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Improvement on probabilistic small-signal stability of power system with large-scale wind farm integration
Electrical Power and Energy Systems 61 (2014) 482–488
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Improvement on probabilistic small-signal stability of power system with large-scale wind farm integration X.Y. Bian a, X.X. Huang a, K.C. Wong b, K.L. Lo c, Yang Fu a,⇑, S.H. Xuan a a
Electric Engineering College, Shanghai University of Electric Power, China Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong c Faculty of Engineering, University of Strathclyde, Glasgow, Scotland, UK b
a r t i c l e
i n f o
Article history: Received 17 January 2014 Received in revised form 31 March 2014 Accepted 2 April 2014
Keywords: Wind farm integration Probabilistic eigenvalue analysis Power system stabilizer (PSS) Participation factor Center frequency method
a b s t r a c t This paper studies probabilistic small-signal stability of power systems with wind farm integration, considering the stochastic uncertainty of system operating conditions. The distribution function of the realpart of system eigenvalue is computed by the method of probabilistic eigenvalue analysis. For improving probabilistic small-signal stability, PSS is adopted. A method for optimizing PSS based on participation factor and center frequency method is proposed. In order to evaluate the above proposed methods, the procedure is applied to a test system. The simulation results show that the stochastic variation of wind generation can induce a higher probability of system instability when compared with one that has no wind generation. With eigenvalues distributing in a wider range, it becomes difficult for PSS tuning. By applying the proposed optimized PSSs approach, probabilistic stability of system can be significantly improved. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction With the rapid development of wind power in China, impacts of large scale wind power penetration on power system stability have been paid more and more attention [1–4]. The random fluctuations of wind power output increase the uncertainty of the system, which will produce adverse effect on the system dynamic stability, especially the small-signal stability. It is meaningful to analyze power system small-signal stability with probabilistic methods considering the effects of uncertainty of wind power farm output and stochastic change of load. There are two kinds of methods to analyze small-signal probabilistic stability of power system with wind power penetration: one is based on the Monte Carlo simulation by which a large number of deterministic samples are generated by Monte Carlo sampling to calculate the probability of stability. This method is employed in [5,6]. Although the Monte Carlo simulation has accurate results, it is a time-demanding method because it requires computation of a large number of deterministic samples. The other one is the numerical analysis method. The probabilistic characteristics of eigenvalues are obtained by a formula to determine the probability of stability which is used only in Refs. [7,8]. The method requires complex
⇑ Corresponding author. Tel.: +86 21 35303121. E-mail address: [email protected] (Y. Fu). http://dx.doi.org/10.1016/j.ijepes.2014.04.005 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
formula derivation, but only one step of calculation to obtain the probabilistic distribution of system eigenvalues. In [7,8], Gram– Charlier series and system eigenvalue sensitivity are applied to study the impact of wind power fluctuations on the small-signal probability stability of the system, and is used to verify the correctness of numerical analysis method. But the wind generation model established in [7,8] is a simple one, and only the output variation of wind farm is considered as system uncertainties. It is verified in [8] that the integration of wind farm would cause system probabilistic small-signal instability, but in their study, no measures are proposed to improve system probabilistic stability. Based on the former studies [9–14], a probabilistic small-signal stability method is proposed in this paper by considering the stochastic variation of wind farm output, the fluctuation of load and the synchronous generator output. The complete DFIG transient model [15–17] is adopted. The probabilistic small-signal stability of power system with wind power integration is analyzed by numerical analysis method, and power system stabilizer (PSS) on synchronous generators is applied to improve the stability probability. The previous study on PSS [18,19] shows that the coordinated PSS can enhance the network damping. The PSS in this previous work is designed without considering the uncertainties of random wind power, generating and loading conditions. It may lose damping performance and fail to stabilize the system when operating condition changes. In order to adapt the PSS parameters in multi-operating conditions of power system, this
X.Y. Bian et al. / Electrical Power and Energy Systems 61 (2014) 482–488
paper presents an approach based on participation factors and center frequency method for PSS parameters tuning and coordination. The paper is organized as follows. In ‘PMT modeling technique’, the Plug-in Modeling Technology (PMT) [20] is introduced to build the complete system model. In ‘Probabilistic distribution of power system eigenvalues in multi-operating conditions’, the expression of the probability distribution of system eigenvalues is obtained by the Gram–Charlier expansion method. In ‘PSS parameter adjustment/setting based on the center frequency method’, the center frequency method based on participation factors is proposed for adjusting the parameters of PSS for improving probabilistic small-signal stability. In ‘Case study’, an example of 4-machine 2area power system with grid-connected wind power source is given. The proposed approach is applied to analyze and improve the probabilistic stability. The results of probabilistic stability analysis on the test system demonstrate that the small-signal stability of power system is indeed affected by the stochastic variation of grid-connected wind generation and can be improved by the proposed installation of PSS.
483
Probabilistic distribution of power system eigenvalues in multioperating conditions The eigenvalues of the state matrix A are used to determine the stability of the power system with respect to small disturbances. The eigenvalues are determined by the operating state given by each set of generation and load. Hence it should be noted that, as the operating state varies with respect to generation and load changes, the eigenvalues also vary. Under multi-operating conditions, with loads and output of wind turbine generator and synchronous generators regarded as random variables which may possess any type of distribution, probabilistic methods are applied to load flow calculation and eigenvalue computation. The method based on Gram–Charlier expansion with a hybrid algorithm using moments and cumulates is employed to obtain the Probabilistic Density Function (PDF) of system critical eigenvalues. For probabilistic load flow computation, all nodal voltages (V) and nodal injections (Y) are regarded as random variables. In an N-node system, Y can be expressed as a quadratic function.
PMT modeling technique
Y ¼ f ðVÞ ¼ gðV 1 V 1 ; . . . ; V i V j ; . . . ; V 2N V 2N Þ
PMT is adopted in this paper to construct state matrix of power system when analyzing small-signal stability of electric power system [9,12,20]. The whole model of the system is shown in Fig. 1. In Fig. 1, each system component is modeled as a module with 4 pins of voltage and current, which can be conveniently plugged into the network module. The models of synchronous generator units in [21,22], which contain the complete six-order generator model, including the excitation system and the prime mover model, are used to represent the general power plants in this paper. Complete transient models [15– 17] of the Doubly Fed Induction Generator (DFIG), which is widely used in wind farm, are adopted. The whole model [15–17] of the doubly fed induction generator including wind turbine, two-mass shaft system, rotor-side converter and grid-side converter and their control system, and pitch angle control system for reflecting the effect of wind farm on power system are all included.
