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Probabilistic assessment of power system transient stability incorporating SMES
Physica C xxx (2012) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Probabilistic assessment of power system transient stability incorporating SMES Jiakun Fang, Wei Yao, Jinyu Wen ⇑, Shijie Cheng, Yuejin Tang, Zhuo Cheng State Key Lab of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, No. 1037, Luoyu Road, Wuhan 430074, China
a r t i c l e
i n f o
Article history: Accepted 29 March 2012 Available online xxxx Keywords: Wind farm SMES Transient stability Monte-Carlo simulation
a b s t r a c t This paper presents a stochastic-based approach to evaluate the probabilistic transient stability index of the power system incorporating the wind farm and the SMES. Uncertain factors include both sequence of disturbance in power grid and stochastic generation of the wind farm. The spectrums of disturbance in the grid as the fault type, the fault location, the fault clearing time and the automatic reclosing process with their probabilities of occurrence are used to calculate the probability indices, while the wind speed statistics and parameters of the wind generator are used in a Monte Carlo simulation to generate samples for the studies. With the proposed method, system stability is ’’measured’’. Quantitative relationship of penetration level, SMES coil size and system stability is established. Considering the stability versus coil size to be the production curve, together with the cost function, the coil size is optimized economically. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction With the increasing wind energy being put into utility power system, the transient stability becomes one of the major problems when subject to the continuous disturbance brought by the wind power. As flexible, reliable and fast responding power compensation equipment, SMES is expected to contribute to dynamic stability of power system consisting wind farms [1–4]. Generally, the SMES with sufficiently large stored energy and power rating capacity is able to provide satisfactory damping effect. However, the superconducting magnetic coil, as the heart of SMES, need bulk load of investment per MJ. Hence, it is essential to evaluate the relationship between system stability and SMES capacity. Some related works considering coil size of SMES are done [5,6], but fail to calculated the stability with certain SMES capacity. The problem is that it is difficult to ‘‘measure’’ system stability quantitatively with a unified specification. Traditionally, power systems have been planned and operated using deterministic transient stability criteria. The systematic index, however, representing stability cannot be given out by this method. Probabilistic analysis of power system stability is to determine the stability index with main probabilistic factors which reflect the stochastic nature of the operating conditions such as parameters of the generator, network topology, and disturbances. Compared to the deterministic method, the probabilistic analysis takes considerably long computation time. But it measures the risk that system loses stability and helps with making decisions such as capacity of the SMES. ⇑ Corresponding author. Tel.: +86 27 87540569; fax: +86 27 87540937. E-mail addresses: [email protected] (J. Fang), [email protected] (J. Wen).
This paper presents a stochastic-based approach to evaluate the probabilistic transient stability indices of the power system incorporating the wind farm and the SMES by combining the MonteCarlo simulation and Bayesian classification. Uncertain factors include both sequence of events in power grid and stochastic generation of the wind farm. The 3rd order model of SMES considering dc current dynamic is proposed and time-domain simulations are given. The effects of the penetration level of the wind farm and the size of the SMES coil to the system stability is investigated. The proposed methodology can be used in system planning, decision for SMES capacity, and security warning. 2. System modeling and description A modified version of a two-area four-machine interconnected power system [7] is used in this paper. As is shown in Fig. 1, the synchronous generator is simulated with the 6th order model, as well as the power system stabilizer and governor. The No. 2 generator in area 1 is replaced with a doubly fed asynchronous generator (DFAG). The SMES is installed at the terminal of the DFAG. The capacitor bank at node 2 is fixed reactive compensator so that the power factor of the wind farm is 1.0. The steady state power flow can be adjusted to represent different penetration level of wind farm. 2.1. Modeling the DFAG The DFAG is the most commonly used device for wind power generation. As is generally known, the rotor terminals are fed with a symmetrical three-phase voltage of variable frequency and amplitude. This voltage is supplied by a voltage source converter
0921-4534/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2012.03.068
Please cite this article in press as: J. Fang et al., Probabilistic assessment of power system transient stability incorporating SMES, Physica C (2012), http:// dx.doi.org/10.1016/j.physc.2012.03.068
2
J. Fang et al. / Physica C xxx (2012) xxx–xxx
1
5
G1
7
6 25km
110km
8
110km
9
10km 110km
110km
10 10km
11 25km
3 G3
2 4
Area 1 WT
SMES
G4
Area 2
Fig. 1. The test system.
set to 1.0. The inner control loop, named current control loop, which is executed by the rotor side converter, decouples the power regulation by controlling the direct and quadrature components of the rotor current. The stator-flux-oriented vector control scheme is used to implement the decoupled control of active and reactive power in modeling the rotor-side converter controller. The low voltage ride through (LVRT) process is considered in control scheme of the DFAG. During and immediately following sustained faults, limiting the real power command with both a cap and a ramp rate limit reduces system stress to the wind generator. As is shown in Fig. 2, the low voltage power logic (LVPL) is from GE’s report [11]. The maximum limit of the reference power PQref is related to the terminal voltage. During fault time, the terminal voltage is less than 1.0 pu, which leads to limitation to the power output of the DFAG. When terminal voltage is under 0.5 pu the crow bar is activated and PQref are set to zero. In this paper, the capacity of the converters is set to 30% of the DFAG’s capacity. Hence, the slip of the DFAG cannot be over 1.3 pu or lower than 0.7 pu, which will lead to splitting of the DFAG. Not only the deviation of rotor angle of the synchronous machine being over 180°, but the slip larger than 1.3 pu or lower than 0.7 pu is the used to judge the system is unstable.
usually equipped with IGBT-based power electronics circuitry [8]. The basic structure is shown in Fig. 2. The electrical part of the DFAG is modeled as the asynchronous generator using motor convention. The transient in the stator and in the branch between the inverters and the grid is neglected. This is widely used simplification in transient stability modeling of the synchronous and asynchronous machines [9]. (1) is derived from the basic equations of the asynchronous machine, taking positive currents as going into the machine (motor convention).
