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An investigation on interarea mode oscillations of interconnected power systems with integrated wind farms
Electrical Power and Energy Systems 78 (2016) 148–157
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
An investigation on interarea mode oscillations of interconnected power systems with integrated wind farms Ping He a, Fushuan Wen b,⇑,1, Gerard Ledwich c, Yusheng Xue d a
College of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China Institut Teknologi Brunei, Bandar Seri Begawan BE1410, Brunei Darussalam c School of Electrical Engineering and Computer Science, Queensland University of Technology, Brisbane, Queensland 4001, Australia d State Grid Electric Power Research Institute, Nanjing 210003, China b
a r t i c l e
i n f o
Article history: Received 26 January 2014 Received in revised form 16 November 2015 Accepted 17 November 2015
Keywords: Power system Doubly fed induction generator (DFIG) Interarea oscillation mode Low-frequency oscillation characteristic Eigenvalue analysis
a b s t r a c t The ever-increasing penetration of wind power integration into a power system can produce significant impacts on the operation of an interconnected power system. As the major energy conversion technology for large wind turbines, the doubly fed induction generator (DFIG) will play an important role in future power systems. Hence, the impacts of the DFIG on the low-frequency oscillations of interconnected power systems have become an important issue with extensive concerns. This paper examines the impacts of several factors, including the DFIG transmission distance, tie-line power of the interconnected system, DFIG capacity, with/without a power system stabilizer (PSS), on the low frequency oscillation characteristic of an interconnected power system using both eigenvalue analysis and dynamic simulations. To investigate the effects of these factors on the interarea oscillation mode, case studies are carried out on two two-area interconnected power systems, and some conclusions are obtained. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction The interconnections of province-level electric power grids have been carried out in China since 2001 [1], as manifested by several interconnection projects such as Northeast China to North China, Chuanyu to Central China, Fujian to East China. Meanwhile, wind energy has been paid more and more attention in many parts over the world [2,3] because of the advantages in relieving energy crisis, protecting environment and promoting sustainable development. With the development of wind energy technology, it is expected that more and more large-scale wind farms will be connected to power systems. In recent years, the doubly-fed induction generator (DFIG) has become the dominant type among the new installed wind turbines due to its capability of controlling reactive power, high energy efficiency, and the fact that the converter rating of appropriately 20–30% of the total machine power is needed. With the increasing penetration level of wind power generation, the impacts of wind power on actual power systems have drawn much attention ⇑ Corresponding author at: Department of Electrical and Electronic Engineering, Institut Teknologi Brunei, Bandar Seri Begawan BE1410, Brunei Darussalam. E-mail address: [email protected] (F. Wen). 1 Taking leave from School of Electrical Engineering, Zhejiang University, Hangzhou 310027, China. http://dx.doi.org/10.1016/j.ijepes.2015.11.052 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
around the globe [4–7]. It is well known that the power network interconnection is useful for optimizing the distribution of resources and improving network reliability, but also bringing some new challenges to power system operation such as low frequency oscillation. Thus, it is necessary to systematically examine how the DFIG penetration affects an actual interconnected largescale power system, especially on low frequency oscillation characteristic [7–10]. An interconnected power system formed by an N-machine independent power system and an M-machine one, has M + N 1 electromechanical oscillation modes, one more than the sum of the two independent power systems. Typically, low frequency oscillations are classified as local mode oscillations and interarea mode oscillations [11]. The former is caused by interactions among a few generators close to each other with frequencies in the range of 0.5–2.0 Hz, while the latter by interactions among large groups of generators with frequencies in the range of 0.1–1.0 Hz. Recently, some research work has been done regarding the interarea mode oscillation of an interconnected power system with wind power integrated. The issues of large wind farm integration and its potential impact on power system damping characteristic were first addressed in 2003 by Slootweg et al. [12]; since then much research work has been carried out by modal analysis or time-domain simulations. In [13,14], the impacts of wind power
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on oscillations and damping were investigated by gradually replacing the power generated by synchronous generators with wind power respectively in New Zealand and Nordic power system. Up to now, the impacts of some important factors, such as the DFIG transmission distance, tie-line power of the interconnected system, DFIG capacity, with/without a power system stabilizer (PSS), on the low frequency oscillation characteristic of an interconnected power system have not yet been systematically investigated, and this is the focus of this paper. Based on the comprehensive model of DFIG, the damping performances of a two-area interconnected system with wind power integration are quantitatively analyzed. The impacts of the above mentioned factors on the interarea mode oscillation are examined for a 2-area 4-unit power system and a 2-area 8-unit 24-bus one by eigenvalue analysis and dynamic simulations. The results of this study are expected to provide some reference for the plan, design, and operation of wind power integration into an interconnected power system, and for the comprehensive understanding of the interarea mode oscillation in this kind of power systems.
Low frequency oscillation in a simple interconnected power system
ð1Þ
where Dx1 and Dx2 are the speed deviations of G1 and G2, respectively; Dd1 and Dd2 are the two generators0 power angle deviations, respectively; M1 and M2 are the two generators0 inertia time constants, respectively; D1 and D2 are the two generators0 damping torque coefficients; DP1m and DP2m are the two generators0 mechanical power deviations; DP1e and DP2e are the electromagnetic power deviations.
E1
3
1
G1 T1 1 Area 1 Pw
P34
Vw
4
T2 L4
2
2 Area 2
Wind Farm
1
Fig. 1. A two-generator interconnected power system.
1
0
0
KM121
MD11
KM222
D2 0M 2
3 2 3 0 2 Dd 1 3 Dd1 7 6 Dd 7 1 76 Dd 2 7 2 7 7 6 76 6 7 ¼ AS 6 7 07 4 5 4 Dx 1 5 5 Dx1 Dx2 Dx 2
ð2Þ
where K ij ¼ @Pei =@dj is generators’ synchronizing torque coefficients at the operating point; AS is the state matrix. The electromagnetic power of G1 and G2 can be calculated as
(
P1e ¼ E21 =jZ 11 j sin a11 þ E1 E2 =jZ 12 j sinðd12 a12 Þ P2e ¼ E22 =jZ 22 j sin a22 þ E1 E2 =jZ 12 j sinðd12 a12 Þ
ð3Þ
where Zii is the input impedances; Zij is the transfer impedance, and Z ii ¼ jZ ii j\uii , Z ij ¼ jZ ij j\uij ; the impedance angles can be expressed as uii ¼ 90 aii , uij ¼ 90 aij . d120 is used to represent the relative angle between G1 and G2 at a given operating point. The synchronizing torque coefficients can be derived from Eq. (3) as
K 11 ¼ K 12 ¼ E1 E2 =jZ 12 j cosðd120 a12 Þ K 22 ¼ K 21 ¼ E1 E2 =jZ 12 j cosðd120 a12 Þ
ð4Þ
Based on matrix transformations and the Schur theorem, the characteristic equation of the state matrix jAS kIj ¼ 0 can be expressed as
D1 D2 3 K 11 K 22 D1 D2 2 k þ k þ þ þ M1 M2 M1 M2 M1 M2 1 þ ðD1 K 22 þ D2 K 11 Þk ¼ 0 M1 M2
k4 þ
ð5Þ
If D1 =M1 ¼ D2 =M 2 ¼ c, i.e. the mechanical damping coefficients are homogeneous, then Eq. (5) can be simplified as
K 11 K 22 ¼0 kðk þ cÞ k2 þ ck þ þ M1 M2
ð6Þ
The eigenvalues of Eq. (6) are: k1 ¼ 0, k2 ¼ c ¼ D1 =M1 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k3 ¼ r jx ¼ D1 =2M1 j ðD1 =M1 Þ2 þ 4ðK 11 =M 1 þ K 22 =M 2 Þ=2. K11, K22, D1 and D2 are all positive under normal operating conditions. It can be proved that Eq. (6) satisfies the Routh stability criterion, and the real parts of three nonzero eigenvalues are negative. k3 represents the electro-mechanical modes of this two-generator power system. If the system is negative damping or zero damping, i.e. D 6 0 and D/M P 0, then the real parts of eigenvalues are situated on or at the right of the imaginary axis, and this demonstrates that the system is small signal unstable. Modeling of DFIG Drive train
8 > < dxr =dt ¼ ðT sh T e Dt xr Þ=2Hg dht =dt ¼ xb ðxt xr Þ > : dxt =dt ¼ ðT m T sh Þ=2Ht
2
1
2
0
The drive train is usually represented by a two-mass model [3] and its dynamics can be expressed by the following differential equations.
G2
w
L3
E2
3 2 0 Dd_ 1 6 Dd_ 7 6 6 0 6 2 7 6 7 ¼ 6 K 11 4 Dx _ 15 6 4 M1 _2 KM21 Dx 2
Relative to the inertia center of a power system, there are always two groups, i.e. one with accelerated generators and the other with decelerated generators, when an oscillation occurs in an interconnected power system. These demonstrates the relative swing of the two groups, and the coherent characteristic in the same group. An interconnected power system can be simplified into two equivalent machines [11] whose speed deviation (Dx1, Dx2) and power angle deviation (Dd1 , Dd2 ) have the characteristic of reversed-phase sinusoidal oscillators with the same frequency, as shown in Fig. 1. The tie line power is transmitted from area 1 to area 2, i.e., areas 1 and 2 are respectively the sending end and the receiving end. To simplify the analysis, the second-order classical model for generators and the constant impedance models for loads are employed. The rotor motion equations of the 2-generator interconnected power system can be expressed as
8 _ Dd 1 ¼ Dx 1 > > > < Dd_ 2 ¼ Dx2 > _ 1 ¼ ðDP1m DP1e D1 Dx1 Þ=M 1 Dx > > : _ 2 ¼ ðDP2m DP2e D2 Dx2 Þ=M 2 Dx
Suppose that DP1m = DP2m = 0, from Eq. (1) the Heffron-Phillips model with the state matrix form can be obtained as follows.
2
ð7Þ
where Ht and Hg are the inertia constants of the turbine and the generator, respectively; xr and xt are the generator and wind turbine speeds, respectively; xb is the reference speed; ht is the shaft twist angle; Dt is the damping coefficient of wind turbine (WT).
