,” IEEE Transactions on Power Electronics, vol. 22, no.2, pp. 613–625, 2007.

[2] P. R.H. Lasseter and P. Paigi, “Microgrid: a conceptual solution,”in Proceedings of the IEEE 35th Annual Power ElectronicsSpecialists Conference (PESC ’04), pp. 4285–4290, June 2004.

[3] E. Antonio, A. Coelho, P. C. Cortizo, P. Francisco, andD.Garcia,“Small-signal stability for parallel-connected inverters in stand-alone AC supply systems,” IEEE Transactions on IndustryApplications, vol. 38, no. 2, pp. 533–542, 2002.

[4] C. K. Sao and P. W. Lehn, “Intentional islanded operation ofconverter fed microgrids,” in Proceedings of the IEEE PowerEngineering Society General Meeting, Quebec, Canada, 2006.

[5] H. Nikkhajoei and R. Iravani, “Steady-state model and powerflow analysis of electronically-coupled fistributed resourceunits,” IEEE Transactions on Power Delivery, vol. 22, no. 1, pp.721–728, 2007.

[6] R. Majumder, A. Ghosh, G. Ledwich, and F. Zare, “Stabilityanalysis and control of multiple converter based autonomousmicrogrid,” in Proceedings of the IEEE International Confer-ence on Control and Automation (ICCA ’09), pp. 1663–1668,Christchurch, New Zealand, December 2009.

[7] F. Katiraei, M. R. Iravani, and P. W. Lehn, “Small-signaldynamic model of a micro-grid including conventional andelectronically interfaced distributed resources,” IET Generation,Transmission and Distribution, vol. 1, no. 3, pp. 369–378, 2007.

[8] Y. Zhang, Z. Jiang, and X. Yu, “Small-signal modeling andanalysis of parallel-connected voltage source inverters,” inProceedings of the IEEE 6th International Power Electronics andMotion Control Conference (IPEMC ’09), pp. 377–383, Wuhan,China, 2009.

[9] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis ofElectric Machinery and Drive Systems, John Wiley & Sons, NewYork, NY, USA, 2002.

[10] S. C. Vegunta, J. V. Milanovic, and S. Z. Djokic, “Modellingof VHz and vector controlled ASDs in PSCAD/EMTDC forvoltage sag studies,” Electric Power Systems Research, vol. 80, no.1, pp. 1–8, 2010.

[11] P. Kundur, Power System Stability and Control, McGraw-Hill,New York, NY, USA, 1994.

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