With expansion at expectation point V using Taylor series, these equations can be represented as in Eq. (2).
Y ¼ f ðVÞ þ J V DV þ f ðDVÞ
ðVÞ where J V is the Jacobin matrix at the expectation point J V ¼ @f@V
ð1Þ
ð2Þ V¼V
þ DY, the expected value of Y is expressed as in With Y ¼ Y Eq. (3).
Y ¼ f ðVÞ þ f ðDVÞ
ð3Þ
where f ðDVÞ ¼ gðC V 1 ;V 1 ; . . . ; C V i ;V J ; . . . ; C V 2N ;V 2N Þ; C V i ;V J can be obtained by Eq. (4)
n h i C V ¼ J 1 C Y E J V DVf T DðVÞ þ f ðDVÞDV T J TV þ f ðDVÞf T ðDVÞ V o T þ f ðDVÞf T ðDVÞ J 1 V
Fig. 1. The power system with DIFG in PMT.
ð4Þ
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where DV ¼ J 1 DY ¼ J 1 ðY YÞ,CY is the covariance of nodal injecV V tions, and are calculated from Y in (2) and Y in (3). These values are substituted into (3) and the expected value of the operation voltage of power system is then iterated until it converges. The covariance matrix of the voltage is:
C V ¼ EðDV DV T Þ ¼ J 1 C Y ðJ 1 Þ V V
T
ð5Þ
According to the nature [23,24] of cumulants, j-order cumulants of voltage deviation are:
PSS parameter adjustment/setting based on the center frequency method Power system stabilizer
j
cj ðDVÞ ¼ ðJ 1 Þ cj ðDYÞ V
ð6Þ
where CV and CY are covariance matrices of voltage vector and injected power respectively. cj(DV) and cj(DY) are J-order cumulants of the voltage and the injected power respectively. A certain eigenvalue of power system can be expressed as a nonlinear function of the node voltage vector:
kk ¼ g k ðVÞ
ð7Þ
By expansion using Taylor Series at the vicinity of the expected value of voltage and keeping the 2-order approximate expression, these equation can be obtained as in Eq. (8):
2N X @g k kk ¼ gðVÞ þ @V
i V¼V
i¼1
DV i
2N X 2N 1X @ 2 gk þ j DV i DV j 2 i¼1 j¼1 @V i @V j V¼V
ð8Þ
The expected value of the eigenvalue can be interpreted as follows:
Eðkk Þ ¼ gðVÞ
ð9Þ
The covariance of the eigenvalue can be got according to the definition of covariance and expressed as Eq. (10).
C km ;kn ¼ Ef½km Eðkm Þ½kn Eðkn Þ ¼
2N X 2N X J km;i J kn;j C v i ;v j
ð10Þ
i¼1 j¼1
By neglecting the higher order term in Eq. (8) and re-arranging the terms yields:
Dk ¼ J k DV
ð11Þ
where Jk is the retained first order derivative term. DV ¼ J Y DY is obtained according to the probabilistic load flow equation and is substituted into the following equation:
Dk ¼ J k J Y DY ¼ JDY
ð12Þ
J1 . V
where J Y ¼ Thus the relation between nodal injection power and the eigenvalue is obtained. According to the nature [24] of cumulants, J-order cumulants of eigenvalue kk can be obtained as follow. 2N X cj ðkk Þ ¼ Jjk;i ck ðY i Þ
ð13Þ
i¼1
The probability density function and cumulative distribution function of the eigenvalue kk can be obtained by applying Gram– Charlier series. It can be expressed in Eqs. (14) and (15) respectively:
c ðkk Þ c ðkk Þ f ðkk Þ ¼ NðxÞ 1 þ 3 3 ðx3 3xÞ þ 4 4 ðx4 6x2 þ 3Þ 6r 24r c5 ðkk Þ 5 ðx 10x3 þ 15xÞ þ 120r5 Pfkk < xc g ¼
where f ðkk Þ is probability density function of the eigenvalue kk , N(x) is the probability density function for the standard normal distribukÞ tion, cj ðkk Þ is a J order semi-invariant variable, x ¼ kk Eðk is a norr malizing variable, and the standard variance of eigenvalues can be obtained by Eq. (10).
Z
xc
NðxÞdx Nðxc Þ
1
þ
Phase lead provided by power system stabilizer (PSS) can compensate the phase lag caused by an excitation control device and as such the damping level of the system can be greatly improved. The linearized transfer function module of the excitation system adopted in this paper is shown in Fig. 2. where DVt is the input voltage of the excitation system, DVPSS is 1 the voltage signal from PSS, 1þpT is the transfer function of the A1 automatic voltage regulator, KA2 is the excitation gain, and DEfd is the excitation voltage output. The angle lag caused by excitation system is as follows.
uf ¼ arctan T A1 x
ð16Þ
The module of the PSS is as follows. where Kp is PSS gain, Tw and T1/T2 are the washout and lead/lag time constants, respectively. PSS parameter adjustment The frequency characteristic of excitation system shows that the phase of excitation system becomes slightly lag or may even lead at low frequency. Therefore, Tw should be set as a high value to make the washout stage as a high-pass filter such that the PSS does not respond to DC offsets and very low frequency modes. When the input signal is Dx, Tw may be set to about 10 s, and lead-lag phase of
pT w 1þpT w
of the excitation system can be obtained
as uf = arctan Twx. When the parameters of the lead-lag stage are T1 and T2 respectively, the relationship of the rotating speed and phase angle of the compensation can be expressed as follows.
uPSS ¼ arctan
ðT 1 T 2 Þx 1 T1T2x
The maximum compensation phase angle is expressed as follows.
ðT T Þ 2 T1T2
1 2 ffiffiffiffiffiffiffiffi uMAX ¼ arctan p
ð18Þ
The center frequency is
fPSS ¼
1 pffiffiffiffiffiffiffiffi 2p T 1 T 2
ð19Þ
The corresponding rotational speed is
pffiffiffiffiffiffiffiffiffiffi
xjuMAX ¼ 1= T 1 T 2
ð20Þ
Let
T 1 ¼ kT 2
ð21Þ
ð14Þ
c ðkk Þ
c3 ðkk Þ 2 xc 1 þ 4 4 x3c 3xc 3 6r 24r
c5 ðkk Þ 4 x 6x2c þ 3 120r5 c
ð17Þ
ð15Þ Fig. 2. The module of the excitation system.