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The electromagnetic torque Te, shaft torque Tsh and mechanical torque Tm can be calculated by
8 2 3 > < T m ¼ Pm =xt ¼ 0:5qpR C P V m =xt T sh ¼ K sh ht þ Dsh xb ðxt xr Þ > : T e ¼ Lm ðids iqr iqs idr Þ
ð8Þ
where Pm is the electromagnetic power of WT; CP is the power coefficient, and is a nonlinear function of the blade pitch angle (b) and the blade tip speed ratio ðk0 Þ; q is the air density; R is the WT blade radius; Vm is the wind speed; Ksh and Dsh are the shaft stiffness coefficient and the damping coefficient, respectively; Lm is the mutual inductance; ids and iqs are the d and q axis stator currents, respectively; idr and iqr are the d and q axis rotor currents, respectively. The power coefficient CP [15] can be expressed as
(
8 > < Pr ¼ Pg þ PDC Pr ¼ v dr idr þ v qr iqr Pg ¼ v dg idg þ v qg iqg > : PDC ¼ v DC iDC ¼ C v DC dv DC =dt
ð12Þ
where Pr is the active power at the AC terminal of the RSC; Pg is the active power at the AC terminal of the GSC; PDC is the active power of the DC link; idr and iqr are d and q axis rotor currents, respectively; idg and iqg are d and q axis currents of the grid-side converter, respectively; vdg and vqg are d and q axis voltages of the grid-side converter, respectively; vDC is the capacitor DC voltage; iDC is the current of the capacitor; C is the capacitance of the capacitor. From Eq. (12), the model of the converter together with the DC link can be obtained as
C v DC dv DC =dt ¼ v dg idg þ v qg iqg ðv dr idr þ v qr iqr Þ
ð13Þ
0
C P ¼ 0:22ð116=k0 0:4b 5Þe12:5=ki 1 k0i
¼
1 k0 þ0:08b
ð9Þ
0:035 b3 þ1
where k0i ¼ 1=ð1=k0 þ 0:002Þ, k0 = xtR/Vm. Based on the Betz theory, the maximum value [7] of CP is 0.593. Generator The modeling of DFIG [16] in the synchronously rotating reference frame, i.e. the direct and quadrature (d–q) frame, can be expressed as follows
8 0 xs =xs dids =dt ¼ r s þ xs x0s =xs T 00 ids þ x0s iqs > > > > > þv ds ð1 sr Þe0ds þ e0qs =xs T 00 v dr Lm =Lr > > > > > < x0s =xs diqs =dt ¼ r s þ xs x0s =xs T 00 iqs x0s ids
þv qs ð1 sr Þe0qs e0ds =xs T 00 v qr Lm =Lr > > > > 0 > deds =dt ¼ sr xs e0qs xs v qr Lm =Lr e0ds þ xs x0s iqs =T 00 > > > h i > > : de0 =dt ¼ s x e0 þ x v L =L e0 x x0 i =T 0 qs
r
s ds
s
dr m
r
qs
s
s
ds
ð10Þ
Damping characteristics of an interconnected system with wind farms As shown in the dotted line box of Fig. 1, wind power generators are connected to the interconnected power system. With the line resistance and the power loss ignored, the generators’ electromagnetic power deviations in Eq. (1) can be obtained by the following equations:
where Vw and dw are the voltage magnitude and phase angle at the connection point, respectively; X13 and X23 are the reactances from G1 and G2 to the connection point, respectively; ‘‘0” in the subscript indicates the initial value. Neglecting the load change and system loss yields
DP1e þ DPw ¼ DP2e
0
where wdr and wqr are the d and q axis rotor flux linkages, respectively; Ls is the stator self-inductance; Lr is the rotor selfinductance; Lm is the mutual inductance; rr is the rotor resistance; sr is the rotor slip; xs is the stator reactance; x0 s is the stator transient reactance; e0 ds and e0 qs are the d and q axis voltages behind the transient reactance, respectively; T0 0 is the rotor circuit time constant; ids and iqs are the d and q axis stator currents, respectively; vds and vqs are the d and q axis stator terminal voltages, respectively; vdr and vqr are the d and q axis rotor voltages, respectively.
The pitch angle of the blade is controlled to optimize the power extraction of a wind turbine as well as to prevent overrated power production with strong wind. The pitch servo [5] is modeled as
ð11Þ
where bref is the reference of the blade pitch angle; Tb is the inertia time constant of the pitch control system. Converter model The converter model in the DFIG system comprises of two pulse width modulation invertors connected back to back via a DC link. The rotor-side converter (RSC) and the grid-side converter (GSC) act as controlled voltage sources. The RSC supplies an AC voltage at the slip frequency to the rotor, whereas the GSC supplies an AC voltage at the power frequency to the grid and maintains the DC-link voltage constant. The power balance equations for the converter model can be formulated as
ð15Þ
where DPw is the dynamic active power of the wind farm. To form the state equation set, DPw must be expressed by state variables. From [17], DPw = g1DxB. Here, g1 is the modulus of the wind farm dynamic frequency characteristic; DxB is the frequency deviation of the wind farm at the connection point, and can be expressed by the speed of the generator [17], i.e. DxB = g2Dx1 (with the wind farm at the sending end), DxB = g2Dx2 (with the wind farm at the receiving end). Thus, DPw can be obtained by the following equations.
Pitch control
db=dt ¼ ðbref bÞ=T b
DP1e ¼ E1 V w0 =X 13 cosðd10 dw0 ÞðDd1 Ddw Þ ¼ k1 ðDd1 Ddw Þ DP2e ¼ E2 V w0 =X 23 cosðdw0 d20 ÞðDdw Dd2 Þ ¼ k2 ðDdw Dd2 Þ ð14Þ
DPw ¼ k3 Dx1 ðArea 1Þ DPw ¼ k4 Dx2 ðArea 2Þ
ð16Þ
By employing Eqs. (1) and (14)–(16), DPw and DxB can be eliminated, and the state space equation sets of the interconnected system with the wind farm at the sending end and at the receiving end respectively can be obtained as Eqs. (17) and (18),
3 2 0 Dd_ 1 6 Dd_ 7 6 0 6 6 2 7 k1 k2 7¼6 6 6 4 Dx _ 1 5 4 M1 ðk1 þk2 Þ k1 k2 _2 M ðk Dx þk Þ 2
2
1
2
0
1
32 3 Dd1 6 1 7 76 Dd2 7 7 7 7 0 76 5 4 D x 1 5 D2 Dx 2 M 0
0
0
k1 k2 M1 ðk1 þk2 Þ
D1 k1 k3 M þ M1 ðk 1 1 þk2 Þ
k1 k2 M2 ðk1 þk2 Þ
k1 k3 M2 ðk1 þk2 Þ
2
ð17Þ 3 2 0 Dd_ 1 6 Dd_ 7 6 0 6 6 2 7 k1 k2 6 7¼6 6 4 Dx _ 1 5 4 M1 ðk1 þk2 Þ k1 k2 _2 M ðk Dx þk Þ 2
2
1
2
0
1
0
0
0
1
k1 k2 M1 ðk1 þk2 Þ
D1 M 1
k2 k4 M1 ðk1 þk2 Þ
k1 k2 M2 ðk1 þk2 Þ
0
D2 k2 k4 M þ M2 ðk 2 1 þk2
32 3 Dd1 76 76 Dd2 7 7 76 74 Dx 7 5 1 5 D x 2 Þ ð18Þ
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P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
To examine the damping properties of the interconnected system with the wind farm integrated, the damping change of Eqs. (17) and (18) and (2) should be investigated, and this can be carried out by employing the concept of total damping as detailed in [18]. If positive damping is manifested by Eqs. (17) and (18), then the wind farm can provide positive damping effect, and vice versa. It is proved mathematically in [19] that the sum of eigenvalues of the state equation set is the same as the sum of the diagonal elements in the state matrix; meanwhile, the complex eigenvalue is conjugate. Thus, the total system damping of all modes is equal of the sum of the diagonal elements in the state matrix. Hence, the total system damping for the scenario without the wind farm can be obtained from Eq. (2), and represented by (D1/M1 + D2/M2); the total system damping for the scenario with the wind farm at the sending end is (D1/M1 + D2/M2) + k1k3/M1(k1 + k2), and this can be derived from Eq. (17); similarly, the total system damping for the scenario with the wind farm at the receiving end is (D1/M1 + D2/M2) + k2k4/M2(k1 + k2), and this can be obtained from Eq. (18). For the scenario with the wind farm at the sending end, the damping increment is k1k3/M1(k1 + k2) over the scenario without the wind farm; here, k1 ¼ E1 V w0 =X 13 cosðd10 dw0 Þ, k2 ¼ E2 V w0 = X 23 cosðdw0 d20 Þ, k3 = g1g2. Therefore, the damping effect with the wind farm integration depends on the magnitude and signs of k1, k2 and k3. Eigenvalue and participation factor The most frequently used approach for studying low frequency oscillation is modal analysis [11]. The linearized model of the power system, mostly expressed in a state space form, is employed in this approach. The canonical description is obtained from the analytical linearization of the differential equation set that defines the state space model of the power system. The eigenvalues of the state matrix are calculated, and then the damping condition associated with each oscillation mode found. For detailed analysis, mode shapes and participations factors can also be used. For a particular eigenvalue k ¼ r þ jx of the state matrix AS, the oscillation frequency is f ¼ x=2p , and the damping ratio for k is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi defined as: n ¼ r= r2 þ x2 . When all eigenvalues are computed, the participation factors can also be obtained so as to examine the correlative degree between each state variable and each system mode. The participation factor of the i-th state variable to the jth eigenvalue can be obtained by the right and left eigenvectors w and v as pij = wijvji/wTjvj, where, for a particular eigenvalue ki, a n-column vector wi satisfying ASwi = kiwi (i = 1,2, . . . , n) is the right eigenvector of ki, and a n-row vector vi satisfying viAS = vi ki (i = 1, 2, . . . , n) is the left eigenvector.
Case study 1 The first sample system is shown in Fig. 2, and is extended from a well-known two-area test system for stability studies [21]. In each area of the system, two synchronous equivalent generators, a load and a capacitor are included. A DFIG equivalent wind generation unit is connected to bus 6 in area 1. Note that each generator represents a group of strongly coupled generators. All data of this system are the same as given in [21]. Each synchronous generator (SG) is represented by a six-order model, with magnetic saturation neglected and voltage regulators included and modeled with an IEEE type 1 model [21]. Each load is modeled as constant impedance. The DFIG is represented by the model described in Section ‘Modeling of DFIG’, the DFIG parameters are: rs = 0.00706 p.u., rr = 0.005 p.u., Lr = 0.156 p.u., Ls = 0.171 p.u., Lm = 3.5 p.u., Dsh = 0.01, Ksh = 0.5, Ht = 3 s, Hg = 0.5 s, Hm = 3.5 s, Tb = 0.25 s. For comparison purposes, the scenario without wind power is considered as the base scenario. There is totally 2800 MW installed generation capacity in this system, while approximately 400 MW of active power is exported from area 1 to area 2. The electromechanical oscillatory modes for the base scenario are shown in Table 1, including the situations without PSS (NPSS) and with PSS equipped in all SGs. The values of eigenvalues, damping and frequency are included. The last column lists the classification of the modes. From Table 1, it can be found that the damping ratios of electromechanical modes are obviously enhanced with PSSs equipped. The contribution of each generator to an oscillatory mode can be assessed by analyzing its participation factor. It is demonstrated in Fig. 3 that mode 1 is characterized by oscillations of G1 against G2 in area 1, mode 2 is characterized by oscillations of G3 against G4 in area 2, and mode 3 is described by oscillations of generators located in area 1 against generators located in area 2. Therefore, mode 1 and mode 2 are the local mode; mode 3 belongs to the interarea oscillation mode. Impacts of the transmission distance of the wind farm on damping characteristics In this section, the impacts of the transmission distance (TD) of the wind farm on damping characteristics are discussed, and two 1
5
Case studies Two sample systems are employed for case studies.
8
9
Area 1
2
L9 C9
C7 L7
3
Area 2
4
13
G4
G2
ð19Þ
10 11
G3
12
Power system stabilizers (PSSs) have been widely used in actual power systems for dynamic stability enhancement. In [20], a typical PSS is modeled by two identical lead/lag networks represented by a gain KPSS, four time constants Tl, T2, T3 and T4, and a washout circuit taking a time constant Tw. The transfer function of this PSS, G(s), can be expressed as Eq. (19), and is employed in this work.
sT w 1 þ sT 1 1 þ sT 3 1 þ sT w 1 þ sT 2 1 þ sT 4
7
G1
Power system stabilizer (PSS)
GðsÞ ¼ K PSS
6
Wind Farm Fig. 2. The two-area four-generator interconnected power system.