8 h i > > pE_ 0 ¼ jsE_ 0 RLrr E_ 0 jðLs X 0 ÞI_s j LRmr U_ r > > > < _s U_ s ¼ Rs I_s þ jxs u > _ _ _ > us ¼ Ls Is þ Lm Ir > > > : _ ur ¼ Lr I_r þ Lm I_s
ð1Þ
Here, E0 is an electrical potential component related to rotor flux u. The subscript s and r represents electrical quantity on the stator and rotor, respectively. The dotted variables are vectors synthesized by direct and quadrature components, i.e. U_ s ¼ uds þ juqs The parameters of the DFAG are given in Table A.1 in Appendix A. In the mechanical part of the DFAG representing the drive train and rotor dynamic, the 2-mass model is adopted as shown in (2).
8 dxM > > < 2HM dt ¼ T M T G DM xM xG 2HG ddt ¼ T G T E DG xG > > : dhs ¼ x0 ðxM xG Þ dt
2.2. Modeling of SMES
ð2Þ
A simplified 3rd order SMES model is proposed in this work, as is shown in Fig. 3. The modulation ratio M and firing angle a is modeled as two 1st order lag
where TG is The mechanical torque of the asynchronous machine, and TG = KShS DS(xG xM). The DFAG control structure consists of 2 cascaded control loops [10]. The outer one, named power control loop, regulates the active and reactive power at the stator side based on the maximum power point tracking (MPPT) strategy. And the power factor is
(
_ ¼ 1 M þ uM M Tc
ð3Þ
a_ ¼ T1c a þ ua
where uM and ua are input signals from the upper level control of the SMES. The parameter Tc is the communication delay and the dy-
LVPL
1.11
V
Low voltage power logic MPPT control
zerox (0.5)
Brkpt (0.9)
Rotor-side converter
Electrical Equations
Power System
Asynchronous generator Wind speed
Windmill aerodynamic
Drive train
Mechanical Equations
Fig. 2. Block diagram for DFAG modeling.
Please cite this article in press as: J. Fang et al., Probabilistic assessment of power system transient stability incorporating SMES, Physica C (2012), http:// dx.doi.org/10.1016/j.physc.2012.03.068
3
J. Fang et al. / Physica C xxx (2012) xxx–xxx dc current dynamic
+ -
Damping controller
Power System
+ Voltage restorer
Converter dynamic
upper level control
Fig. 3. The upper level controller of SMES.
namic of the converter. Meanwhile, the dynamic of the dc current of the SMES can be calculated by integration of SMES active power output,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rt 2 ð1 þ nÞPsm dt Id ðtÞ ¼ I2d0 0 L
ð4Þ 2.3. Upper level control strategy of SMES
where Psm, Id0 and L are SMES output, initial state of the dc current, and inductance of the superconducting coil, respectively. n represents the loss during power transferring. With M, a and Id(t), the SMES output can be
(
pffiffi Psm ¼ 32 U s Is cos a ¼ 3 4 3 U s MId ðtÞ cos a pffiffi Q sm ¼ 32 U s Is sin a ¼ 3 4 3 U s MId ðtÞ sin a
in PQ frame, whose radius is proportional to dc current Id. The coil size and dc current also determine the maximum energy to be stored in SMES. Hence, with proposed 3rd order SMES model, the capacity of the SMES is represented by single specification L.
ð5Þ
For the ideal converter for SMES, the modulation ration can be no more than 1.0. Consequently, the active and reactive power regulation is limited by dc current Id. Neglecting the impedance of the transformer and filter circuit, the power regulation range is a circle
Owing to the decoupled 4-quadrant power regulation characteristics, the SMES improves system stability in two ways: damping controller in active power control loop and voltage restorer in reactive power control loop, represented in Fig. 3. In this paper, the mixed output-feedback control with regional pole placement is applied to design the damping controller [12,13]. This multi-objective control overcomes the limitations of single objective synthesis technique functioned with pole placement and incorporate the H1 and H2 performance by selecting weight of each indicator. Fig. 4 illustrates the effectiveness of the controller. In this case, a 3-phase fault lasting 200 ms is simulated in node 7
Fig. 4. Active power damping controller.
Please cite this article in press as: J. Fang et al., Probabilistic assessment of power system transient stability incorporating SMES, Physica C (2012), http:// dx.doi.org/10.1016/j.physc.2012.03.068
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1.2 1
1
SMES capacity = 30 H SMES capacity = 20 H SMES capacity = 10 H
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
SMES capacity = 30 H SMES capacity = 20 H SMES capacity = 10 H
0.8
QSMES (p.u.)
Terminal voltage (pu)
1.5
0.6 0.4 0.2 0
4.5
−0.2
0
0.5
1
1.5
2
time (sec)
2.5
3
3.5
4
4.5
5
time (sec)
Fig. 5. Time domain simulation: SMES voltage restorer.
and penetration level is 50%. The capacity of SMES is 50H. Note that in this case, SMES damps the power oscillation only by active power regulation, and hence improves stability. The voltage restorer is a simple PI controller adopting terminal voltage deviation as feedback, which helps the DFAG to ‘‘ride through’’ the low voltage during fault time. From time domain simulation in Fig. 5, one can see the SMES reactive power output during fault is much larger than post fault. In this case, a 3-phase fault lasting 300 ms is simulated in node 7 and penetration level is 50%. SMES acts as dynamic voltage compensator such as STATCOM. Its capacity determines how the voltage can be improved so that the DFAG can be and hence whether the system is stable to this fault. Time-domain simulations illustrating the influence of SMES capacity and wind farm penetration level are given in Fig. 6. A 3phase fault lasting 300 ms is simulated in node 8. From Fig. 6a, one can see that SMES with larger capacity can achieve better damping effect. We adjust wind farm penetration by increasing power output of DFAG while holding No. 1 synchronous generator’s output invariant. Meantime, the Nos. 3 and 4 synchronous generators’ outputs are reduced to maintain power balance. Fig. 6b shows that under the same fault, system with higher penetration level is easier to be unstable.