X.Y. Bian et al. / Electrical Power and Energy Systems 61 (2014) 482–488
xjuMAX ¼
n X qi xi =qR
485
ð23Þ
1
Fig. 3. Module of the PSS.
uPSS
Then
uMAX
ðk 1ÞT 2 ðk 1Þ ¼ arctan qffiffiffiffiffiffiffiffi ¼ arctan pffiffiffi 2 2 k 2 kT 2
ð22Þ
In a single-machine system, PSS provides damping to the generator at system oscillation frequency. The lead phase and center frequency can be obtained according to the system oscillation modes and phase frequency characteristics of the excitation system. Suppose an oscillation mode of a single-machine system is pffiffiffiffiffiffiffiffiffiffi q + jx, then x ¼ xjuMAX ¼ 1= T 1 T 2 . With uMAX = ufd, k is obtained according to Eq. (22). T1 and T2 are obtained by substituting Eqs. (22) into Eq. (21). For multi-machine system, each oscillation mode is contributed by several machines. Participation factor is adopted for evaluating the participation degree of each machine to each mode. Suppose one machine participates in two oscillation modes q1 + jx1 and q2 + jx2 with participation factors q1 and q2 respectively, the setting of the machine PSS parameter should involve the two oscillation modes. The higher the participation degree of a machine to a mode, the closer the center frequency is to this mode oscillation frequency. Namely, the center frequency should consider the weight of the participation factor of machines in each mode, and are adjusted as follows,
qi u qR fd
ð24Þ
where qi is the participation factor of the variable Dx of the machine involving in an oscillation mode. qR is the sum of all Dx participation factors of machines in this mode. ufd and uPSS are the lead-lag angle of excitation system and the phase angle of PSS compensation respectively. PSS parameters obtained by Eqs. (23) and (24) can give the best compensation effect to the oscillation mode to which the machine has the biggest participation factor, and can produce possible lesser effect on the other modes. Since coordination is needed in the multi-machine system, the most suitable PSS compensation phase angle could not be determined directly. Eq. (24) is the constraints of the PSS compensation phase angle of this generator. By adopting Eq. (24), the most suitable PSS parameter can be obtained. Case study Test system A two-area system [21] shown in Fig. 4, which is widely used for studies of power system oscillations, is adopted in this paper to demonstrate the proposed method. The network and generator data can be obtained from [21]. G5 represents a wind farm with
Fig. 4. A two-area system with wind farm integration.
Fig. 5. Operating curves of wind farm.
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a capacity of 120 MW, or 80 DFIG with 1.5 MW each, details of the system and DFIG parameters are listed in the Appendix. The original system [21] without wind farm and the test system with DFIGtype wind farm are both simulated and compared. A single equivalent model was used to represent all individual units within the wind farm and it is connected to the system using a typical layout as shown in the appendix. All synchronous generators are represented using 6-order models and are extended to include exciter and power system stabilizer (PSS) models. The whole system model is represented using the aforementioned PMT modeling method. The standardized daily operating curves of load1, load2, G1, G2, G3 and G4 are given in [9]. The standardized daily operating curves of wind farm output and voltage are shown in Fig. 5. The small-signal probabilistic stability analyses By adopting the proposed probabilistic eigenvalue analysis method in section III, the simulation results on the original system is obtained. There are three electromechanical oscillation modes in the original system without the wind farm. These mode shapes are shown in Fig. 6. From the mode shapes, it is observed that Mode 1 is a local mode and with generator G1 swinging against G2; Mode 2 is also a local mode and with generator G3 swinging against G4; whereas Mode 3 is an inter-area mode and with generators G1 and G2 in area 1 swinging against generators G3 and G4 in area 2. Table 1 gives information about the distribution of these 3 modes. It shows that the stability problem is likely to occur because Mode 3 has a probability of 81.78% to remain in the left
Table 1 Electromechanical modes of the original system. Mode
Eigenvalues r ± jx
Frequency (Hz)
Damping n
P{r < 0} (%)
1 2 3
1.0293 ± j5.8877 0.9726 ± j5.5302 0.90122 ± j2.9196
0.94 0.88 0.46
0.172 0.173 0.004
100 100 81.78
half-plane and it is poorly damped, with the mean of damping ratio is 0.0004 and the probability of more than 0.1 is 0. For the test system with the wind farm integration, the statistical result of electromechanical oscillation modes are shown in Table 2. There is a new electromechanical oscillation mode (Mode 4) caused by the wind farm. By analyzing the participation factor, the DFIG-type G5 did not swing against the other synchronous generators although the frequency of Mode 5 is within the range of inter-area oscillation. It can also be concluded that DFIG-type G5 did not participate in the pre-existing electromechanical oscillations. But DFIG-type G5 indeed reduces the probabilistic stability of local modes. The probability that local modes (Mode1 and Mode 2) remain in the left half-plane is reduced dramatically. Conversely, the integration of DFIG can have a positive impact on the probability of inter-area mode (Mode 3) as well as the damping performance as shown in Table 2.
PSS parameters adjustments for the improvement of the probabilistic small-signal stability To improve the probabilistic stability of the test system, PSSs are installed in four synchronous generators, with structures as shown in Fig. 3, and with initial parameters as shown in [21], and Dx as the input signal. The statistics of electromechanical oscillation modes of the test system with PSSs are shown in Table 3. By comparing Tables 2 and 3, the probability of all modes (Mode 1 and 2) to remain in the left half-plane is raised slightly in Table 3, and so is the probability of the damping ratio. The inter-area mode (Mode 3) has some improvement with its damping ratio increasing to 0.04 from 0.027, and stability probability increasing to 98.84% from 90.15%. Because all synchronous generators do not participate in Mode 4, installation of PSS in all synchronous generators does not affect this mode. The result shows that PSSs on synchronous generators can improve the stability of small-signal probability of the system. The plot of the Probabilistic Density Function (PDF) of the real part of Mode 3 under different scenarios is shown in Fig. 7. From this plot, it is observed that the PDF became scattered with the connection of wind farm. Therefore the real part of Mode 3 varies
Table 2 Electromechanical modes of the system with wind farm connected. Mode
Eigenvalues r ± jx
Frequency(Hz)
Damping n
P{r < 0} (%)
1 2 3 4
1.0266 ± j5.8909 1.0557 ± j5.4197 0.0818 ± j3.0797 1.8312 ± j2.8679
0.94 0.86 0.49 0.46
0.172 0.191 0.027 0.538
67.24 78.68 90.15 100
Table 3 Electromechanical modes of the system with wind farm connected (with PSSs).