Table 1 Electro-mechanical oscillatory modes without wind farms. No.
k
n
f/Hz
Notes
NPSS
1 2 3
0.9546 ± j6.2353 1.0704 ± j6.4774 0.2619 ± j3.6842
0.1513 0.1630 0.0709
0.9924 1.0309 0.5864
Local mode of area 1 Local mode of area 2 Interarea mode
PSS
1 2 3
3.0515 ± j7.5937 3.2942 ± j8.1625 0.8405 ± j3.6049
0.3729 0.3743 0.2271
1.2086 1.2991 0.5737
Local mode of area 1 Local mode of area 2 Interarea mode
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P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
0.3
Mode 1
Mode 2
Mode 3 0.2
0.1
0
G1
G2
G3
G4
Fig. 3. The participation factor of each generator to different electromechanical modes for the scenario without PSS.
different wind farm active power outputs (14 MW and 50 MW) are considered. For the first case, the wind farm has a total active power output of 14 MW, and 414 MW are exported from area 1 to area 2. Table 2 lists the electro-mechanical oscillatory modes in the system with DFIG integration under different transmission distances, including the situations with and without PSSs. For TD = 50 km, the participation factors for all units are shown in Fig. 4. From the results in Table 2, it is known that there is one more interarea mode related to DFIG, i.e. Mode 4, and this can also be explained by Fig. 4. Mode 4 is characterized by oscillations among generators G1, G2, G3, G4 and DFIG. From the results shown in Table 2, the changes of the oscillation frequency and damping ratio of local modes k1,2 are not significant with the increase of TD from 50 km to 300 km, although the oscillation frequency tends to increase; the changes of f and n of interarea mode 1 are not noticeable, while n tends to decrease, and f tends to increase/decrease in the situation with/without PSSs. There are relatively obvious changes in f and n of k4 (i.e. interarea mode 2, as shown in bold in Table 2), f decreases gradually and n tends to increase. Table 3 gives the results of oscillation modes for the system with 50 MW DFIG active power output integrated under different
transmission distances, and 448.9 MW of active power is transmitted from area 1 to area 2. With the increase of the transmission distances, f and n of local modes k1,2 do not change significantly although f still shows an increasing tendency; there are relatively obvious changes in f and n of the interarea mode k3,4 especially k4 (i.e. interarea mode 2, as shown in bold in Table 3), f decreases gradually and n increases from 0.5930 to 0.6325. The oscillation frequency and damping ratios in Tables 2 and 3 show a similar trend, while n shown in Table 3 is significantly smaller than that in Table 2, and this means that the increasing wind power integration level worsens the oscillation modes, especially the interarea mode. In order to systematically analyze the impacts of different DFIG transmission distances on power system dynamic response, suppose that a three-phase grounding short-circuit fault occurred on one of the double lines between buses 8 and 9 (the fault time tf = 1.0 s and clearing time tc = 1.2 s). The relative rotor angle curves between G1 and G4, the voltage on bus 7, and the response of the rotor current of DFIG denoted by idr are shown in Figs. 5–7, respectively. It can be seen from Figs. 5–7 that the change of TD has less impact on the system dynamic response, and the profiles of these response curves are similar, as illustrated in Fig. 5. With the increase of TD, the voltage tends to increase and idr tends to decrease. In addition, as can be observed from Figs. 5–7, the effectiveness of PSSs is obvious. Impacts of tie-line power on inter-area oscillation mode The amount and even the direction of the tie-line power flow are subjected to changes under various system operating conditions. In this section, the impacts of the tie-line power flow on the inter-area oscillation mode are examined by adjusting the outputs of generators under various operating conditions, with wind farm output as 30 MW, TD as 25 km, and PSSs equipped. By adjusting the overall generation output in area 1, the tie-line power transmitted from area 1 to area 2 changes. In this way, the changes of the damping performances of the inter-area oscillation modes can be examined, and the results are shown in Table 4. It can be seen from the results in Table 4 that with the increase of the tie-line power from 46 MW to 618 MW, f and n of the
Table 2 Electro-mechanical oscillatory modes in the system with DFIG integration under different transmission distances (with 14 MW DFIG output). TD (km)
No.
Without PSS
With PSS
k
n
f/Hz
k
n
f/Hz
Dominant machines
Notes
5
1 2 3 4
0.9542 ± j6.2393 1.0864 ± j6.4665 0.2671 ± j3.6937 0.3523 ± j0.7567
0.1512 0.1657 0.0721 0.4221
0.9930 1.0292 0.5878 0.1204
3.0468 ± j7.5846 3.3151 ± j8.1507 0.8432 ± j3.6066 0.5898 ± j0.6346
0.3728 0.3768 0.2277 0.6808
1.2071 1.2972 0.5740 0.1012
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
50
1 2 3 4
0.9544 ± j6.2403 1.0864 ± j6.4665 0.2668 ± j3.6928 0.3475 ± j0.7443
0.1512 0.1657 0.0721 0.4231
0.9932 1.0292 0.5877 0.1185
3.0493 ± j7.5982 3.3153 ± j8.1510 0.8433 ± j3.6085 0.5890 ± j0.6335
0.3724 0.3768 0.2276 0.6809
1.2093 1.2973 0.5743 0.1008
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
100
1 2 3 4
0.9547 ± j6.2416 1.0864 ± j6.4665 0.2664 ± j3.6921 0.3424 ± j0.7315
0.1512 0.1657 0.0720 0.4239
0.9934 1.0292 0.5876 0.1164
3.0551 ± j7.6171 3.3154 ± j8.1511 0.8433 ± j3.6095 0.5889 ± j0.6325
0.3723 0.3768 0.2275 0.6814
1.2123 1.2973 0.5744 0.1006
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
200
1 2 3 4
0.9552 ± j6.2446 1.0864 ± j6.4666 0.2656 ± j3.6913 0.3326 ± j0.7072
0.1512 0.1657 0.0718 0.4256
0.9939 1.0292 0.5875 0.1126
3.0613 ± j7.6483 3.3156 ± j8.1519 0.8433 ± j3.6112 0.5886 ± j0.6291
0.3716 0.3768 0.2274 0.6832
1.2173 1.2974 0.5747 0.1001
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
300
1 2 3 4
0.9557 ± j6.2477 1.0865 ± j6.4666 0.2648 ± j3.6907 0.3220 ± j0.6831
0.1512 0.1657 0.0716 0.4264
0.9944 1.0292 0.5874 0.1087
3.0697 ± j7.6866 3.3158 ± j8.1522 0.8432 ± j3.6154 0.5884 ± j0.6251
0.3708 0.3768 0.2271 0.6854
1.2234 1.2975 0.5754 0.0994
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
153
P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
0.2
0.25
Mode 1
Mode 1 0.2
Mode 2
Mode 2 0.15
Mode 3
Mode 3 0.1
Mode 4
0.1
Mode 4
0.05
0
G1
G2
G3
G4
DFIG
0
G1
(a) Without PSS
G2
G3
G4
DFIG
(b) With PSS
Fig. 4. The participation factors of each generator in different electromechanical modes.
Table 3 Electro-mechanical modes in the system with DFIG integration under different transmission distances (with 50 MW DFIG output). TD (km)
No.
Without PSS
With PSS
k
n
f/Hz
k
n
f/Hz
Dominant machines
Notes
5
1 2 3 4
0.9548 ± j6.2332 1.1266 ± j6.4366 0.2732 ± j3.6863 0.3246 ± j0.8653
0.1514 0.1724 0.0739 0.3512
0.9921 1.0244 0.5867 0.1377
3.0454 ± j7.5783 3.3684 ± j8.1149 0.8444 ± j3.6092 0.5689 ± j0.7723
0.3729 0.3834 0.2278 0.5930
1.2061 1.2915 0.5744 0.1229
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
50
1 2 3 4
0.9549 ± j6.2335 1.1264 ± j6.4367 0.2732 ± j3.6848 0.3218 ± j0.8503
0.1514 0.1724 0.0740 0.3540
0.9921 1.0244 0.5865 0.1353
3.0478 ± j7.5791 3.3682 ± j8.1152 0.8445 ± j3.6071 0.5686 ± j0.7521
0.3731 0.3833 0.2280 0.6031
1.2063 1.2916 0.5741 0.1197
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
100
1 2 3 4
0.9550 ± j6.2342 1.1263 ± j6.4368 0.2731 ± j3.6838 0.3193 ± j0.8362
0.1514 0.1724 0.0739 0.3567
0.9922 1.0244 0.5863 0.1331
3.0469 ± j7.5843 3.3681 ± j8.1153 0.8446 ± j3.6066 0.5684 ± j0.7426
0.3728 0.3833 0.2280 0.6078
1.2071 1.2916 0.5740 0.1182
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
200
1 2 3 4
0.9555 ± j6.2362 1.1261 ± j6.4371 0.2727 ± j3.6825 0.3151 ± j0.8118
0.1515 0.1723 0.0739 0.3618
0.9925 1.0245 0.5861 0.1292
3.0485 ± j7.5912 3.3678 ± j8.1158 0.8446 ± j3.6051 0.5675 ± j0.7137
0.3727 0.3833 0.2281 0.6224
1.2082 1.2917 0.5738 0.1136
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area1 Local mode of area 2 Interarea mode 1 Interarea mode 2
300
1 2 3 4
0.9559 ± j6.2386 1.1259 ± j6.4372 0.2723 ± j3.6817 0.3115 ± j0.7897
0.1515 0.1723 0.0738 0.3669
0.9929 1.0245 0.5859 0.1257
3.0522 ± j7.6094 3.3675 ± j8.1161 0.8447 ± j3.6042 0.5666 ± j0.6939
0.3728 0.3832 0.2281 0.6325
1.2111 1.2917 0.5736 0.1104
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
(a) TD=50km
(b) TD=200km
Fig. 5. The relative rotor angle curves between G1 and G4. 1. Wind farm active power output 14 MW without PSS; 2. Wind farm active power output 50 MW without PSS; 3. Wind farm active power output 14 MW with PSS; 4. Wind farm active power output 50 MW with PSS.
P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
VBus7 (p.u.)
VBus7 (p.u.)
154
(a)
(b)
idrDFIG (p.u.)
idrDFIG (p.u.)
Fig. 6. The voltage curve on bus 7. (The meanings of a, b, 1–4 are the same as those in Fig. 5.)
(a)
(b)
Fig. 7. The d axis rotor current of DFIG. (The meanings of a, b, 1–4 are the same as those in Fig. 5).
inter-area oscillation mode 1 increase initially and then decrease, while f of the inter-area oscillation mode 2 shows an increasing tendency and n decreases gradually. Under the same fault condition as that in Section ‘Impacts of the transmission distance of the wind farm on damping characteristics’, Fig. 8 shows the response curves under different tie-line power flows, from left to right: the relative rotor angle between G1 and G4, the voltage on bus 7 and the rotor current of DFIG denoted by idr. With the increase of the tie-line power flow, it can be observed from the curves that dG1,G4 and idr gradually increase, VBus7 gradually decreases. In addition, the amplitudes of the three curves increase after a fault occurs, and longer time is needed for the system to stabilize. By adjusting the generation outputs of the units including the wind farm at the receiving end, the tie-line power flow can be exported from area 2 to area 1. The inter-area oscillation modes under different tie-line power flows are given in Table 5. From this table, it can be observed that with the increase of the tie-line power flow, f and n of the inter-area oscillation mode 1 increase initially and then decrease; Both f and n of the inter-area oscillation mode 2 decreases initially and then increases. When the transmitted power from area 2 to area 1 is 474 MW, the interarea mode 2 has a negative real eigenvalue. Impacts of the DFIG penetration level on low frequency characteristics The studied operating condition is described in Section ‘Impacts of tie-line power on inter-area oscillation mode’, i.e. the output of the wind farm is 30 MW, TD = 25 km, and with PSSs equipped. To keep the tie-line power flow from area 1 to area 2 maintained at
Table 4 The interarea oscillation mode under various tie-line power flows (from area 1 to area 2). No.