type, fault location,fault duration, and successful/unsuccessful reclosing [14]. The stochastic model is quite consistent with the IEEE Power Systems Relaying Committee Working Group [15], saying that the single phase-to-ground, double phase-to-ground, phase-to-phase and the 3-phase fault types take 93%, 2%, 4% and 1%, respectively. The transmission line is divided into three parts: the first 20% near the sending end, that is, area 1 in this paper, the middle 60% called mid-line and 20% near the receiving end. At each part of the line the fault location is evenly distributed sharing the probability of 0.1307, 0.7021 and 0.1672, respectively, shown in Table 1. In each part, the fault location meets the uniform distribution. Through historical fault events from the BC Hydro system, it was noted that in over 90% of the faults caused by lightning, automatic reclosing was successful, while that was the case in less than 50% of faults due to all other causes. And 82.51% of faults are due to lightning. The probability of successful automatic reclosing was thus estimated by 0.8251 0.9 + 0.1749 0.5 = 0.83. For the fault clearing time, a linear combination of two normal distributions is used in this paper. The probability density function used is represented by Fig. 7.
Table 1 Fault location probabilities.
3. Probabilistic transient stability assessment 3.1. Probabilistic factor models Historical data from the BC Hydro system was used to develop stochastic model for the disturbance sequence, including fault
Fault location
Range
Probability
Close-in Mid-line Far-end
First 20% of the line Middle 60% of the line Last 20% of the line
0.1307 0.7021 0.1672
−3
10
SMES capacity = 30 H SMES capacity = 20 H SMES capacity = 10 H
2
Rotor speed deviation of the DFAG (rad)
deviation of rotor angle (rad)
3
1 0 −1 −2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
Penetration level of 41.18% Penetration level of 50% Penetration level of 56.52%
5
0
−5
0
time (sec)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time (sec)
Fig. 6. Time-domain simulation.
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J. Fang et al. / Physica C xxx (2012) xxx–xxx
0.1
the index ‘‘measuring’’ stability on the system level, not just based on one or two selected faults. The calculation process consists of two loops, which divides uncertain factors into two types, as is shown in Fig. 8. The outer loop generates a series of disturbance sequences, taking fault type, range of fault location and successful/unsuccessful reclosing into consideration. For every disturbance sequence, the bisection algorithm is used to search the critical clearing time (CCT) for every steady state and fault. Together with the probability density function shown in Fig. 7, we can get the probability of stability, represented as Pmn j in (6). Then with probability of their occurrence Amn j, the probabilistic stability can be combined using the Bayesian probability formula.
0.09
Probability density
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0
0
5
10
15
20
25
Ps ¼
3.2. Procedures for probabilistic transient stability studies In traditional transient stability studies, the fault type, fault location, and reclosing are deterministic. While probabilistic studies, take into account the stochastic nature of the power system. Fig. 8 shows the calculation procedure for including probabilistic factors in transient stability studies, which is implemented by repeated deterministic simulation many times according to a spectrum of different states with their probabilities of occurrence. Additionally, assessing probabilistic transient stability can give
Start
Monte Carlo simulation
Bayesian classification
Select fault type, fault location, and reclosing
Generate the wind speed sequence Locate fault precisely
Search CCT using Bisection algorithm
m¼1n¼1
Cycles (60 Hz) Fig. 7. Probability density function used for fault clearing time.
M P N P
J P
Pmnj Amnj
ð6Þ
j¼1
The second class of uncertain factors are calculated in the inner loop using the Monte-Carlo method. The precise location of the fault in the range determined in the outer loop, parameters of the wind farm and wind speed are sampled to make every Pmn j accurate enough (variance is less than 0.01). With processing the uncertain factors mentioned above, we can get the stability index of the system by proposed method. 4. Optimal coil sizing With probabilistic transient stability study, we give the stability index of the system incorporating wind farm and SMES, shown in Fig. 9. Wind farm penetration is adjusted by increasing power output of DFAG and reducing the Nos. 3 and 4 synchronous generators’ outputs. Hence, increase of penetration level leads to increase of transmission burden of the tie-line. Form Fig. 9, we can see the system stability increases with the capacity of the SMES, while decreases with the penetration of wind power.This matches the time domain simulation results mentioned in former sections. From transient stability point of view, more energy storage is required by higher penetration of the wind power to maintain system stable. Here we use simple theory of economics to determine the optimal coil size. The cost function of SMES is proposed in [16,17].
Cost ¼ 0:95 ½EnergyðMJÞ0:67
ð7Þ
Under certain penetration level of wind farm, a production curve representing system stability versus SMES coil size can be obtained from Fig. 9. Taking 30% penetration for example, we draw both the production curve and cost function in Fig. 10a with dark and blue solid line,respectively. Here we assume that the rated dc current
Accurate enough?
Record the average CCT for this disturbance
N all disturbances covered?
Output results Fig. 8. Simulation flowchart.
Probability of stability (%)
N 100 80 60 40 20 0 52
50
48
46
44
Penitration level (%) 42
0
20
60
40
80
100
ES capacity (H)
Fig. 9. Index of transient stability.