Fig. 6. Mode shape of rotor angle modes.
Mode
Eigenvalues r ± jx
Frequency(Hz)
Damping n
P{r < 0} (%)
1 2 3 4
1.1013 ± j5.9181 1.1215 ± j5.4389 0.1046 ± j3.0798 1.8312 ± j2.8679
0.96 0.885 0.5 0.46
0.177 0.197 0.04 0.538
68.08 86.65 98.84 100
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X.Y. Bian et al. / Electrical Power and Energy Systems 61 (2014) 482–488
this plot, it is observed that the PDF became scattered with the connection of wind farm. Therefore the real part of Mode 3 varies in a larger range than that of the original system without wind farm. It can also be also observed that the curve moves to the left of the imaginary axis with wind farm connected and it moves further again with PSSs installed. It shows that the integration of wind farm and the addition of PSS improved the probabilistic stability of Mode 3. While with wind farm connected, the system eigenvalues became distributed in a wider range, making PSS tuning more difficult. The proposed PSS parameter design method in ‘PSS parameter adjustment/setting based on the center frequency method’ is applied in the next test for improving the probabilistic stability of electromechanical modes. The participation factors of x which are state variables of synchronous generators are obtained first and shown in Table 4. In this table, xMAX is the rotational speed at the central frequency which is obtained according to Eq. (23). The PSS parameters can be calculated by Eq. (20), (21) and with Eq. (23). After minor adjustments for coordination by try and error method, the final PSS parameters are obtained as shown in Table 5. The corresponding statistics of electromechanical oscillation modes are shown in Table 6. By comparing Tables 6, 3 and 2 respectively, it can be concluded that, except those of Mode 4 which remain unchanged, the damping ratio and the probability of all the other oscillation modes are improved greatly. The probabilistic stability of Mode 3 of inter-area oscillation reaches 100% and its damping ratio is increased to greater than 0.1 and with the PDF plot of the real part shown in Fig. 7. Mode 2 a local oscillation one is also close to 100%. Although the probabilistic stability of Mode 1 does not achieve 100%, yet its improvement is obvious when compared to that in Table 3.
Fig. 7. Real part PDF of the Mode 3.
Table 4 Participation factor of concerned modes before PSS added. Mode
Eigenvalues r ± jx
G1
G2
G3
G4
1 2 3 4
1.0266 ± j5.8909 1.0557 ± j5.4197 0.0818 ± j3.0797 1.8312 ± j2.8679
0 0.257 0.284 0 4.19
0 0.369 0.19 0 4.62
0.4 0 0.019 0 5.76
0.213 0 0.024 0 5.61
xMAX
Table 5 Parameters of the PSS in synchronous generators.
T1 T2 K
G1
G2
G3
G4
0.237 0.176 10.5
0.282 0.171 4
0.183 0.167 8
0.195 0.166 8
Conclusion
Table 6 Concerned modes of the system with wind farm connected (optimized PSS added). Mode
Eigenvalues r ± jx
Frequency(Hz)
Damping n
P{r < 0} (%)
1 2 3 4
2.3048 ± j6.5328 2.2825 ± j5.5868 0.4003 ± j3.0278 1.8312 ± j2.8679
1.04 0.89 0.48 0.46
0.333 0.378 0.131 0.538
76.42 99.55 100 100
in a larger range than that of the original system without wind farm. It can also be also observed that the curve moves to the left of the imaginary axis with wind farm connected and it moves further again with PSSs installed. It shows that the integration of wind farm and the addition of PSS improved the probabilistic stability of Mode 3. While with wind farm connected, the system eigenvalues became distributed in a wider range, making PSS tuning more difficult. The plot of the Probabilistic Density Function (PDF) of the real part of Mode 3 under different scenarios is shown in Fig. 7. From Point of Interconnection Bus Transmisson Line
5
Acknowledgements This work is supported by National Natural Science Foundation of China (51007054), Innovation Program of Shanghai Municipal Education Commission (12ZZ172), Shanghai Green Energy Grid Connected Technology Engineering Research Center (13DZ2251900).
High Side Bus Terminal Bus Collector Equivalent transmission line
Substation Transformer
4
A probabilistic method to evaluate the small-signal stability of power system with wind farm integration is proposed in this paper and it considers multi-operating conditions including variations in power output of wind-driven generators and load. The simulation results based on the test system shows that the stochastic output of wind farms deteriorates the probabilistic small signal stability of power systems. With a wind farm connected, the distribution of system eigenvalues cover a wider range, and it makes PSS tuning more difficult. Consequently the central frequency and participation factor method is proposed to solve this problem. The designed PSS at synchronous generators based on the proposed method increases the probabilistic stability of all electromechanical modes except the one mode relative to DFIG.
3
Unit transformer
2
Fig. A1. Typical layout of integration system for the wind farm.
1
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X.Y. Bian et al. / Electrical Power and Energy Systems 61 (2014) 482–488
Table A1 Data of integration system. Bus no.
Bus no.
Resistance
Reactance
Susceptance (Half)
Off-normal ratio(Trans)
5 4 3 2
4 3 2 1
0.025 – 0.015 –
0.25 0.1 0.025 0.05
0.0025 – 0.005 –
– 1.0 – 1.0
Table A2 Data of DFIG. Wound-rotor induction generator Back-to-back converter Converter controller
Shaft system Pitch controller Wind turbine aerodynamics
Rs Rr C Kp1 Ki2 Kpdg Kig Ht Dsh Kp4
q
vwout
0.00706 pu 0.005 pu 0.001 F 0.6 5.1 0.012 131 3s 3.14 10 1.225 kg/m3 25 m/s
Appendix The integrated system with the wind farm is shown as Fig. A1. The data of the system in per unit on 100 MVA and rated kV base are given in Table A1. The data of DFIG and its controllers on 1.5 MW base are given in Table A2.