Tie-line power flow/ MW
k
n
f/Hz
Interarea mode 1
46 143 239 335 430 618
0.7459 ± j3.6074 0.8545 ± j3.6252 0.8534 ± j3.6384 0.8501 ± j3.6342 0.8439 ± j3.6134 0.8195 ± j3.5233
0.2025 0.2294 0.2284 0.2278 0.2274 0.2266
0.5741 0.5769 0.5791 0.5784 0.5751 0.5608
Interarea mode 2
46 143 239 335 430 618
0.5106 ± j0.4424 0.5906 ± j0.5931 0.5842 ± j0.6355 0.5793 ± j0.6756 0.5753 ± j0.7156 0.5693 ± j0.8041
0.7558 0.7056 0.6768 0.6509 0.6266 0.5778
0.0704 0.0944 0.1011 0.1075 0.1139 0.1281
400 MW, the power outputs of G1 and G2 are adjusted when the power output of the wind farm changes. The interarea oscillation modes at various DFIG penetration levels are given in Table 6, where the wind power output (%) is the percentage of the DFIG output over the total output power of area 1. Under the same fault condition with Section ‘Impacts of the transmission distance of the wind farm on damping characteristics’, the responses curves under various DFIG penetration levels are shown in Fig. 9, and the meanings of the curves are the same as those in Fig. 8. It can be seen from Table 6 that with the increase of the DFIG output power, f and n of the inter-area oscillation mode 1 tend to
155
VBus7 (p.u.)
idrDFIG (p.u.)
P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
Fig. 8. Response curves under different tie-line power flows.
Table 5 Interarea oscillation modes under various tie-line power flows (from area 2 to area 1). No.
Tie-line power flow/ MW
k
n
f/Hz
Interarea mode 1
51 153 255 361 474
0.7431 ± j3.5482 0.7263 ± j3.4717 0.6943 ± j3.3415 0.5601 ± j3.1347 0.3250 ± j2.6001
0.2049 0.2048 0.2034 0.1759 0.1240
0.5647 0.5525 0.5318 0.4989 0.4138
Interarea mode 2
51 153 255 361 474
0.5071 ± j0.3675 0.4953 ± j0.2384 0.2814 ± j0.1315 0.5494 ± j0.3639 0.56784
0.8097 0.9010 0.9059 0.8337 –
0.0585 0.0379 0.0209 0.0579 –
the wind power output reaches 6%, d14 decreases and VBus7 and idr tend to increase. Case study 2
No.
Wind power output (%)
k
n
f/Hz
Interarea mode 1
0 1 3 5 6
0.8405 ± j3.6049 0.8437 ± j3.6221 0.8458 ± j3.6222 0.8481 ± j3.6231 0.8499 ± j3.6268
0.2271 0.2269 0.2274 0.2279 0.2281
0.5737 0.5765 0.5765 0.5766 0.5772
Interarea mode 2
0 1 3 5 6
0.5692 ± j0.5288 0.5949 ± j0.6752 0.5729 ± j0.7247 0.5662 ± j0.7797 0.5569 ± j0.7863
0.7326 0.6611 0.6202 0.5876 0.5779
0.0842 0.1075 0.1153 0.1241 0.1251
VBus7 (p.u.)
increase, while f of the inter-area oscillation mode 2 tends to increase and n decreases gradually. When the wind power output varies from 1% to 5%, the relative rotor angle between G1 and G4 and the voltage on bus 7 do not change significantly, while when
Fig. 9. Response curves at various DFIG output levels.
idrDFIG (p.u.)
Table 6 Interarea oscillation modes at various DFIG penetration levels.
The second sample system is shown in Fig. 10, and is a 2-area 8-unit 24-bus [22] power system. In this system, each generator is represented by a sixth-order model without PSS installed. G6–G8 are included in area 1, while G1–G5 in area 2. The regional tie-lines can be classified into two channels, i.e. the single-circuit channel branch 6–7, and the double-circuit channel branches 4–10 and 4–11. Usually, the loading level in the single-circuit channel is heavier than that in the double-circuit channel. With this in mind, the following analysis on the tie-line power flow will focus on branch 6–7, and tie-line power flow is from area 1 to area 2. The simulation results of this sample system are shown in Table 7. There are four interarea oscillation modes, and can be described by oscillations of generators located in area 1 against generators located in area 2. A DFIG equivalent wind generation unit is connected to bus 2 in area 1. Note that each generator does not represent a single unit but a group of strongly coupled generators. There is one more interarea mode related to the DFIG with the oscillation frequency around 0.54 Hz. The impacts of different factors on the five oscillation modes are discussed in this section. The total active power output of the wind farm is around 50 MW and 247 MW are exported from area 1 to area 2 through the tie-line 6–7. The results of electro-mechanical modes of the system with DFIG integration under different transmission distances are given in Table 8. The power flow on the tie-line 6–7 can be adjusted by changing the outputs of generators in area 1 given TD = 50 km, and the interarea oscillation modes under different tie-line power flows are listed in Table 9. Similarly, the interarea oscillation modes at various DFIG penetration levels are given in Table 10, in which the
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P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
G8 24
Table 10 Interarea oscillation modes under various DFIG penetration levels.
~
6 1
2
7
section
13 G1
10 L36
3
11
~
G6
L7
L5
~
4 23 G7
~
12
L3
Area 1
5
14 9 18 L8 G3 G2
20
~
~
8 L4
22
L1 L2
16
17
~
Wind power output (%)
0
1
3
5
7
1
n f/Hz
0.0796 1.6708
0.0781 1.6785
0.0775 1.6815
0.0769 1.6841
0.0762 1.687
2
n f/Hz
0.0992 1.5070
0.1008 1.5103
0.1013 1.5123
0.1017 1.5145
0.1023 1.5159
3
n f/Hz
0.0889 1.2426
0.0892 1.2422
0.0892 1.2422
0.0892 1.2422
0.0892 1.2423
4
n f/Hz
0.0776 1.0342
0.0779 1.0344
0.0778 1.0351
0.0777 1.0357
0.0777 1.0363
5
n f/Hz
0.0723 0.5331
0.0558 0.5302
0.0421 0.5429
0.0334 0.5542
0.0271 0.5633
15
21
G4
~
19
No.
L9
G5
Area 2
Fig. 10. A 2-area 8-machine 24-node system.
Table 7 Interconnected oscillation modes. No.
k
n
f/Hz
Dominant generators
1 2 3 4
0.8378 ± j10.4979 0.9438 ± j9.4691 0.6970 ± j7.8073 0.5062 ± j6.4982
0.0796 0.0992 0.0889 0.0776
1.6708 1.5070 1.2426 1.0342
G1, G1, G5, G1,
G7 G2, G7 G6 G5, G6
Table 8 Electro-mechanical modes for the system with DFIG integration under different transmission distances. No.
TD (km)
50
100
200
300
1
n f/Hz
0.0775 1.6810
0.0772 1.6825
0.0769 1.6854
0.0765 1.6881
2
n f/Hz
0.1011 1.513
0.1014 1.5132
0.1019 1.5134
0.1025 1.5136
3
n f/Hz
0.0891 1.2423
0.0892 1.2422
0.0892 1.2421
0.0892 1.2421
4
n f/Hz
0.0777 1.0354
0.0778 1.0353
0.0779 1.0353
0.0781 1.0352
5
n f/Hz
0.0372 0.5491
0.0382 0.5475
0.0402 0.5445
0.0422 0.5414
Table 9 Interarea oscillation modes under various tie-line power flows. No.
Tie line power/MW
56
163
247
292
475
1
n f/Hz
0.1113 1.5988
0.1095 1.6104
0.0772 1.6825
0.0852 1.6765
0.0977 1.6351
2
n f/Hz
0.0912 1.4568
0.0941 1.4576
0.1014 1.5132
0.1062 1.4952
0.1033 1.4511
3
n f/Hz
0.0884 1.2462
0.0888 1.2442
0.0892 1.2422
0.0891 1.2422
0.0887 1.2444
4
n f/Hz
0.0690 1.036
0.0716 1.042
0.0778 1.0353
0.0794 1.0292
0.0831 0.9989
5
n f/Hz
0.0737 0.6294
0.0181 0.6431
0.0382 0.5475
0.2548 0.4610
0.2351 0.1954
wind power output levels (%) are the same as those stated in Section ‘Impacts of the DFIG penetration level on low frequency characteristics’. As shown in Table 8, with the increase of TD from 50 km to 300 km, the oscillation frequency and damping ratio of the four
interarea oscillation modes do not change significantly. However, the oscillation frequencies of modes 1 and 2 tend to increase, while those of modes 3 and 4 tend to decrease. For the new interarea mode 5, f tends to decrease and n tends to increase, and this is consistent with the analysis in Section ‘Impacts of the transmission distance of the wind farm on damping characteristics’. It can be seen from Table 9 that with the increase of the tie-line power flow from 56 MW to 475 MW, the f0 s of the inter-area oscillation modes 1, 2 and 4 increase initially and then decrease, while f of the mode 5 tends to decrease. With the increase of the DFIG power output, it can be seen from Table 10 that the n0 s of modes 1, 3 and 5 tend to decrease, while those of modes 2 and 4 increases gradually. Conclusions In this work, the impacts of several factors, including the DFIG transmission distance, DFIG penetration level, tie-line power flow between areas and PSSs, on the interarea mode oscillation of interconnected power systems are investigated. A 2-area 4-unit 13-bus power system and a 2-area 8-unit 24-bus power system are employed for eigenvalue analysis and dynamic simulations. There is one more interarea oscillatory mode related to the integrated DFIG Simulation results show that the impacts of the DFIG transmission distances on the oscillating characteristics of the local mode and the interarea mode are not very significant, but with the increase of the transmission distances, the oscillation frequencies of local modes gradually increases, the damping ratio of the new interarea mode tends to increase. Thus, an appropriate DFIG transmission distance should be considered in planning the DFIG integration to the power system concerned. If the wind farm is located at the sending end, the frequency of the inter-area oscillation mode tends to increase with the increase of the tie-line power flow, and the frequency of the interarea oscillation mode related to DFIG decreases at a certain value of the tieline power flow. If the wind farm is located at the receiving end, the frequency of the inter-area oscillation mode tends to decrease with the increase of the tie-line power flow, and the frequency of the interarea mode related to DFIG increases at a certain value of the tie-line power flow. Thus, the impacts of the wind farm on the interarea oscillation mode should be taken into account in determining the common coupling point. The increase of the DFIG power output has a less impact on the local oscillation mode, while a relatively significant effect on the inter-area oscillation mode. When the DFIG penetration level reaches a certain value, the oscillation modes of the system will be deteriorated. Thus, an appropriate DFIG penetration level should be considered in the DFIG integration planning and operation so as to the small signal stability of the power system concerned.
P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
It is clearly shown, by the simulation results of eigenvalue analysis and dynamic simulations, that the damping and dynamic response characteristics of power systems with wind farms can be improved by installing PSSs.
Acknowledgements This work is jointly supported by National High Technology Research and Development Program of China (863 Program) (No. 2015AA050202), the National Natural Science Foundation of China (NSFC) (No. 51507157), and the Doctoral Scientific Research Foundation of Zhengzhou University of Light Industry (No. 2014BSJJ043).