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J. Fang et al. / Physica C xxx (2012) xxx–xxx
70 3
60 50
2
40 30 20 10 0
1
Cost functions 0
0 10 20 30 40 50 60 70 80 90 100
100
5
Optimized coil size
90 4
80
Cost (M$)
4
80
Cost (M$)
Stable Probability (%)
90
Stable Probability (%)
5
100
70 3
60 20
30 L*
Inductance (H)
40
50
Inductance (H)
Fig. 10. Optimal coil sizing.
is 100A. Then the cost function is translated parallel to being tangent with the production curve, as is shown in dash blue line in this figure. Fig. 10b is zoomed into depict the optimal coil size L⁄. It can be seen from these figures that when SMES capacity is less than L⁄, the slope of production curve is always larger than that of cost function, indicating that additional investment on SMES with larger size is repaid abundantly. When coil size L > L⁄, the increment of improvement of system stability is less than increment of investment, indicating that other types of energy storage or FACTs devices instead of SMES may be considered to meet the system requirements. 5. Conclusion Probabilistic study of power system incorporating wind farm and SMES is proposed in this paper. It extends and illustrates a basic procedure for calculating the probability of transient stability. Electro-mechanical model of both DFAG and SMES are proposed and time domain simulation results are given. The stability index of the power system is evaluated, using both bisection algorithm and Monte-Carlo simulation. Quantitative relationship between system stability with SMES capacity and penetration level of the wind power is given. Considering the stability versus coil size to be the production curve, together with the cost function, the coil size is optimized. This study confirms the practical merits in terms of stability and economics of SMES. Appendix A. The wind turbine parameters Both the electrical and mechanical parameters of the DFAG used in this paper are shown in Table A.1. Note that Ls and Lr are sum of the stator linkage reactance and the mutual reactance. The gust wind speed fluctuation is considered and modeled according to [18], shown in (A.1). Table A.1 Parameters of the DFAG. Electrical parameters
Value
Rs/Rr, stator/rotor resistance Ls, total reactance of the stator circuit Lr, total reactance of the rotor circuit Lm, mutual reactance
0.01499/0.01534 8.6929 8.6336 8.504
Mechanical parameters HM/HG, inertia of the wind mill and generator DM/DG, damping ratio of the wind mill Ks, windmill shaft stiffness
3.72/0.68 1.5/0.04 0.1
8 N P > > < V w ¼ 2 ½SV ðxi ÞDx1=2 cosðxi t þ /i Þ i¼1
> > : SV ð x i Þ ¼
ðA:1Þ
2K N F 2 jxi j p2 ½1þðF xi =lpÞ2 4=3
Here, KN, F and l are surface drag coefficient, turbulance scale and mean speed of wind, respectively. The specification N = 50 and Dx = 1.5. References [1] M.G. Molina, P.E. Mercado, E.H. Watanabe, Improved superconducting magnetic energy storage (SMES) controller for high-power utility applications, IEEE Trans. Energy Convers. 26 (2011) 444–456. [2] J. Shi, Y. Tang, Y. Xia, L. Ren, J. Li, SMES based excitation system for doubly-fed induction generator in wind power application, IEEE Trans. Appl. Supercond. 21 (2011) 1105–1108. [3] J. Shi, Y. Tang, K. Yang, L. Chen, L. Ren, J. Li, S. Cheng, SMES based dynamic voltage restorer for voltage fluctuations compensation, IEEE Trans. Appl. Supercond. 20 (2010) 1360–1364. [4] L. Wang, S.S. Chen, W.J. Lee, Z. Chen, Dynamic stability enhancement and power flow control of a hybrid wind and marine-current farm using SMES, IEEE Trans. Energy Convers. 24 (2009) 626–639. [5] I. Ngamroo, Robust SMES controller design for stabilization of inter-area oscillation considering coil size and system uncertainties, Physica C 470 (2010) 1986–1993. [6] J. Shi, Y. Tang, T. Dai, L. Ren, J. Li, S. Cheng, Determination of SMES capacity to enhance the dynamic stability of power system, Physica C 470 (2010) 1707– 1710. [7] K. Prabha, Power System Stability and Control, McGraw-Hill, New York, 1994. [8] M. Kayikci, J.V. Milanovic, Assessing transient response of DFIG-based wind plantsthe influence of model simplifications and parameters, IEEE Trans. Power Syst. 23 (2008) 545–554. [9] A. Feijoo, J. Cidras, C. Carrillo, A third order model for the doubly-fed induction machine, Electric Power Systems Research 56 (2000) 121–127. [10] A. Tapia, G. Tapia, J.X. Ostolaza, J.R. Saenz, Modeling and control of a wind turbine driven doubly fed induction generator, IEEE Trans. Energy Convers. 18 (2003) 194–204. [11] K. Clark, N.W. Miller, J.J. Sanchez-Gasca, Modeling of GE Wind TurbineGenerators for Grid Studies (Version 4.2), Tech. rep., General Electric International, Inc., 2008. [12] G. Balas, R. Chiang, A. Packard, M. Safonov, Robust Control Toolbox for Use with MATLAB, 2006. [13] Z. Yang, A. Bose, Design of wide-area damping controllers for interarea oscillations, IEEE Trans. Power Syst. 23 (2008) 1136–1143. [14] E. Vaahedi, W. Li, T. Chia, H. Dommel, Large scale probabilistic transient stability assessment using bc hydro’s on-line tool, IEEE Trans. Power Syst. 15 (2000) 661–667. [15] Esztergalyos, J. Chmn, Single phase tripping and auto reclosing of transmission lines – IEEE committee report, IEEE Trans. Power Del. 7 (1992) 182–192. [16] M.A. Green, B.P. Strauss, The cost of superconducting magnets as a function of stored energy and design magnetic induction times the field volume, IEEE Trans. Appl. Supercond. 18 (2008) 248–251. [17] S. Nomura, T. Shintomi, S. Akita, T. Nitta, R. Shimada, S. Meguro, Technical and cost evaluation on SMES for electric power compensation, IEEE Trans. Appl. Supercond. 20 (2010) 1373–1378. [18] P.M. Anderson, A. Bose, Stability simulation of wind turbine system, IEEE Trans. Power Appl. Syst. PAS-10 (1983) 3791–3795.