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Improvement on probabilistic small-signal stability of power system with large-scale wind farm integration X.Y. Bian a, X.X. Huang a, K.C. Wong b, K.L. Lo c, Yang Fu a,⇑, S.H. Xuan a a
Electric Engineering College, Shanghai University of Electric Power, China Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong c Faculty of Engineering, University of Strathclyde, Glasgow, Scotland, UK b
a r t i c l e
i n f o
Article history: Received 17 January 2014 Received in revised form 31 March 2014 Accepted 2 April 2014
Keywords: Wind farm integration Probabilistic eigenvalue analysis Power system stabilizer (PSS) Participation factor Center frequency method
a b s t r a c t This paper studies probabilistic small-signal stability of power systems with wind farm integration, considering the stochastic uncertainty of system operating conditions. The distribution function of the realpart of system eigenvalue is computed by the method of probabilistic eigenvalue analysis. For improving probabilistic small-signal stability, PSS is adopted. A method for optimizing PSS based on participation factor and center frequency method is proposed. In order to evaluate the above proposed methods, the procedure is applied to a test system. The simulation results show that the stochastic variation of wind generation can induce a higher probability of system instability when compared with one that has no wind generation. With eigenvalues distributing in a wider range, it becomes difficult for PSS tuning. By applying the proposed optimized PSSs approach, probabilistic stability of system can be significantly improved. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction With the rapid development of wind power in China, impacts of large scale wind power penetration on power system stability have been paid more and more attention [1–4]. The random fluctuations of wind power output increase the uncertainty of the system, which will produce adverse effect on the system dynamic stability, especially the small-signal stability. It is meaningful to analyze power system small-signal stability with probabilistic methods considering the effects of uncertainty of wind power farm output and stochastic change of load. There are two kinds of methods to analyze small-signal probabilistic stability of power system with wind power penetration: one is based on the Monte Carlo simulation by which a large number of deterministic samples are generated by Monte Carlo sampling to calculate the probability of stability. This method is employed in [5,6]. Although the Monte Carlo simulation has accurate results, it is a time-demanding method because it requires computation of a large number of deterministic samples. The other one is the numerical analysis method. The probabilistic characteristics of eigenvalues are obtained by a formula to determine the probability of stability which is used only in Refs. [7,8]. The method requires complex
⇑ Corresponding author. Tel.: +86 21 35303121. E-mail address: [email protected] (Y. Fu). http://dx.doi.org/10.1016/j.ijepes.2014.04.005 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
formula derivation, but only one step of calculation to obtain the probabilistic distribution of system eigenvalues. In [7,8], Gram– Charlier series and system eigenvalue sensitivity are applied to study the impact of wind power fluctuations on the small-signal probability stability of the system, and is used to verify the correctness of numerical analysis method. But the wind generation model established in [7,8] is a simple one, and only the output variation of wind farm is considered as system uncertainties. It is verified in [8] that the integration of wind farm would cause system probabilistic small-signal instability, but in their study, no measures are proposed to improve system probabilistic stability. Based on the former studies [9–14], a probabilistic small-signal stability method is proposed in this paper by considering the stochastic variation of wind farm output, the fluctuation of load and the synchronous generator output. The complete DFIG transient model [15–17] is adopted. The probabilistic small-signal stability of power system with wind power integration is analyzed by numerical analysis method, and power system stabilizer (PSS) on synchronous generators is applied to improve the stability probability. The previous study on PSS [18,19] shows that the coordinated PSS can enhance the network damping. The PSS in this previous work is designed without considering the uncertainties of random wind power, generating and loading conditions. It may lose damping performance and fail to stabilize the system when operating condition changes. In order to adapt the PSS parameters in multi-operating conditions of power system, this
X.Y. Bian et al. / Electrical Power and Energy Systems 61 (2014) 482–488
paper presents an approach based on participation factors and center frequency method for PSS parameters tuning and coordination. The paper is organized as follows. In ‘PMT modeling technique’, the Plug-in Modeling Technology (PMT) [20] is introduced to build the complete system model. In ‘Probabilistic distribution of power system eigenvalues in multi-operating conditions’, the expression of the probability distribution of system eigenvalues is obtained by the Gram–Charlier expansion method. In ‘PSS parameter adjustment/setting based on the center frequency method’, the center frequency method based on participation factors is proposed for adjusting the parameters of PSS for improving probabilistic small-signal stability. In ‘Case study’, an example of 4-machine 2area power system with grid-connected wind power source is given. The proposed approach is applied to analyze and improve the probabilistic stability. The results of probabilistic stability analysis on the test system demonstrate that the small-signal stability of power system is indeed affected by the stochastic variation of grid-connected wind generation and can be improved by the proposed installation of PSS.
483
Probabilistic distribution of power system eigenvalues in multioperating conditions The eigenvalues of the state matrix A are used to determine the stability of the power system with respect to small disturbances. The eigenvalues are determined by the operating state given by each set of generation and load. Hence it should be noted that, as the operating state varies with respect to generation and load changes, the eigenvalues also vary. Under multi-operating conditions, with loads and output of wind turbine generator and synchronous generators regarded as random variables which may possess any type of distribution, probabilistic methods are applied to load flow calculation and eigenvalue computation. The method based on Gram–Charlier expansion with a hybrid algorithm using moments and cumulates is employed to obtain the Probabilistic Density Function (PDF) of system critical eigenvalues. For probabilistic load flow computation, all nodal voltages (V) and nodal injections (Y) are regarded as random variables. In an N-node system, Y can be expressed as a quadratic function.
PMT modeling technique
Y ¼ f ðVÞ ¼ gðV 1 V 1 ; . . . ; V i V j ; . . . ; V 2N V 2N Þ
PMT is adopted in this paper to construct state matrix of power system when analyzing small-signal stability of electric power system [9,12,20]. The whole model of the system is shown in Fig. 1. In Fig. 1, each system component is modeled as a module with 4 pins of voltage and current, which can be conveniently plugged into the network module. The models of synchronous generator units in [21,22], which contain the complete six-order generator model, including the excitation system and the prime mover model, are used to represent the general power plants in this paper. Complete transient models [15– 17] of the Doubly Fed Induction Generator (DFIG), which is widely used in wind farm, are adopted. The whole model [15–17] of the doubly fed induction generator including wind turbine, two-mass shaft system, rotor-side converter and grid-side converter and their control system, and pitch angle control system for reflecting the effect of wind farm on power system are all included.