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Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
An investigation on interarea mode oscillations of interconnected power systems with integrated wind farms Ping He a, Fushuan Wen b,⇑,1, Gerard Ledwich c, Yusheng Xue d a
College of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China Institut Teknologi Brunei, Bandar Seri Begawan BE1410, Brunei Darussalam c School of Electrical Engineering and Computer Science, Queensland University of Technology, Brisbane, Queensland 4001, Australia d State Grid Electric Power Research Institute, Nanjing 210003, China b
a r t i c l e
i n f o
Article history: Received 26 January 2014 Received in revised form 16 November 2015 Accepted 17 November 2015
Keywords: Power system Doubly fed induction generator (DFIG) Interarea oscillation mode Low-frequency oscillation characteristic Eigenvalue analysis
a b s t r a c t The ever-increasing penetration of wind power integration into a power system can produce significant impacts on the operation of an interconnected power system. As the major energy conversion technology for large wind turbines, the doubly fed induction generator (DFIG) will play an important role in future power systems. Hence, the impacts of the DFIG on the low-frequency oscillations of interconnected power systems have become an important issue with extensive concerns. This paper examines the impacts of several factors, including the DFIG transmission distance, tie-line power of the interconnected system, DFIG capacity, with/without a power system stabilizer (PSS), on the low frequency oscillation characteristic of an interconnected power system using both eigenvalue analysis and dynamic simulations. To investigate the effects of these factors on the interarea oscillation mode, case studies are carried out on two two-area interconnected power systems, and some conclusions are obtained. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction The interconnections of province-level electric power grids have been carried out in China since 2001 [1], as manifested by several interconnection projects such as Northeast China to North China, Chuanyu to Central China, Fujian to East China. Meanwhile, wind energy has been paid more and more attention in many parts over the world [2,3] because of the advantages in relieving energy crisis, protecting environment and promoting sustainable development. With the development of wind energy technology, it is expected that more and more large-scale wind farms will be connected to power systems. In recent years, the doubly-fed induction generator (DFIG) has become the dominant type among the new installed wind turbines due to its capability of controlling reactive power, high energy efficiency, and the fact that the converter rating of appropriately 20–30% of the total machine power is needed. With the increasing penetration level of wind power generation, the impacts of wind power on actual power systems have drawn much attention ⇑ Corresponding author at: Department of Electrical and Electronic Engineering, Institut Teknologi Brunei, Bandar Seri Begawan BE1410, Brunei Darussalam. E-mail address: [email protected] (F. Wen). 1 Taking leave from School of Electrical Engineering, Zhejiang University, Hangzhou 310027, China. http://dx.doi.org/10.1016/j.ijepes.2015.11.052 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
around the globe [4–7]. It is well known that the power network interconnection is useful for optimizing the distribution of resources and improving network reliability, but also bringing some new challenges to power system operation such as low frequency oscillation. Thus, it is necessary to systematically examine how the DFIG penetration affects an actual interconnected largescale power system, especially on low frequency oscillation characteristic [7–10]. An interconnected power system formed by an N-machine independent power system and an M-machine one, has M + N 1 electromechanical oscillation modes, one more than the sum of the two independent power systems. Typically, low frequency oscillations are classified as local mode oscillations and interarea mode oscillations [11]. The former is caused by interactions among a few generators close to each other with frequencies in the range of 0.5–2.0 Hz, while the latter by interactions among large groups of generators with frequencies in the range of 0.1–1.0 Hz. Recently, some research work has been done regarding the interarea mode oscillation of an interconnected power system with wind power integrated. The issues of large wind farm integration and its potential impact on power system damping characteristic were first addressed in 2003 by Slootweg et al. [12]; since then much research work has been carried out by modal analysis or time-domain simulations. In [13,14], the impacts of wind power
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on oscillations and damping were investigated by gradually replacing the power generated by synchronous generators with wind power respectively in New Zealand and Nordic power system. Up to now, the impacts of some important factors, such as the DFIG transmission distance, tie-line power of the interconnected system, DFIG capacity, with/without a power system stabilizer (PSS), on the low frequency oscillation characteristic of an interconnected power system have not yet been systematically investigated, and this is the focus of this paper. Based on the comprehensive model of DFIG, the damping performances of a two-area interconnected system with wind power integration are quantitatively analyzed. The impacts of the above mentioned factors on the interarea mode oscillation are examined for a 2-area 4-unit power system and a 2-area 8-unit 24-bus one by eigenvalue analysis and dynamic simulations. The results of this study are expected to provide some reference for the plan, design, and operation of wind power integration into an interconnected power system, and for the comprehensive understanding of the interarea mode oscillation in this kind of power systems.
Low frequency oscillation in a simple interconnected power system
ð1Þ
where Dx1 and Dx2 are the speed deviations of G1 and G2, respectively; Dd1 and Dd2 are the two generators0 power angle deviations, respectively; M1 and M2 are the two generators0 inertia time constants, respectively; D1 and D2 are the two generators0 damping torque coefficients; DP1m and DP2m are the two generators0 mechanical power deviations; DP1e and DP2e are the electromagnetic power deviations.
E1
3
1
G1 T1 1 Area 1 Pw
P34
Vw
4
T2 L4
2
2 Area 2
Wind Farm
1
Fig. 1. A two-generator interconnected power system.
1
0
0
KM121
MD11
KM222
D2 0M 2
3 2 3 0 2 Dd 1 3 Dd1 7 6 Dd 7 1 76 Dd 2 7 2 7 7 6 76 6 7 ¼ AS 6 7 07 4 5 4 Dx 1 5 5 Dx1 Dx2 Dx 2
ð2Þ
where K ij ¼ @Pei =@dj is generators’ synchronizing torque coefficients at the operating point; AS is the state matrix. The electromagnetic power of G1 and G2 can be calculated as
(
P1e ¼ E21 =jZ 11 j sin a11 þ E1 E2 =jZ 12 j sinðd12 a12 Þ P2e ¼ E22 =jZ 22 j sin a22 þ E1 E2 =jZ 12 j sinðd12 a12 Þ
ð3Þ
where Zii is the input impedances; Zij is the transfer impedance, and Z ii ¼ jZ ii j\uii , Z ij ¼ jZ ij j\uij ; the impedance angles can be expressed as uii ¼ 90 aii , uij ¼ 90 aij . d120 is used to represent the relative angle between G1 and G2 at a given operating point. The synchronizing torque coefficients can be derived from Eq. (3) as
K 11 ¼ K 12 ¼ E1 E2 =jZ 12 j cosðd120 a12 Þ K 22 ¼ K 21 ¼ E1 E2 =jZ 12 j cosðd120 a12 Þ
ð4Þ
Based on matrix transformations and the Schur theorem, the characteristic equation of the state matrix jAS kIj ¼ 0 can be expressed as
D1 D2 3 K 11 K 22 D1 D2 2 k þ k þ þ þ M1 M2 M1 M2 M1 M2 1 þ ðD1 K 22 þ D2 K 11 Þk ¼ 0 M1 M2
k4 þ
ð5Þ
If D1 =M1 ¼ D2 =M 2 ¼ c, i.e. the mechanical damping coefficients are homogeneous, then Eq. (5) can be simplified as
K 11 K 22 ¼0 kðk þ cÞ k2 þ ck þ þ M1 M2
ð6Þ
The eigenvalues of Eq. (6) are: k1 ¼ 0, k2 ¼ c ¼ D1 =M1 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k3 ¼ r jx ¼ D1 =2M1 j ðD1 =M1 Þ2 þ 4ðK 11 =M 1 þ K 22 =M 2 Þ=2. K11, K22, D1 and D2 are all positive under normal operating conditions. It can be proved that Eq. (6) satisfies the Routh stability criterion, and the real parts of three nonzero eigenvalues are negative. k3 represents the electro-mechanical modes of this two-generator power system. If the system is negative damping or zero damping, i.e. D 6 0 and D/M P 0, then the real parts of eigenvalues are situated on or at the right of the imaginary axis, and this demonstrates that the system is small signal unstable. Modeling of DFIG Drive train
8 > < dxr =dt ¼ ðT sh T e Dt xr Þ=2Hg dht =dt ¼ xb ðxt xr Þ > : dxt =dt ¼ ðT m T sh Þ=2Ht
2
1
2
0
The drive train is usually represented by a two-mass model [3] and its dynamics can be expressed by the following differential equations.
G2
w
L3
E2
3 2 0 Dd_ 1 6 Dd_ 7 6 6 0 6 2 7 6 7 ¼ 6 K 11 4 Dx _ 15 6 4 M1 _2 KM21 Dx 2
Relative to the inertia center of a power system, there are always two groups, i.e. one with accelerated generators and the other with decelerated generators, when an oscillation occurs in an interconnected power system. These demonstrates the relative swing of the two groups, and the coherent characteristic in the same group. An interconnected power system can be simplified into two equivalent machines [11] whose speed deviation (Dx1, Dx2) and power angle deviation (Dd1 , Dd2 ) have the characteristic of reversed-phase sinusoidal oscillators with the same frequency, as shown in Fig. 1. The tie line power is transmitted from area 1 to area 2, i.e., areas 1 and 2 are respectively the sending end and the receiving end. To simplify the analysis, the second-order classical model for generators and the constant impedance models for loads are employed. The rotor motion equations of the 2-generator interconnected power system can be expressed as
8 _ Dd 1 ¼ Dx 1 > > > < Dd_ 2 ¼ Dx2 > _ 1 ¼ ðDP1m DP1e D1 Dx1 Þ=M 1 Dx > > : _ 2 ¼ ðDP2m DP2e D2 Dx2 Þ=M 2 Dx
Suppose that DP1m = DP2m = 0, from Eq. (1) the Heffron-Phillips model with the state matrix form can be obtained as follows.
2
ð7Þ
where Ht and Hg are the inertia constants of the turbine and the generator, respectively; xr and xt are the generator and wind turbine speeds, respectively; xb is the reference speed; ht is the shaft twist angle; Dt is the damping coefficient of wind turbine (WT).