Please cite this article in press as: J. Fang et al., Probabilistic assessment of power system transient stability incorporating SMES, Physica C (2012), http:// dx.doi.org/10.1016/j.physc.2012.03.068
Contents lists available at SciVerse ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Probabilistic assessment of power system transient stability incorporating SMES Jiakun Fang, Wei Yao, Jinyu Wen ⇑, Shijie Cheng, Yuejin Tang, Zhuo Cheng State Key Lab of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, No. 1037, Luoyu Road, Wuhan 430074, China
a r t i c l e
i n f o
Article history: Accepted 29 March 2012 Available online xxxx Keywords: Wind farm SMES Transient stability Monte-Carlo simulation
a b s t r a c t This paper presents a stochastic-based approach to evaluate the probabilistic transient stability index of the power system incorporating the wind farm and the SMES. Uncertain factors include both sequence of disturbance in power grid and stochastic generation of the wind farm. The spectrums of disturbance in the grid as the fault type, the fault location, the fault clearing time and the automatic reclosing process with their probabilities of occurrence are used to calculate the probability indices, while the wind speed statistics and parameters of the wind generator are used in a Monte Carlo simulation to generate samples for the studies. With the proposed method, system stability is ’’measured’’. Quantitative relationship of penetration level, SMES coil size and system stability is established. Considering the stability versus coil size to be the production curve, together with the cost function, the coil size is optimized economically. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction With the increasing wind energy being put into utility power system, the transient stability becomes one of the major problems when subject to the continuous disturbance brought by the wind power. As flexible, reliable and fast responding power compensation equipment, SMES is expected to contribute to dynamic stability of power system consisting wind farms [1–4]. Generally, the SMES with sufficiently large stored energy and power rating capacity is able to provide satisfactory damping effect. However, the superconducting magnetic coil, as the heart of SMES, need bulk load of investment per MJ. Hence, it is essential to evaluate the relationship between system stability and SMES capacity. Some related works considering coil size of SMES are done [5,6], but fail to calculated the stability with certain SMES capacity. The problem is that it is difficult to ‘‘measure’’ system stability quantitatively with a unified specification. Traditionally, power systems have been planned and operated using deterministic transient stability criteria. The systematic index, however, representing stability cannot be given out by this method. Probabilistic analysis of power system stability is to determine the stability index with main probabilistic factors which reflect the stochastic nature of the operating conditions such as parameters of the generator, network topology, and disturbances. Compared to the deterministic method, the probabilistic analysis takes considerably long computation time. But it measures the risk that system loses stability and helps with making decisions such as capacity of the SMES. ⇑ Corresponding author. Tel.: +86 27 87540569; fax: +86 27 87540937. E-mail addresses: [email protected] (J. Fang), [email protected] (J. Wen).
This paper presents a stochastic-based approach to evaluate the probabilistic transient stability indices of the power system incorporating the wind farm and the SMES by combining the MonteCarlo simulation and Bayesian classification. Uncertain factors include both sequence of events in power grid and stochastic generation of the wind farm. The 3rd order model of SMES considering dc current dynamic is proposed and time-domain simulations are given. The effects of the penetration level of the wind farm and the size of the SMES coil to the system stability is investigated. The proposed methodology can be used in system planning, decision for SMES capacity, and security warning. 2. System modeling and description A modified version of a two-area four-machine interconnected power system [7] is used in this paper. As is shown in Fig. 1, the synchronous generator is simulated with the 6th order model, as well as the power system stabilizer and governor. The No. 2 generator in area 1 is replaced with a doubly fed asynchronous generator (DFAG). The SMES is installed at the terminal of the DFAG. The capacitor bank at node 2 is fixed reactive compensator so that the power factor of the wind farm is 1.0. The steady state power flow can be adjusted to represent different penetration level of wind farm. 2.1. Modeling the DFAG The DFAG is the most commonly used device for wind power generation. As is generally known, the rotor terminals are fed with a symmetrical three-phase voltage of variable frequency and amplitude. This voltage is supplied by a voltage source converter
0921-4534/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2012.03.068
Please cite this article in press as: J. Fang et al., Probabilistic assessment of power system transient stability incorporating SMES, Physica C (2012), http:// dx.doi.org/10.1016/j.physc.2012.03.068
2
J. Fang et al. / Physica C xxx (2012) xxx–xxx
1
5
G1
7
6 25km
110km
8
110km
9
10km 110km
110km
10 10km
11 25km
3 G3
2 4
Area 1 WT
SMES
G4
Area 2
Fig. 1. The test system.
set to 1.0. The inner control loop, named current control loop, which is executed by the rotor side converter, decouples the power regulation by controlling the direct and quadrature components of the rotor current. The stator-flux-oriented vector control scheme is used to implement the decoupled control of active and reactive power in modeling the rotor-side converter controller. The low voltage ride through (LVRT) process is considered in control scheme of the DFAG. During and immediately following sustained faults, limiting the real power command with both a cap and a ramp rate limit reduces system stress to the wind generator. As is shown in Fig. 2, the low voltage power logic (LVPL) is from GE’s report [11]. The maximum limit of the reference power PQref is related to the terminal voltage. During fault time, the terminal voltage is less than 1.0 pu, which leads to limitation to the power output of the DFAG. When terminal voltage is under 0.5 pu the crow bar is activated and PQref are set to zero. In this paper, the capacity of the converters is set to 30% of the DFAG’s capacity. Hence, the slip of the DFAG cannot be over 1.3 pu or lower than 0.7 pu, which will lead to splitting of the DFAG. Not only the deviation of rotor angle of the synchronous machine being over 180°, but the slip larger than 1.3 pu or lower than 0.7 pu is the used to judge the system is unstable.
usually equipped with IGBT-based power electronics circuitry [8]. The basic structure is shown in Fig. 2. The electrical part of the DFAG is modeled as the asynchronous generator using motor convention. The transient in the stator and in the branch between the inverters and the grid is neglected. This is widely used simplification in transient stability modeling of the synchronous and asynchronous machines [9]. (1) is derived from the basic equations of the asynchronous machine, taking positive currents as going into the machine (motor convention).