With expansion at expectation point V using Taylor series, these equations can be represented as in Eq. (2).
Y ¼ f ðVÞ þ J V DV þ f ðDVÞ
ðVÞ where J V is the Jacobin matrix at the expectation point J V ¼ @f@V
ð1Þ
ð2Þ V¼V
þ DY, the expected value of Y is expressed as in With Y ¼ Y Eq. (3).
Y ¼ f ðVÞ þ f ðDVÞ
ð3Þ
where f ðDVÞ ¼ gðC V 1 ;V 1 ; . . . ; C V i ;V J ; . . . ; C V 2N ;V 2N Þ; C V i ;V J can be obtained by Eq. (4)
n h i C V ¼ J 1 C Y E J V DVf T DðVÞ þ f ðDVÞDV T J TV þ f ðDVÞf T ðDVÞ V o T þ f ðDVÞf T ðDVÞ J 1 V
Fig. 1. The power system with DIFG in PMT.
ð4Þ
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where DV ¼ J 1 DY ¼ J 1 ðY YÞ,CY is the covariance of nodal injecV V tions, and are calculated from Y in (2) and Y in (3). These values are substituted into (3) and the expected value of the operation voltage of power system is then iterated until it converges. The covariance matrix of the voltage is:
C V ¼ EðDV DV T Þ ¼ J 1 C Y ðJ 1 Þ V V
T
ð5Þ
According to the nature [23,24] of cumulants, j-order cumulants of voltage deviation are:
PSS parameter adjustment/setting based on the center frequency method Power system stabilizer
j
cj ðDVÞ ¼ ðJ 1 Þ cj ðDYÞ V
ð6Þ
where CV and CY are covariance matrices of voltage vector and injected power respectively. cj(DV) and cj(DY) are J-order cumulants of the voltage and the injected power respectively. A certain eigenvalue of power system can be expressed as a nonlinear function of the node voltage vector:
kk ¼ g k ðVÞ
ð7Þ
By expansion using Taylor Series at the vicinity of the expected value of voltage and keeping the 2-order approximate expression, these equation can be obtained as in Eq. (8):
2N X @g k kk ¼ gðVÞ þ @V
i V¼V
i¼1
DV i
2N X 2N 1X @ 2 gk þ j DV i DV j 2 i¼1 j¼1 @V i @V j V¼V
ð8Þ
The expected value of the eigenvalue can be interpreted as follows:
Eðkk Þ ¼ gðVÞ
ð9Þ
The covariance of the eigenvalue can be got according to the definition of covariance and expressed as Eq. (10).
C km ;kn ¼ Ef½km Eðkm Þ½kn Eðkn Þ ¼
2N X 2N X J km;i J kn;j C v i ;v j
ð10Þ
i¼1 j¼1
By neglecting the higher order term in Eq. (8) and re-arranging the terms yields:
Dk ¼ J k DV
ð11Þ
where Jk is the retained first order derivative term. DV ¼ J Y DY is obtained according to the probabilistic load flow equation and is substituted into the following equation:
Dk ¼ J k J Y DY ¼ JDY
ð12Þ
J1 . V
where J Y ¼ Thus the relation between nodal injection power and the eigenvalue is obtained. According to the nature [24] of cumulants, J-order cumulants of eigenvalue kk can be obtained as follow. 2N X cj ðkk Þ ¼ Jjk;i ck ðY i Þ
ð13Þ
i¼1
The probability density function and cumulative distribution function of the eigenvalue kk can be obtained by applying Gram– Charlier series. It can be expressed in Eqs. (14) and (15) respectively:
c ðkk Þ c ðkk Þ f ðkk Þ ¼ NðxÞ 1 þ 3 3 ðx3 3xÞ þ 4 4 ðx4 6x2 þ 3Þ 6r 24r c5 ðkk Þ 5 ðx 10x3 þ 15xÞ þ 120r5 Pfkk < xc g ¼
where f ðkk Þ is probability density function of the eigenvalue kk , N(x) is the probability density function for the standard normal distribukÞ tion, cj ðkk Þ is a J order semi-invariant variable, x ¼ kk Eðk is a norr malizing variable, and the standard variance of eigenvalues can be obtained by Eq. (10).
Z
xc
NðxÞdx Nðxc Þ
1
þ
Phase lead provided by power system stabilizer (PSS) can compensate the phase lag caused by an excitation control device and as such the damping level of the system can be greatly improved. The linearized transfer function module of the excitation system adopted in this paper is shown in Fig. 2. where DVt is the input voltage of the excitation system, DVPSS is 1 the voltage signal from PSS, 1þpT is the transfer function of the A1 automatic voltage regulator, KA2 is the excitation gain, and DEfd is the excitation voltage output. The angle lag caused by excitation system is as follows.
uf ¼ arctan T A1 x
ð16Þ
The module of the PSS is as follows. where Kp is PSS gain, Tw and T1/T2 are the washout and lead/lag time constants, respectively. PSS parameter adjustment The frequency characteristic of excitation system shows that the phase of excitation system becomes slightly lag or may even lead at low frequency. Therefore, Tw should be set as a high value to make the washout stage as a high-pass filter such that the PSS does not respond to DC offsets and very low frequency modes. When the input signal is Dx, Tw may be set to about 10 s, and lead-lag phase of
pT w 1þpT w
of the excitation system can be obtained
as uf = arctan Twx. When the parameters of the lead-lag stage are T1 and T2 respectively, the relationship of the rotating speed and phase angle of the compensation can be expressed as follows.
uPSS ¼ arctan
ðT 1 T 2 Þx 1 T1T2x
The maximum compensation phase angle is expressed as follows.
ðT T Þ 2 T1T2
1 2 ffiffiffiffiffiffiffiffi uMAX ¼ arctan p
ð18Þ
The center frequency is
fPSS ¼
1 pffiffiffiffiffiffiffiffi 2p T 1 T 2
ð19Þ
The corresponding rotational speed is
pffiffiffiffiffiffiffiffiffiffi
xjuMAX ¼ 1= T 1 T 2
ð20Þ
Let
T 1 ¼ kT 2
ð21Þ
ð14Þ
c ðkk Þ
c3 ðkk Þ 2 xc 1 þ 4 4 x3c 3xc 3 6r 24r
c5 ðkk Þ 4 x 6x2c þ 3 120r5 c
ð17Þ
ð15Þ Fig. 2. The module of the excitation system.