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The electromagnetic torque Te, shaft torque Tsh and mechanical torque Tm can be calculated by
8 2 3 > < T m ¼ Pm =xt ¼ 0:5qpR C P V m =xt T sh ¼ K sh ht þ Dsh xb ðxt xr Þ > : T e ¼ Lm ðids iqr iqs idr Þ
ð8Þ
where Pm is the electromagnetic power of WT; CP is the power coefficient, and is a nonlinear function of the blade pitch angle (b) and the blade tip speed ratio ðk0 Þ; q is the air density; R is the WT blade radius; Vm is the wind speed; Ksh and Dsh are the shaft stiffness coefficient and the damping coefficient, respectively; Lm is the mutual inductance; ids and iqs are the d and q axis stator currents, respectively; idr and iqr are the d and q axis rotor currents, respectively. The power coefficient CP [15] can be expressed as
(
8 > < Pr ¼ Pg þ PDC Pr ¼ v dr idr þ v qr iqr Pg ¼ v dg idg þ v qg iqg > : PDC ¼ v DC iDC ¼ C v DC dv DC =dt
ð12Þ
where Pr is the active power at the AC terminal of the RSC; Pg is the active power at the AC terminal of the GSC; PDC is the active power of the DC link; idr and iqr are d and q axis rotor currents, respectively; idg and iqg are d and q axis currents of the grid-side converter, respectively; vdg and vqg are d and q axis voltages of the grid-side converter, respectively; vDC is the capacitor DC voltage; iDC is the current of the capacitor; C is the capacitance of the capacitor. From Eq. (12), the model of the converter together with the DC link can be obtained as
C v DC dv DC =dt ¼ v dg idg þ v qg iqg ðv dr idr þ v qr iqr Þ
ð13Þ
0
C P ¼ 0:22ð116=k0 0:4b 5Þe12:5=ki 1 k0i
¼
1 k0 þ0:08b
ð9Þ
0:035 b3 þ1
where k0i ¼ 1=ð1=k0 þ 0:002Þ, k0 = xtR/Vm. Based on the Betz theory, the maximum value [7] of CP is 0.593. Generator The modeling of DFIG [16] in the synchronously rotating reference frame, i.e. the direct and quadrature (d–q) frame, can be expressed as follows
8 0 xs =xs dids =dt ¼ r s þ xs x0s =xs T 00 ids þ x0s iqs > > > > > þv ds ð1 sr Þe0ds þ e0qs =xs T 00 v dr Lm =Lr > > > > > < x0s =xs diqs =dt ¼ r s þ xs x0s =xs T 00 iqs x0s ids
þv qs ð1 sr Þe0qs e0ds =xs T 00 v qr Lm =Lr > > > > 0 > deds =dt ¼ sr xs e0qs xs v qr Lm =Lr e0ds þ xs x0s iqs =T 00 > > > h i > > : de0 =dt ¼ s x e0 þ x v L =L e0 x x0 i =T 0 qs
r
s ds
s
dr m
r
qs
s
s
ds
ð10Þ
Damping characteristics of an interconnected system with wind farms As shown in the dotted line box of Fig. 1, wind power generators are connected to the interconnected power system. With the line resistance and the power loss ignored, the generators’ electromagnetic power deviations in Eq. (1) can be obtained by the following equations:
where Vw and dw are the voltage magnitude and phase angle at the connection point, respectively; X13 and X23 are the reactances from G1 and G2 to the connection point, respectively; ‘‘0” in the subscript indicates the initial value. Neglecting the load change and system loss yields
DP1e þ DPw ¼ DP2e
0
where wdr and wqr are the d and q axis rotor flux linkages, respectively; Ls is the stator self-inductance; Lr is the rotor selfinductance; Lm is the mutual inductance; rr is the rotor resistance; sr is the rotor slip; xs is the stator reactance; x0 s is the stator transient reactance; e0 ds and e0 qs are the d and q axis voltages behind the transient reactance, respectively; T0 0 is the rotor circuit time constant; ids and iqs are the d and q axis stator currents, respectively; vds and vqs are the d and q axis stator terminal voltages, respectively; vdr and vqr are the d and q axis rotor voltages, respectively.
The pitch angle of the blade is controlled to optimize the power extraction of a wind turbine as well as to prevent overrated power production with strong wind. The pitch servo [5] is modeled as
ð11Þ
where bref is the reference of the blade pitch angle; Tb is the inertia time constant of the pitch control system. Converter model The converter model in the DFIG system comprises of two pulse width modulation invertors connected back to back via a DC link. The rotor-side converter (RSC) and the grid-side converter (GSC) act as controlled voltage sources. The RSC supplies an AC voltage at the slip frequency to the rotor, whereas the GSC supplies an AC voltage at the power frequency to the grid and maintains the DC-link voltage constant. The power balance equations for the converter model can be formulated as
ð15Þ
where DPw is the dynamic active power of the wind farm. To form the state equation set, DPw must be expressed by state variables. From [17], DPw = g1DxB. Here, g1 is the modulus of the wind farm dynamic frequency characteristic; DxB is the frequency deviation of the wind farm at the connection point, and can be expressed by the speed of the generator [17], i.e. DxB = g2Dx1 (with the wind farm at the sending end), DxB = g2Dx2 (with the wind farm at the receiving end). Thus, DPw can be obtained by the following equations.
Pitch control
db=dt ¼ ðbref bÞ=T b
DP1e ¼ E1 V w0 =X 13 cosðd10 dw0 ÞðDd1 Ddw Þ ¼ k1 ðDd1 Ddw Þ DP2e ¼ E2 V w0 =X 23 cosðdw0 d20 ÞðDdw Dd2 Þ ¼ k2 ðDdw Dd2 Þ ð14Þ
DPw ¼ k3 Dx1 ðArea 1Þ DPw ¼ k4 Dx2 ðArea 2Þ
ð16Þ
By employing Eqs. (1) and (14)–(16), DPw and DxB can be eliminated, and the state space equation sets of the interconnected system with the wind farm at the sending end and at the receiving end respectively can be obtained as Eqs. (17) and (18),
3 2 0 Dd_ 1 6 Dd_ 7 6 0 6 6 2 7 k1 k2 7¼6 6 6 4 Dx _ 1 5 4 M1 ðk1 þk2 Þ k1 k2 _2 M ðk Dx þk Þ 2
2
1
2
0
1
32 3 Dd1 6 1 7 76 Dd2 7 7 7 7 0 76 5 4 D x 1 5 D2 Dx 2 M 0
0
0
k1 k2 M1 ðk1 þk2 Þ
D1 k1 k3 M þ M1 ðk 1 1 þk2 Þ
k1 k2 M2 ðk1 þk2 Þ
k1 k3 M2 ðk1 þk2 Þ
2
ð17Þ 3 2 0 Dd_ 1 6 Dd_ 7 6 0 6 6 2 7 k1 k2 6 7¼6 6 4 Dx _ 1 5 4 M1 ðk1 þk2 Þ k1 k2 _2 M ðk Dx þk Þ 2
2
1
2
0
1
0
0
0
1
k1 k2 M1 ðk1 þk2 Þ
D1 M 1
k2 k4 M1 ðk1 þk2 Þ
k1 k2 M2 ðk1 þk2 Þ
0
D2 k2 k4 M þ M2 ðk 2 1 þk2
32 3 Dd1 76 76 Dd2 7 7 76 74 Dx 7 5 1 5 D x 2 Þ ð18Þ
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P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
To examine the damping properties of the interconnected system with the wind farm integrated, the damping change of Eqs. (17) and (18) and (2) should be investigated, and this can be carried out by employing the concept of total damping as detailed in [18]. If positive damping is manifested by Eqs. (17) and (18), then the wind farm can provide positive damping effect, and vice versa. It is proved mathematically in [19] that the sum of eigenvalues of the state equation set is the same as the sum of the diagonal elements in the state matrix; meanwhile, the complex eigenvalue is conjugate. Thus, the total system damping of all modes is equal of the sum of the diagonal elements in the state matrix. Hence, the total system damping for the scenario without the wind farm can be obtained from Eq. (2), and represented by (D1/M1 + D2/M2); the total system damping for the scenario with the wind farm at the sending end is (D1/M1 + D2/M2) + k1k3/M1(k1 + k2), and this can be derived from Eq. (17); similarly, the total system damping for the scenario with the wind farm at the receiving end is (D1/M1 + D2/M2) + k2k4/M2(k1 + k2), and this can be obtained from Eq. (18). For the scenario with the wind farm at the sending end, the damping increment is k1k3/M1(k1 + k2) over the scenario without the wind farm; here, k1 ¼ E1 V w0 =X 13 cosðd10 dw0 Þ, k2 ¼ E2 V w0 = X 23 cosðdw0 d20 Þ, k3 = g1g2. Therefore, the damping effect with the wind farm integration depends on the magnitude and signs of k1, k2 and k3. Eigenvalue and participation factor The most frequently used approach for studying low frequency oscillation is modal analysis [11]. The linearized model of the power system, mostly expressed in a state space form, is employed in this approach. The canonical description is obtained from the analytical linearization of the differential equation set that defines the state space model of the power system. The eigenvalues of the state matrix are calculated, and then the damping condition associated with each oscillation mode found. For detailed analysis, mode shapes and participations factors can also be used. For a particular eigenvalue k ¼ r þ jx of the state matrix AS, the oscillation frequency is f ¼ x=2p , and the damping ratio for k is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi defined as: n ¼ r= r2 þ x2 . When all eigenvalues are computed, the participation factors can also be obtained so as to examine the correlative degree between each state variable and each system mode. The participation factor of the i-th state variable to the jth eigenvalue can be obtained by the right and left eigenvectors w and v as pij = wijvji/wTjvj, where, for a particular eigenvalue ki, a n-column vector wi satisfying ASwi = kiwi (i = 1,2, . . . , n) is the right eigenvector of ki, and a n-row vector vi satisfying viAS = vi ki (i = 1, 2, . . . , n) is the left eigenvector.
Case study 1 The first sample system is shown in Fig. 2, and is extended from a well-known two-area test system for stability studies [21]. In each area of the system, two synchronous equivalent generators, a load and a capacitor are included. A DFIG equivalent wind generation unit is connected to bus 6 in area 1. Note that each generator represents a group of strongly coupled generators. All data of this system are the same as given in [21]. Each synchronous generator (SG) is represented by a six-order model, with magnetic saturation neglected and voltage regulators included and modeled with an IEEE type 1 model [21]. Each load is modeled as constant impedance. The DFIG is represented by the model described in Section ‘Modeling of DFIG’, the DFIG parameters are: rs = 0.00706 p.u., rr = 0.005 p.u., Lr = 0.156 p.u., Ls = 0.171 p.u., Lm = 3.5 p.u., Dsh = 0.01, Ksh = 0.5, Ht = 3 s, Hg = 0.5 s, Hm = 3.5 s, Tb = 0.25 s. For comparison purposes, the scenario without wind power is considered as the base scenario. There is totally 2800 MW installed generation capacity in this system, while approximately 400 MW of active power is exported from area 1 to area 2. The electromechanical oscillatory modes for the base scenario are shown in Table 1, including the situations without PSS (NPSS) and with PSS equipped in all SGs. The values of eigenvalues, damping and frequency are included. The last column lists the classification of the modes. From Table 1, it can be found that the damping ratios of electromechanical modes are obviously enhanced with PSSs equipped. The contribution of each generator to an oscillatory mode can be assessed by analyzing its participation factor. It is demonstrated in Fig. 3 that mode 1 is characterized by oscillations of G1 against G2 in area 1, mode 2 is characterized by oscillations of G3 against G4 in area 2, and mode 3 is described by oscillations of generators located in area 1 against generators located in area 2. Therefore, mode 1 and mode 2 are the local mode; mode 3 belongs to the interarea oscillation mode. Impacts of the transmission distance of the wind farm on damping characteristics In this section, the impacts of the transmission distance (TD) of the wind farm on damping characteristics are discussed, and two 1
5
Case studies Two sample systems are employed for case studies.
8
9
Area 1
2
L9 C9
C7 L7
3
Area 2
4
13
G4
G2
ð19Þ
10 11
G3
12
Power system stabilizers (PSSs) have been widely used in actual power systems for dynamic stability enhancement. In [20], a typical PSS is modeled by two identical lead/lag networks represented by a gain KPSS, four time constants Tl, T2, T3 and T4, and a washout circuit taking a time constant Tw. The transfer function of this PSS, G(s), can be expressed as Eq. (19), and is employed in this work.
sT w 1 þ sT 1 1 þ sT 3 1 þ sT w 1 þ sT 2 1 þ sT 4
7
G1
Power system stabilizer (PSS)
GðsÞ ¼ K PSS
6
Wind Farm Fig. 2. The two-area four-generator interconnected power system.