8 h i > > pE_ 0 ¼ jsE_ 0 RLrr E_ 0 jðLs X 0 ÞI_s j LRmr U_ r > > > < _s U_ s ¼ Rs I_s þ jxs u > _ _ _ > us ¼ Ls Is þ Lm Ir > > > : _ ur ¼ Lr I_r þ Lm I_s
ð1Þ
Here, E0 is an electrical potential component related to rotor flux u. The subscript s and r represents electrical quantity on the stator and rotor, respectively. The dotted variables are vectors synthesized by direct and quadrature components, i.e. U_ s ¼ uds þ juqs The parameters of the DFAG are given in Table A.1 in Appendix A. In the mechanical part of the DFAG representing the drive train and rotor dynamic, the 2-mass model is adopted as shown in (2).
8 dxM > > < 2HM dt ¼ T M T G DM xM xG 2HG ddt ¼ T G T E DG xG > > : dhs ¼ x0 ðxM xG Þ dt
2.2. Modeling of SMES
ð2Þ
A simplified 3rd order SMES model is proposed in this work, as is shown in Fig. 3. The modulation ratio M and firing angle a is modeled as two 1st order lag
where TG is The mechanical torque of the asynchronous machine, and TG = KShS DS(xG xM). The DFAG control structure consists of 2 cascaded control loops [10]. The outer one, named power control loop, regulates the active and reactive power at the stator side based on the maximum power point tracking (MPPT) strategy. And the power factor is
(
_ ¼ 1 M þ uM M Tc
ð3Þ
a_ ¼ T1c a þ ua
where uM and ua are input signals from the upper level control of the SMES. The parameter Tc is the communication delay and the dy-
LVPL
1.11
V
Low voltage power logic MPPT control
zerox (0.5)
Brkpt (0.9)
Rotor-side converter
Electrical Equations
Power System
Asynchronous generator Wind speed
Windmill aerodynamic
Drive train
Mechanical Equations
Fig. 2. Block diagram for DFAG modeling.
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J. Fang et al. / Physica C xxx (2012) xxx–xxx dc current dynamic
+ -
Damping controller
Power System
+ Voltage restorer
Converter dynamic
upper level control
Fig. 3. The upper level controller of SMES.
namic of the converter. Meanwhile, the dynamic of the dc current of the SMES can be calculated by integration of SMES active power output,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rt 2 ð1 þ nÞPsm dt Id ðtÞ ¼ I2d0 0 L
ð4Þ 2.3. Upper level control strategy of SMES
where Psm, Id0 and L are SMES output, initial state of the dc current, and inductance of the superconducting coil, respectively. n represents the loss during power transferring. With M, a and Id(t), the SMES output can be
(
pffiffi Psm ¼ 32 U s Is cos a ¼ 3 4 3 U s MId ðtÞ cos a pffiffi Q sm ¼ 32 U s Is sin a ¼ 3 4 3 U s MId ðtÞ sin a
in PQ frame, whose radius is proportional to dc current Id. The coil size and dc current also determine the maximum energy to be stored in SMES. Hence, with proposed 3rd order SMES model, the capacity of the SMES is represented by single specification L.
ð5Þ
For the ideal converter for SMES, the modulation ration can be no more than 1.0. Consequently, the active and reactive power regulation is limited by dc current Id. Neglecting the impedance of the transformer and filter circuit, the power regulation range is a circle
Owing to the decoupled 4-quadrant power regulation characteristics, the SMES improves system stability in two ways: damping controller in active power control loop and voltage restorer in reactive power control loop, represented in Fig. 3. In this paper, the mixed output-feedback control with regional pole placement is applied to design the damping controller [12,13]. This multi-objective control overcomes the limitations of single objective synthesis technique functioned with pole placement and incorporate the H1 and H2 performance by selecting weight of each indicator. Fig. 4 illustrates the effectiveness of the controller. In this case, a 3-phase fault lasting 200 ms is simulated in node 7
Fig. 4. Active power damping controller.
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J. Fang et al. / Physica C xxx (2012) xxx–xxx
1.2 1
1
SMES capacity = 30 H SMES capacity = 20 H SMES capacity = 10 H
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
SMES capacity = 30 H SMES capacity = 20 H SMES capacity = 10 H
0.8
QSMES (p.u.)
Terminal voltage (pu)
1.5
0.6 0.4 0.2 0
4.5
−0.2
0
0.5
1
1.5
2
time (sec)
2.5
3
3.5
4
4.5
5
time (sec)
Fig. 5. Time domain simulation: SMES voltage restorer.
and penetration level is 50%. The capacity of SMES is 50H. Note that in this case, SMES damps the power oscillation only by active power regulation, and hence improves stability. The voltage restorer is a simple PI controller adopting terminal voltage deviation as feedback, which helps the DFAG to ‘‘ride through’’ the low voltage during fault time. From time domain simulation in Fig. 5, one can see the SMES reactive power output during fault is much larger than post fault. In this case, a 3-phase fault lasting 300 ms is simulated in node 7 and penetration level is 50%. SMES acts as dynamic voltage compensator such as STATCOM. Its capacity determines how the voltage can be improved so that the DFAG can be and hence whether the system is stable to this fault. Time-domain simulations illustrating the influence of SMES capacity and wind farm penetration level are given in Fig. 6. A 3phase fault lasting 300 ms is simulated in node 8. From Fig. 6a, one can see that SMES with larger capacity can achieve better damping effect. We adjust wind farm penetration by increasing power output of DFAG while holding No. 1 synchronous generator’s output invariant. Meantime, the Nos. 3 and 4 synchronous generators’ outputs are reduced to maintain power balance. Fig. 6b shows that under the same fault, system with higher penetration level is easier to be unstable.