X.Y. Bian et al. / Electrical Power and Energy Systems 61 (2014) 482–488
xjuMAX ¼
n X qi xi =qR
485
ð23Þ
1
Fig. 3. Module of the PSS.
uPSS
Then
uMAX
ðk 1ÞT 2 ðk 1Þ ¼ arctan qffiffiffiffiffiffiffiffi ¼ arctan pffiffiffi 2 2 k 2 kT 2
ð22Þ
In a single-machine system, PSS provides damping to the generator at system oscillation frequency. The lead phase and center frequency can be obtained according to the system oscillation modes and phase frequency characteristics of the excitation system. Suppose an oscillation mode of a single-machine system is pffiffiffiffiffiffiffiffiffiffi q + jx, then x ¼ xjuMAX ¼ 1= T 1 T 2 . With uMAX = ufd, k is obtained according to Eq. (22). T1 and T2 are obtained by substituting Eqs. (22) into Eq. (21). For multi-machine system, each oscillation mode is contributed by several machines. Participation factor is adopted for evaluating the participation degree of each machine to each mode. Suppose one machine participates in two oscillation modes q1 + jx1 and q2 + jx2 with participation factors q1 and q2 respectively, the setting of the machine PSS parameter should involve the two oscillation modes. The higher the participation degree of a machine to a mode, the closer the center frequency is to this mode oscillation frequency. Namely, the center frequency should consider the weight of the participation factor of machines in each mode, and are adjusted as follows,
qi u qR fd
ð24Þ
where qi is the participation factor of the variable Dx of the machine involving in an oscillation mode. qR is the sum of all Dx participation factors of machines in this mode. ufd and uPSS are the lead-lag angle of excitation system and the phase angle of PSS compensation respectively. PSS parameters obtained by Eqs. (23) and (24) can give the best compensation effect to the oscillation mode to which the machine has the biggest participation factor, and can produce possible lesser effect on the other modes. Since coordination is needed in the multi-machine system, the most suitable PSS compensation phase angle could not be determined directly. Eq. (24) is the constraints of the PSS compensation phase angle of this generator. By adopting Eq. (24), the most suitable PSS parameter can be obtained. Case study Test system A two-area system [21] shown in Fig. 4, which is widely used for studies of power system oscillations, is adopted in this paper to demonstrate the proposed method. The network and generator data can be obtained from [21]. G5 represents a wind farm with
Fig. 4. A two-area system with wind farm integration.
Fig. 5. Operating curves of wind farm.
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a capacity of 120 MW, or 80 DFIG with 1.5 MW each, details of the system and DFIG parameters are listed in the Appendix. The original system [21] without wind farm and the test system with DFIGtype wind farm are both simulated and compared. A single equivalent model was used to represent all individual units within the wind farm and it is connected to the system using a typical layout as shown in the appendix. All synchronous generators are represented using 6-order models and are extended to include exciter and power system stabilizer (PSS) models. The whole system model is represented using the aforementioned PMT modeling method. The standardized daily operating curves of load1, load2, G1, G2, G3 and G4 are given in [9]. The standardized daily operating curves of wind farm output and voltage are shown in Fig. 5. The small-signal probabilistic stability analyses By adopting the proposed probabilistic eigenvalue analysis method in section III, the simulation results on the original system is obtained. There are three electromechanical oscillation modes in the original system without the wind farm. These mode shapes are shown in Fig. 6. From the mode shapes, it is observed that Mode 1 is a local mode and with generator G1 swinging against G2; Mode 2 is also a local mode and with generator G3 swinging against G4; whereas Mode 3 is an inter-area mode and with generators G1 and G2 in area 1 swinging against generators G3 and G4 in area 2. Table 1 gives information about the distribution of these 3 modes. It shows that the stability problem is likely to occur because Mode 3 has a probability of 81.78% to remain in the left
Table 1 Electromechanical modes of the original system. Mode
Eigenvalues r ± jx
Frequency (Hz)
Damping n
P{r < 0} (%)
1 2 3
1.0293 ± j5.8877 0.9726 ± j5.5302 0.90122 ± j2.9196
0.94 0.88 0.46
0.172 0.173 0.004
100 100 81.78
half-plane and it is poorly damped, with the mean of damping ratio is 0.0004 and the probability of more than 0.1 is 0. For the test system with the wind farm integration, the statistical result of electromechanical oscillation modes are shown in Table 2. There is a new electromechanical oscillation mode (Mode 4) caused by the wind farm. By analyzing the participation factor, the DFIG-type G5 did not swing against the other synchronous generators although the frequency of Mode 5 is within the range of inter-area oscillation. It can also be concluded that DFIG-type G5 did not participate in the pre-existing electromechanical oscillations. But DFIG-type G5 indeed reduces the probabilistic stability of local modes. The probability that local modes (Mode1 and Mode 2) remain in the left half-plane is reduced dramatically. Conversely, the integration of DFIG can have a positive impact on the probability of inter-area mode (Mode 3) as well as the damping performance as shown in Table 2.
PSS parameters adjustments for the improvement of the probabilistic small-signal stability To improve the probabilistic stability of the test system, PSSs are installed in four synchronous generators, with structures as shown in Fig. 3, and with initial parameters as shown in [21], and Dx as the input signal. The statistics of electromechanical oscillation modes of the test system with PSSs are shown in Table 3. By comparing Tables 2 and 3, the probability of all modes (Mode 1 and 2) to remain in the left half-plane is raised slightly in Table 3, and so is the probability of the damping ratio. The inter-area mode (Mode 3) has some improvement with its damping ratio increasing to 0.04 from 0.027, and stability probability increasing to 98.84% from 90.15%. Because all synchronous generators do not participate in Mode 4, installation of PSS in all synchronous generators does not affect this mode. The result shows that PSSs on synchronous generators can improve the stability of small-signal probability of the system. The plot of the Probabilistic Density Function (PDF) of the real part of Mode 3 under different scenarios is shown in Fig. 7. From this plot, it is observed that the PDF became scattered with the connection of wind farm. Therefore the real part of Mode 3 varies
Table 2 Electromechanical modes of the system with wind farm connected. Mode
Eigenvalues r ± jx
Frequency(Hz)
Damping n
P{r < 0} (%)
1 2 3 4
1.0266 ± j5.8909 1.0557 ± j5.4197 0.0818 ± j3.0797 1.8312 ± j2.8679
0.94 0.86 0.49 0.46
0.172 0.191 0.027 0.538
67.24 78.68 90.15 100
Table 3 Electromechanical modes of the system with wind farm connected (with PSSs).