Table 1 Electro-mechanical oscillatory modes without wind farms. No.
k
n
f/Hz
Notes
NPSS
1 2 3
0.9546 ± j6.2353 1.0704 ± j6.4774 0.2619 ± j3.6842
0.1513 0.1630 0.0709
0.9924 1.0309 0.5864
Local mode of area 1 Local mode of area 2 Interarea mode
PSS
1 2 3
3.0515 ± j7.5937 3.2942 ± j8.1625 0.8405 ± j3.6049
0.3729 0.3743 0.2271
1.2086 1.2991 0.5737
Local mode of area 1 Local mode of area 2 Interarea mode
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P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
0.3
Mode 1
Mode 2
Mode 3 0.2
0.1
0
G1
G2
G3
G4
Fig. 3. The participation factor of each generator to different electromechanical modes for the scenario without PSS.
different wind farm active power outputs (14 MW and 50 MW) are considered. For the first case, the wind farm has a total active power output of 14 MW, and 414 MW are exported from area 1 to area 2. Table 2 lists the electro-mechanical oscillatory modes in the system with DFIG integration under different transmission distances, including the situations with and without PSSs. For TD = 50 km, the participation factors for all units are shown in Fig. 4. From the results in Table 2, it is known that there is one more interarea mode related to DFIG, i.e. Mode 4, and this can also be explained by Fig. 4. Mode 4 is characterized by oscillations among generators G1, G2, G3, G4 and DFIG. From the results shown in Table 2, the changes of the oscillation frequency and damping ratio of local modes k1,2 are not significant with the increase of TD from 50 km to 300 km, although the oscillation frequency tends to increase; the changes of f and n of interarea mode 1 are not noticeable, while n tends to decrease, and f tends to increase/decrease in the situation with/without PSSs. There are relatively obvious changes in f and n of k4 (i.e. interarea mode 2, as shown in bold in Table 2), f decreases gradually and n tends to increase. Table 3 gives the results of oscillation modes for the system with 50 MW DFIG active power output integrated under different
transmission distances, and 448.9 MW of active power is transmitted from area 1 to area 2. With the increase of the transmission distances, f and n of local modes k1,2 do not change significantly although f still shows an increasing tendency; there are relatively obvious changes in f and n of the interarea mode k3,4 especially k4 (i.e. interarea mode 2, as shown in bold in Table 3), f decreases gradually and n increases from 0.5930 to 0.6325. The oscillation frequency and damping ratios in Tables 2 and 3 show a similar trend, while n shown in Table 3 is significantly smaller than that in Table 2, and this means that the increasing wind power integration level worsens the oscillation modes, especially the interarea mode. In order to systematically analyze the impacts of different DFIG transmission distances on power system dynamic response, suppose that a three-phase grounding short-circuit fault occurred on one of the double lines between buses 8 and 9 (the fault time tf = 1.0 s and clearing time tc = 1.2 s). The relative rotor angle curves between G1 and G4, the voltage on bus 7, and the response of the rotor current of DFIG denoted by idr are shown in Figs. 5–7, respectively. It can be seen from Figs. 5–7 that the change of TD has less impact on the system dynamic response, and the profiles of these response curves are similar, as illustrated in Fig. 5. With the increase of TD, the voltage tends to increase and idr tends to decrease. In addition, as can be observed from Figs. 5–7, the effectiveness of PSSs is obvious. Impacts of tie-line power on inter-area oscillation mode The amount and even the direction of the tie-line power flow are subjected to changes under various system operating conditions. In this section, the impacts of the tie-line power flow on the inter-area oscillation mode are examined by adjusting the outputs of generators under various operating conditions, with wind farm output as 30 MW, TD as 25 km, and PSSs equipped. By adjusting the overall generation output in area 1, the tie-line power transmitted from area 1 to area 2 changes. In this way, the changes of the damping performances of the inter-area oscillation modes can be examined, and the results are shown in Table 4. It can be seen from the results in Table 4 that with the increase of the tie-line power from 46 MW to 618 MW, f and n of the
Table 2 Electro-mechanical oscillatory modes in the system with DFIG integration under different transmission distances (with 14 MW DFIG output). TD (km)
No.
Without PSS
With PSS
k
n
f/Hz
k
n
f/Hz
Dominant machines
Notes
5
1 2 3 4
0.9542 ± j6.2393 1.0864 ± j6.4665 0.2671 ± j3.6937 0.3523 ± j0.7567
0.1512 0.1657 0.0721 0.4221
0.9930 1.0292 0.5878 0.1204
3.0468 ± j7.5846 3.3151 ± j8.1507 0.8432 ± j3.6066 0.5898 ± j0.6346
0.3728 0.3768 0.2277 0.6808
1.2071 1.2972 0.5740 0.1012
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
50
1 2 3 4
0.9544 ± j6.2403 1.0864 ± j6.4665 0.2668 ± j3.6928 0.3475 ± j0.7443
0.1512 0.1657 0.0721 0.4231
0.9932 1.0292 0.5877 0.1185
3.0493 ± j7.5982 3.3153 ± j8.1510 0.8433 ± j3.6085 0.5890 ± j0.6335
0.3724 0.3768 0.2276 0.6809
1.2093 1.2973 0.5743 0.1008
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
100
1 2 3 4
0.9547 ± j6.2416 1.0864 ± j6.4665 0.2664 ± j3.6921 0.3424 ± j0.7315
0.1512 0.1657 0.0720 0.4239
0.9934 1.0292 0.5876 0.1164
3.0551 ± j7.6171 3.3154 ± j8.1511 0.8433 ± j3.6095 0.5889 ± j0.6325
0.3723 0.3768 0.2275 0.6814
1.2123 1.2973 0.5744 0.1006
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
200
1 2 3 4
0.9552 ± j6.2446 1.0864 ± j6.4666 0.2656 ± j3.6913 0.3326 ± j0.7072
0.1512 0.1657 0.0718 0.4256
0.9939 1.0292 0.5875 0.1126
3.0613 ± j7.6483 3.3156 ± j8.1519 0.8433 ± j3.6112 0.5886 ± j0.6291
0.3716 0.3768 0.2274 0.6832
1.2173 1.2974 0.5747 0.1001
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
300
1 2 3 4
0.9557 ± j6.2477 1.0865 ± j6.4666 0.2648 ± j3.6907 0.3220 ± j0.6831
0.1512 0.1657 0.0716 0.4264
0.9944 1.0292 0.5874 0.1087
3.0697 ± j7.6866 3.3158 ± j8.1522 0.8432 ± j3.6154 0.5884 ± j0.6251
0.3708 0.3768 0.2271 0.6854
1.2234 1.2975 0.5754 0.0994
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
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0.2
0.25
Mode 1
Mode 1 0.2
Mode 2
Mode 2 0.15
Mode 3
Mode 3 0.1
Mode 4
0.1
Mode 4
0.05
0
G1
G2
G3
G4
DFIG
0
G1
(a) Without PSS
G2
G3
G4
DFIG
(b) With PSS
Fig. 4. The participation factors of each generator in different electromechanical modes.
Table 3 Electro-mechanical modes in the system with DFIG integration under different transmission distances (with 50 MW DFIG output). TD (km)
No.
Without PSS
With PSS
k
n
f/Hz
k
n
f/Hz
Dominant machines
Notes
5
1 2 3 4
0.9548 ± j6.2332 1.1266 ± j6.4366 0.2732 ± j3.6863 0.3246 ± j0.8653
0.1514 0.1724 0.0739 0.3512
0.9921 1.0244 0.5867 0.1377
3.0454 ± j7.5783 3.3684 ± j8.1149 0.8444 ± j3.6092 0.5689 ± j0.7723
0.3729 0.3834 0.2278 0.5930
1.2061 1.2915 0.5744 0.1229
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
50
1 2 3 4
0.9549 ± j6.2335 1.1264 ± j6.4367 0.2732 ± j3.6848 0.3218 ± j0.8503
0.1514 0.1724 0.0740 0.3540
0.9921 1.0244 0.5865 0.1353
3.0478 ± j7.5791 3.3682 ± j8.1152 0.8445 ± j3.6071 0.5686 ± j0.7521
0.3731 0.3833 0.2280 0.6031
1.2063 1.2916 0.5741 0.1197
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
100
1 2 3 4
0.9550 ± j6.2342 1.1263 ± j6.4368 0.2731 ± j3.6838 0.3193 ± j0.8362
0.1514 0.1724 0.0739 0.3567
0.9922 1.0244 0.5863 0.1331
3.0469 ± j7.5843 3.3681 ± j8.1153 0.8446 ± j3.6066 0.5684 ± j0.7426
0.3728 0.3833 0.2280 0.6078
1.2071 1.2916 0.5740 0.1182
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
200
1 2 3 4
0.9555 ± j6.2362 1.1261 ± j6.4371 0.2727 ± j3.6825 0.3151 ± j0.8118
0.1515 0.1723 0.0739 0.3618
0.9925 1.0245 0.5861 0.1292
3.0485 ± j7.5912 3.3678 ± j8.1158 0.8446 ± j3.6051 0.5675 ± j0.7137
0.3727 0.3833 0.2281 0.6224
1.2082 1.2917 0.5738 0.1136
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area1 Local mode of area 2 Interarea mode 1 Interarea mode 2
300
1 2 3 4
0.9559 ± j6.2386 1.1259 ± j6.4372 0.2723 ± j3.6817 0.3115 ± j0.7897
0.1515 0.1723 0.0738 0.3669
0.9929 1.0245 0.5859 0.1257
3.0522 ± j7.6094 3.3675 ± j8.1161 0.8447 ± j3.6042 0.5666 ± j0.6939
0.3728 0.3832 0.2281 0.6325
1.2111 1.2917 0.5736 0.1104
G1, G3, G1, G1,
G2 G4 G2, G3, G4 G2, G3, G4, DFIG
Local mode of area 1 Local mode of area 2 Interarea mode 1 Interarea mode 2
(a) TD=50km
(b) TD=200km
Fig. 5. The relative rotor angle curves between G1 and G4. 1. Wind farm active power output 14 MW without PSS; 2. Wind farm active power output 50 MW without PSS; 3. Wind farm active power output 14 MW with PSS; 4. Wind farm active power output 50 MW with PSS.
P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
VBus7 (p.u.)
VBus7 (p.u.)
154
(a)
(b)
idrDFIG (p.u.)
idrDFIG (p.u.)
Fig. 6. The voltage curve on bus 7. (The meanings of a, b, 1–4 are the same as those in Fig. 5.)
(a)
(b)
Fig. 7. The d axis rotor current of DFIG. (The meanings of a, b, 1–4 are the same as those in Fig. 5).
inter-area oscillation mode 1 increase initially and then decrease, while f of the inter-area oscillation mode 2 shows an increasing tendency and n decreases gradually. Under the same fault condition as that in Section ‘Impacts of the transmission distance of the wind farm on damping characteristics’, Fig. 8 shows the response curves under different tie-line power flows, from left to right: the relative rotor angle between G1 and G4, the voltage on bus 7 and the rotor current of DFIG denoted by idr. With the increase of the tie-line power flow, it can be observed from the curves that dG1,G4 and idr gradually increase, VBus7 gradually decreases. In addition, the amplitudes of the three curves increase after a fault occurs, and longer time is needed for the system to stabilize. By adjusting the generation outputs of the units including the wind farm at the receiving end, the tie-line power flow can be exported from area 2 to area 1. The inter-area oscillation modes under different tie-line power flows are given in Table 5. From this table, it can be observed that with the increase of the tie-line power flow, f and n of the inter-area oscillation mode 1 increase initially and then decrease; Both f and n of the inter-area oscillation mode 2 decreases initially and then increases. When the transmitted power from area 2 to area 1 is 474 MW, the interarea mode 2 has a negative real eigenvalue. Impacts of the DFIG penetration level on low frequency characteristics The studied operating condition is described in Section ‘Impacts of tie-line power on inter-area oscillation mode’, i.e. the output of the wind farm is 30 MW, TD = 25 km, and with PSSs equipped. To keep the tie-line power flow from area 1 to area 2 maintained at
Table 4 The interarea oscillation mode under various tie-line power flows (from area 1 to area 2). No.