type, fault location,fault duration, and successful/unsuccessful reclosing [14]. The stochastic model is quite consistent with the IEEE Power Systems Relaying Committee Working Group [15], saying that the single phase-to-ground, double phase-to-ground, phase-to-phase and the 3-phase fault types take 93%, 2%, 4% and 1%, respectively. The transmission line is divided into three parts: the first 20% near the sending end, that is, area 1 in this paper, the middle 60% called mid-line and 20% near the receiving end. At each part of the line the fault location is evenly distributed sharing the probability of 0.1307, 0.7021 and 0.1672, respectively, shown in Table 1. In each part, the fault location meets the uniform distribution. Through historical fault events from the BC Hydro system, it was noted that in over 90% of the faults caused by lightning, automatic reclosing was successful, while that was the case in less than 50% of faults due to all other causes. And 82.51% of faults are due to lightning. The probability of successful automatic reclosing was thus estimated by 0.8251 0.9 + 0.1749 0.5 = 0.83. For the fault clearing time, a linear combination of two normal distributions is used in this paper. The probability density function used is represented by Fig. 7.
Table 1 Fault location probabilities.
3. Probabilistic transient stability assessment 3.1. Probabilistic factor models Historical data from the BC Hydro system was used to develop stochastic model for the disturbance sequence, including fault
Fault location
Range
Probability
Close-in Mid-line Far-end
First 20% of the line Middle 60% of the line Last 20% of the line
0.1307 0.7021 0.1672
−3
10
SMES capacity = 30 H SMES capacity = 20 H SMES capacity = 10 H
2
Rotor speed deviation of the DFAG (rad)
deviation of rotor angle (rad)
3
1 0 −1 −2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
Penetration level of 41.18% Penetration level of 50% Penetration level of 56.52%
5
0
−5
0
time (sec)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time (sec)
Fig. 6. Time-domain simulation.
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J. Fang et al. / Physica C xxx (2012) xxx–xxx
0.1
the index ‘‘measuring’’ stability on the system level, not just based on one or two selected faults. The calculation process consists of two loops, which divides uncertain factors into two types, as is shown in Fig. 8. The outer loop generates a series of disturbance sequences, taking fault type, range of fault location and successful/unsuccessful reclosing into consideration. For every disturbance sequence, the bisection algorithm is used to search the critical clearing time (CCT) for every steady state and fault. Together with the probability density function shown in Fig. 7, we can get the probability of stability, represented as Pmn j in (6). Then with probability of their occurrence Amn j, the probabilistic stability can be combined using the Bayesian probability formula.
0.09
Probability density
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0
0
5
10
15
20
25
Ps ¼
3.2. Procedures for probabilistic transient stability studies In traditional transient stability studies, the fault type, fault location, and reclosing are deterministic. While probabilistic studies, take into account the stochastic nature of the power system. Fig. 8 shows the calculation procedure for including probabilistic factors in transient stability studies, which is implemented by repeated deterministic simulation many times according to a spectrum of different states with their probabilities of occurrence. Additionally, assessing probabilistic transient stability can give
Start
Monte Carlo simulation
Bayesian classification
Select fault type, fault location, and reclosing
Generate the wind speed sequence Locate fault precisely
Search CCT using Bisection algorithm
m¼1n¼1
Cycles (60 Hz) Fig. 7. Probability density function used for fault clearing time.
M P N P
J P
Pmnj Amnj
ð6Þ
j¼1
The second class of uncertain factors are calculated in the inner loop using the Monte-Carlo method. The precise location of the fault in the range determined in the outer loop, parameters of the wind farm and wind speed are sampled to make every Pmn j accurate enough (variance is less than 0.01). With processing the uncertain factors mentioned above, we can get the stability index of the system by proposed method. 4. Optimal coil sizing With probabilistic transient stability study, we give the stability index of the system incorporating wind farm and SMES, shown in Fig. 9. Wind farm penetration is adjusted by increasing power output of DFAG and reducing the Nos. 3 and 4 synchronous generators’ outputs. Hence, increase of penetration level leads to increase of transmission burden of the tie-line. Form Fig. 9, we can see the system stability increases with the capacity of the SMES, while decreases with the penetration of wind power.This matches the time domain simulation results mentioned in former sections. From transient stability point of view, more energy storage is required by higher penetration of the wind power to maintain system stable. Here we use simple theory of economics to determine the optimal coil size. The cost function of SMES is proposed in [16,17].
Cost ¼ 0:95 ½EnergyðMJÞ0:67
ð7Þ
Under certain penetration level of wind farm, a production curve representing system stability versus SMES coil size can be obtained from Fig. 9. Taking 30% penetration for example, we draw both the production curve and cost function in Fig. 10a with dark and blue solid line,respectively. Here we assume that the rated dc current
Accurate enough?
Record the average CCT for this disturbance
N all disturbances covered?
Output results Fig. 8. Simulation flowchart.
Probability of stability (%)
N 100 80 60 40 20 0 52
50
48
46
44
Penitration level (%) 42
0
20
60
40
80
100
ES capacity (H)
Fig. 9. Index of transient stability.