Fig. 6. Mode shape of rotor angle modes.
Mode
Eigenvalues r ± jx
Frequency(Hz)
Damping n
P{r < 0} (%)
1 2 3 4
1.1013 ± j5.9181 1.1215 ± j5.4389 0.1046 ± j3.0798 1.8312 ± j2.8679
0.96 0.885 0.5 0.46
0.177 0.197 0.04 0.538
68.08 86.65 98.84 100
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X.Y. Bian et al. / Electrical Power and Energy Systems 61 (2014) 482–488
this plot, it is observed that the PDF became scattered with the connection of wind farm. Therefore the real part of Mode 3 varies in a larger range than that of the original system without wind farm. It can also be also observed that the curve moves to the left of the imaginary axis with wind farm connected and it moves further again with PSSs installed. It shows that the integration of wind farm and the addition of PSS improved the probabilistic stability of Mode 3. While with wind farm connected, the system eigenvalues became distributed in a wider range, making PSS tuning more difficult. The proposed PSS parameter design method in ‘PSS parameter adjustment/setting based on the center frequency method’ is applied in the next test for improving the probabilistic stability of electromechanical modes. The participation factors of x which are state variables of synchronous generators are obtained first and shown in Table 4. In this table, xMAX is the rotational speed at the central frequency which is obtained according to Eq. (23). The PSS parameters can be calculated by Eq. (20), (21) and with Eq. (23). After minor adjustments for coordination by try and error method, the final PSS parameters are obtained as shown in Table 5. The corresponding statistics of electromechanical oscillation modes are shown in Table 6. By comparing Tables 6, 3 and 2 respectively, it can be concluded that, except those of Mode 4 which remain unchanged, the damping ratio and the probability of all the other oscillation modes are improved greatly. The probabilistic stability of Mode 3 of inter-area oscillation reaches 100% and its damping ratio is increased to greater than 0.1 and with the PDF plot of the real part shown in Fig. 7. Mode 2 a local oscillation one is also close to 100%. Although the probabilistic stability of Mode 1 does not achieve 100%, yet its improvement is obvious when compared to that in Table 3.
Fig. 7. Real part PDF of the Mode 3.
Table 4 Participation factor of concerned modes before PSS added. Mode
Eigenvalues r ± jx
G1
G2
G3
G4
1 2 3 4
1.0266 ± j5.8909 1.0557 ± j5.4197 0.0818 ± j3.0797 1.8312 ± j2.8679
0 0.257 0.284 0 4.19
0 0.369 0.19 0 4.62
0.4 0 0.019 0 5.76
0.213 0 0.024 0 5.61
xMAX
Table 5 Parameters of the PSS in synchronous generators.
T1 T2 K
G1
G2
G3
G4
0.237 0.176 10.5
0.282 0.171 4
0.183 0.167 8
0.195 0.166 8
Conclusion
Table 6 Concerned modes of the system with wind farm connected (optimized PSS added). Mode
Eigenvalues r ± jx
Frequency(Hz)
Damping n
P{r < 0} (%)
1 2 3 4
2.3048 ± j6.5328 2.2825 ± j5.5868 0.4003 ± j3.0278 1.8312 ± j2.8679
1.04 0.89 0.48 0.46
0.333 0.378 0.131 0.538
76.42 99.55 100 100
in a larger range than that of the original system without wind farm. It can also be also observed that the curve moves to the left of the imaginary axis with wind farm connected and it moves further again with PSSs installed. It shows that the integration of wind farm and the addition of PSS improved the probabilistic stability of Mode 3. While with wind farm connected, the system eigenvalues became distributed in a wider range, making PSS tuning more difficult. The plot of the Probabilistic Density Function (PDF) of the real part of Mode 3 under different scenarios is shown in Fig. 7. From Point of Interconnection Bus Transmisson Line
5
Acknowledgements This work is supported by National Natural Science Foundation of China (51007054), Innovation Program of Shanghai Municipal Education Commission (12ZZ172), Shanghai Green Energy Grid Connected Technology Engineering Research Center (13DZ2251900).
High Side Bus Terminal Bus Collector Equivalent transmission line
Substation Transformer
4
A probabilistic method to evaluate the small-signal stability of power system with wind farm integration is proposed in this paper and it considers multi-operating conditions including variations in power output of wind-driven generators and load. The simulation results based on the test system shows that the stochastic output of wind farms deteriorates the probabilistic small signal stability of power systems. With a wind farm connected, the distribution of system eigenvalues cover a wider range, and it makes PSS tuning more difficult. Consequently the central frequency and participation factor method is proposed to solve this problem. The designed PSS at synchronous generators based on the proposed method increases the probabilistic stability of all electromechanical modes except the one mode relative to DFIG.
3
Unit transformer
2
Fig. A1. Typical layout of integration system for the wind farm.
1
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X.Y. Bian et al. / Electrical Power and Energy Systems 61 (2014) 482–488
Table A1 Data of integration system. Bus no.
Bus no.
Resistance
Reactance
Susceptance (Half)
Off-normal ratio(Trans)
5 4 3 2
4 3 2 1
0.025 – 0.015 –
0.25 0.1 0.025 0.05
0.0025 – 0.005 –
– 1.0 – 1.0
Table A2 Data of DFIG. Wound-rotor induction generator Back-to-back converter Converter controller
Shaft system Pitch controller Wind turbine aerodynamics
Rs Rr C Kp1 Ki2 Kpdg Kig Ht Dsh Kp4
q
vwout
0.00706 pu 0.005 pu 0.001 F 0.6 5.1 0.012 131 3s 3.14 10 1.225 kg/m3 25 m/s
Appendix The integrated system with the wind farm is shown as Fig. A1. The data of the system in per unit on 100 MVA and rated kV base are given in Table A1. The data of DFIG and its controllers on 1.5 MW base are given in Table A2.
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