Tie-line power flow/ MW
k
n
f/Hz
Interarea mode 1
46 143 239 335 430 618
0.7459 ± j3.6074 0.8545 ± j3.6252 0.8534 ± j3.6384 0.8501 ± j3.6342 0.8439 ± j3.6134 0.8195 ± j3.5233
0.2025 0.2294 0.2284 0.2278 0.2274 0.2266
0.5741 0.5769 0.5791 0.5784 0.5751 0.5608
Interarea mode 2
46 143 239 335 430 618
0.5106 ± j0.4424 0.5906 ± j0.5931 0.5842 ± j0.6355 0.5793 ± j0.6756 0.5753 ± j0.7156 0.5693 ± j0.8041
0.7558 0.7056 0.6768 0.6509 0.6266 0.5778
0.0704 0.0944 0.1011 0.1075 0.1139 0.1281
400 MW, the power outputs of G1 and G2 are adjusted when the power output of the wind farm changes. The interarea oscillation modes at various DFIG penetration levels are given in Table 6, where the wind power output (%) is the percentage of the DFIG output over the total output power of area 1. Under the same fault condition with Section ‘Impacts of the transmission distance of the wind farm on damping characteristics’, the responses curves under various DFIG penetration levels are shown in Fig. 9, and the meanings of the curves are the same as those in Fig. 8. It can be seen from Table 6 that with the increase of the DFIG output power, f and n of the inter-area oscillation mode 1 tend to
155
VBus7 (p.u.)
idrDFIG (p.u.)
P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
Fig. 8. Response curves under different tie-line power flows.
Table 5 Interarea oscillation modes under various tie-line power flows (from area 2 to area 1). No.
Tie-line power flow/ MW
k
n
f/Hz
Interarea mode 1
51 153 255 361 474
0.7431 ± j3.5482 0.7263 ± j3.4717 0.6943 ± j3.3415 0.5601 ± j3.1347 0.3250 ± j2.6001
0.2049 0.2048 0.2034 0.1759 0.1240
0.5647 0.5525 0.5318 0.4989 0.4138
Interarea mode 2
51 153 255 361 474
0.5071 ± j0.3675 0.4953 ± j0.2384 0.2814 ± j0.1315 0.5494 ± j0.3639 0.56784
0.8097 0.9010 0.9059 0.8337 –
0.0585 0.0379 0.0209 0.0579 –
the wind power output reaches 6%, d14 decreases and VBus7 and idr tend to increase. Case study 2
No.
Wind power output (%)
k
n
f/Hz
Interarea mode 1
0 1 3 5 6
0.8405 ± j3.6049 0.8437 ± j3.6221 0.8458 ± j3.6222 0.8481 ± j3.6231 0.8499 ± j3.6268
0.2271 0.2269 0.2274 0.2279 0.2281
0.5737 0.5765 0.5765 0.5766 0.5772
Interarea mode 2
0 1 3 5 6
0.5692 ± j0.5288 0.5949 ± j0.6752 0.5729 ± j0.7247 0.5662 ± j0.7797 0.5569 ± j0.7863
0.7326 0.6611 0.6202 0.5876 0.5779
0.0842 0.1075 0.1153 0.1241 0.1251
VBus7 (p.u.)
increase, while f of the inter-area oscillation mode 2 tends to increase and n decreases gradually. When the wind power output varies from 1% to 5%, the relative rotor angle between G1 and G4 and the voltage on bus 7 do not change significantly, while when
Fig. 9. Response curves at various DFIG output levels.
idrDFIG (p.u.)
Table 6 Interarea oscillation modes at various DFIG penetration levels.
The second sample system is shown in Fig. 10, and is a 2-area 8-unit 24-bus [22] power system. In this system, each generator is represented by a sixth-order model without PSS installed. G6–G8 are included in area 1, while G1–G5 in area 2. The regional tie-lines can be classified into two channels, i.e. the single-circuit channel branch 6–7, and the double-circuit channel branches 4–10 and 4–11. Usually, the loading level in the single-circuit channel is heavier than that in the double-circuit channel. With this in mind, the following analysis on the tie-line power flow will focus on branch 6–7, and tie-line power flow is from area 1 to area 2. The simulation results of this sample system are shown in Table 7. There are four interarea oscillation modes, and can be described by oscillations of generators located in area 1 against generators located in area 2. A DFIG equivalent wind generation unit is connected to bus 2 in area 1. Note that each generator does not represent a single unit but a group of strongly coupled generators. There is one more interarea mode related to the DFIG with the oscillation frequency around 0.54 Hz. The impacts of different factors on the five oscillation modes are discussed in this section. The total active power output of the wind farm is around 50 MW and 247 MW are exported from area 1 to area 2 through the tie-line 6–7. The results of electro-mechanical modes of the system with DFIG integration under different transmission distances are given in Table 8. The power flow on the tie-line 6–7 can be adjusted by changing the outputs of generators in area 1 given TD = 50 km, and the interarea oscillation modes under different tie-line power flows are listed in Table 9. Similarly, the interarea oscillation modes at various DFIG penetration levels are given in Table 10, in which the
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P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
G8 24
Table 10 Interarea oscillation modes under various DFIG penetration levels.
~
6 1
2
7
section
13 G1
10 L36
3
11
~
G6
L7
L5
~
4 23 G7
~
12
L3
Area 1
5
14 9 18 L8 G3 G2
20
~
~
8 L4
22
L1 L2
16
17
~
Wind power output (%)
0
1
3
5
7
1
n f/Hz
0.0796 1.6708
0.0781 1.6785
0.0775 1.6815
0.0769 1.6841
0.0762 1.687
2
n f/Hz
0.0992 1.5070
0.1008 1.5103
0.1013 1.5123
0.1017 1.5145
0.1023 1.5159
3
n f/Hz
0.0889 1.2426
0.0892 1.2422
0.0892 1.2422
0.0892 1.2422
0.0892 1.2423
4
n f/Hz
0.0776 1.0342
0.0779 1.0344
0.0778 1.0351
0.0777 1.0357
0.0777 1.0363
5
n f/Hz
0.0723 0.5331
0.0558 0.5302
0.0421 0.5429
0.0334 0.5542
0.0271 0.5633
15
21
G4
~
19
No.
L9
G5
Area 2
Fig. 10. A 2-area 8-machine 24-node system.
Table 7 Interconnected oscillation modes. No.
k
n
f/Hz
Dominant generators
1 2 3 4
0.8378 ± j10.4979 0.9438 ± j9.4691 0.6970 ± j7.8073 0.5062 ± j6.4982
0.0796 0.0992 0.0889 0.0776
1.6708 1.5070 1.2426 1.0342
G1, G1, G5, G1,
G7 G2, G7 G6 G5, G6
Table 8 Electro-mechanical modes for the system with DFIG integration under different transmission distances. No.
TD (km)
50
100
200
300
1
n f/Hz
0.0775 1.6810
0.0772 1.6825
0.0769 1.6854
0.0765 1.6881
2
n f/Hz
0.1011 1.513
0.1014 1.5132
0.1019 1.5134
0.1025 1.5136
3
n f/Hz
0.0891 1.2423
0.0892 1.2422
0.0892 1.2421
0.0892 1.2421
4
n f/Hz
0.0777 1.0354
0.0778 1.0353
0.0779 1.0353
0.0781 1.0352
5
n f/Hz
0.0372 0.5491
0.0382 0.5475
0.0402 0.5445
0.0422 0.5414
Table 9 Interarea oscillation modes under various tie-line power flows. No.
Tie line power/MW
56
163
247
292
475
1
n f/Hz
0.1113 1.5988
0.1095 1.6104
0.0772 1.6825
0.0852 1.6765
0.0977 1.6351
2
n f/Hz
0.0912 1.4568
0.0941 1.4576
0.1014 1.5132
0.1062 1.4952
0.1033 1.4511
3
n f/Hz
0.0884 1.2462
0.0888 1.2442
0.0892 1.2422
0.0891 1.2422
0.0887 1.2444
4
n f/Hz
0.0690 1.036
0.0716 1.042
0.0778 1.0353
0.0794 1.0292
0.0831 0.9989
5
n f/Hz
0.0737 0.6294
0.0181 0.6431
0.0382 0.5475
0.2548 0.4610
0.2351 0.1954
wind power output levels (%) are the same as those stated in Section ‘Impacts of the DFIG penetration level on low frequency characteristics’. As shown in Table 8, with the increase of TD from 50 km to 300 km, the oscillation frequency and damping ratio of the four
interarea oscillation modes do not change significantly. However, the oscillation frequencies of modes 1 and 2 tend to increase, while those of modes 3 and 4 tend to decrease. For the new interarea mode 5, f tends to decrease and n tends to increase, and this is consistent with the analysis in Section ‘Impacts of the transmission distance of the wind farm on damping characteristics’. It can be seen from Table 9 that with the increase of the tie-line power flow from 56 MW to 475 MW, the f0 s of the inter-area oscillation modes 1, 2 and 4 increase initially and then decrease, while f of the mode 5 tends to decrease. With the increase of the DFIG power output, it can be seen from Table 10 that the n0 s of modes 1, 3 and 5 tend to decrease, while those of modes 2 and 4 increases gradually. Conclusions In this work, the impacts of several factors, including the DFIG transmission distance, DFIG penetration level, tie-line power flow between areas and PSSs, on the interarea mode oscillation of interconnected power systems are investigated. A 2-area 4-unit 13-bus power system and a 2-area 8-unit 24-bus power system are employed for eigenvalue analysis and dynamic simulations. There is one more interarea oscillatory mode related to the integrated DFIG Simulation results show that the impacts of the DFIG transmission distances on the oscillating characteristics of the local mode and the interarea mode are not very significant, but with the increase of the transmission distances, the oscillation frequencies of local modes gradually increases, the damping ratio of the new interarea mode tends to increase. Thus, an appropriate DFIG transmission distance should be considered in planning the DFIG integration to the power system concerned. If the wind farm is located at the sending end, the frequency of the inter-area oscillation mode tends to increase with the increase of the tie-line power flow, and the frequency of the interarea oscillation mode related to DFIG decreases at a certain value of the tieline power flow. If the wind farm is located at the receiving end, the frequency of the inter-area oscillation mode tends to decrease with the increase of the tie-line power flow, and the frequency of the interarea mode related to DFIG increases at a certain value of the tie-line power flow. Thus, the impacts of the wind farm on the interarea oscillation mode should be taken into account in determining the common coupling point. The increase of the DFIG power output has a less impact on the local oscillation mode, while a relatively significant effect on the inter-area oscillation mode. When the DFIG penetration level reaches a certain value, the oscillation modes of the system will be deteriorated. Thus, an appropriate DFIG penetration level should be considered in the DFIG integration planning and operation so as to the small signal stability of the power system concerned.
P. He et al. / Electrical Power and Energy Systems 78 (2016) 148–157
It is clearly shown, by the simulation results of eigenvalue analysis and dynamic simulations, that the damping and dynamic response characteristics of power systems with wind farms can be improved by installing PSSs.
Acknowledgements This work is jointly supported by National High Technology Research and Development Program of China (863 Program) (No. 2015AA050202), the National Natural Science Foundation of China (NSFC) (No. 51507157), and the Doctoral Scientific Research Foundation of Zhengzhou University of Light Industry (No. 2014BSJJ043).
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