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J. Fang et al. / Physica C xxx (2012) xxx–xxx
70 3
60 50
2
40 30 20 10 0
1
Cost functions 0
0 10 20 30 40 50 60 70 80 90 100
100
5
Optimized coil size
90 4
80
Cost (M$)
4
80
Cost (M$)
Stable Probability (%)
90
Stable Probability (%)
5
100
70 3
60 20
30 L*
Inductance (H)
40
50
Inductance (H)
Fig. 10. Optimal coil sizing.
is 100A. Then the cost function is translated parallel to being tangent with the production curve, as is shown in dash blue line in this figure. Fig. 10b is zoomed into depict the optimal coil size L⁄. It can be seen from these figures that when SMES capacity is less than L⁄, the slope of production curve is always larger than that of cost function, indicating that additional investment on SMES with larger size is repaid abundantly. When coil size L > L⁄, the increment of improvement of system stability is less than increment of investment, indicating that other types of energy storage or FACTs devices instead of SMES may be considered to meet the system requirements. 5. Conclusion Probabilistic study of power system incorporating wind farm and SMES is proposed in this paper. It extends and illustrates a basic procedure for calculating the probability of transient stability. Electro-mechanical model of both DFAG and SMES are proposed and time domain simulation results are given. The stability index of the power system is evaluated, using both bisection algorithm and Monte-Carlo simulation. Quantitative relationship between system stability with SMES capacity and penetration level of the wind power is given. Considering the stability versus coil size to be the production curve, together with the cost function, the coil size is optimized. This study confirms the practical merits in terms of stability and economics of SMES. Appendix A. The wind turbine parameters Both the electrical and mechanical parameters of the DFAG used in this paper are shown in Table A.1. Note that Ls and Lr are sum of the stator linkage reactance and the mutual reactance. The gust wind speed fluctuation is considered and modeled according to [18], shown in (A.1). Table A.1 Parameters of the DFAG. Electrical parameters
Value
Rs/Rr, stator/rotor resistance Ls, total reactance of the stator circuit Lr, total reactance of the rotor circuit Lm, mutual reactance
0.01499/0.01534 8.6929 8.6336 8.504
Mechanical parameters HM/HG, inertia of the wind mill and generator DM/DG, damping ratio of the wind mill Ks, windmill shaft stiffness
3.72/0.68 1.5/0.04 0.1
8 N P > > < V w ¼ 2 ½SV ðxi ÞDx1=2 cosðxi t þ /i Þ i¼1
> > : SV ð x i Þ ¼
ðA:1Þ
2K N F 2 jxi j p2 ½1þðF xi =lpÞ2 4=3
Here, KN, F and l are surface drag coefficient, turbulance scale and mean speed of wind, respectively. The specification N = 50 and Dx = 1.5. References [1] M.G. Molina, P.E. Mercado, E.H. Watanabe, Improved superconducting magnetic energy storage (SMES) controller for high-power utility applications, IEEE Trans. Energy Convers. 26 (2011) 444–456. [2] J. Shi, Y. Tang, Y. Xia, L. Ren, J. Li, SMES based excitation system for doubly-fed induction generator in wind power application, IEEE Trans. Appl. Supercond. 21 (2011) 1105–1108. [3] J. Shi, Y. Tang, K. Yang, L. Chen, L. Ren, J. Li, S. Cheng, SMES based dynamic voltage restorer for voltage fluctuations compensation, IEEE Trans. Appl. Supercond. 20 (2010) 1360–1364. [4] L. Wang, S.S. Chen, W.J. Lee, Z. Chen, Dynamic stability enhancement and power flow control of a hybrid wind and marine-current farm using SMES, IEEE Trans. Energy Convers. 24 (2009) 626–639. [5] I. Ngamroo, Robust SMES controller design for stabilization of inter-area oscillation considering coil size and system uncertainties, Physica C 470 (2010) 1986–1993. [6] J. Shi, Y. Tang, T. Dai, L. Ren, J. Li, S. Cheng, Determination of SMES capacity to enhance the dynamic stability of power system, Physica C 470 (2010) 1707– 1710. [7] K. Prabha, Power System Stability and Control, McGraw-Hill, New York, 1994. [8] M. Kayikci, J.V. Milanovic, Assessing transient response of DFIG-based wind plantsthe influence of model simplifications and parameters, IEEE Trans. Power Syst. 23 (2008) 545–554. [9] A. Feijoo, J. Cidras, C. Carrillo, A third order model for the doubly-fed induction machine, Electric Power Systems Research 56 (2000) 121–127. [10] A. Tapia, G. Tapia, J.X. Ostolaza, J.R. Saenz, Modeling and control of a wind turbine driven doubly fed induction generator, IEEE Trans. Energy Convers. 18 (2003) 194–204. [11] K. Clark, N.W. Miller, J.J. Sanchez-Gasca, Modeling of GE Wind TurbineGenerators for Grid Studies (Version 4.2), Tech. rep., General Electric International, Inc., 2008. [12] G. Balas, R. Chiang, A. Packard, M. Safonov, Robust Control Toolbox for Use with MATLAB, 2006. [13] Z. Yang, A. Bose, Design of wide-area damping controllers for interarea oscillations, IEEE Trans. Power Syst. 23 (2008) 1136–1143. [14] E. Vaahedi, W. Li, T. Chia, H. Dommel, Large scale probabilistic transient stability assessment using bc hydro’s on-line tool, IEEE Trans. Power Syst. 15 (2000) 661–667. [15] Esztergalyos, J. Chmn, Single phase tripping and auto reclosing of transmission lines – IEEE committee report, IEEE Trans. Power Del. 7 (1992) 182–192. [16] M.A. Green, B.P. Strauss, The cost of superconducting magnets as a function of stored energy and design magnetic induction times the field volume, IEEE Trans. Appl. Supercond. 18 (2008) 248–251. [17] S. Nomura, T. Shintomi, S. Akita, T. Nitta, R. Shimada, S. Meguro, Technical and cost evaluation on SMES for electric power compensation, IEEE Trans. Appl. Supercond. 20 (2010) 1373–1378. [18] P.M. Anderson, A. Bose, Stability simulation of wind turbine system, IEEE Trans. Power Appl. Syst. PAS-10 (1983) 3791–3795.
Please cite this article in press as: J. Fang et al., Probabilistic assessment of power system transient stability incorporating SMES, Physica C (2012), http:// dx.doi.org/10.1016/j.physc.2012.03.068