使用快速審視法分析大型電力系統動態穩定度問題

行政院國家科學委員會專題研究計畫 成果報告

使用快速審視法分析大型電力系統動態穩定度問題 研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 97-2221-E-011-144-

執 行 期 間 ： 97 年 08 月 01 日至 98 年 09 月 30 日 執 行 單 位 ： 國立臺灣科技大學電機工程系

計 畫 主 持 人 ： 郭明哲

計畫參與人員： 碩士班研究生-兼任助理人員：莊智軒 碩士班研究生-兼任助理人員：翁聖傑

處 理 方 式 ： 本計畫涉及專利或其他智慧財產權，1 年後可公開查詢

中 華 民 國 98 年 10 月 27 日

行政院國家科學委員會補助專題研究計畫 ■ 成 果 報 告

□期中進度報告 使用快速審視法分析大型電力系統動態穩定度問題

計畫類別：■ 個別型計畫 □ 整合型計畫 計畫編號：NSC 97-2221-E-011-144-

執行期間： 97 年 8 月 1 日至 98 年 9 月 30 日

計畫主持人：郭明哲 共同主持人：

計畫參與人員：莊智軒、翁聖傑

成果報告類型(依經費核定清單規定繳交)：■精簡報告 □完整報告

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□赴國外出差或研習心得報告一份

□赴大陸地區出差或研習心得報告一份

□出席國際學術會議心得報告及發表之論文各一份

□國際合作研究計畫國外研究報告書一份

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列管計畫及下列情形者外，得立即公開查詢

■涉及專利或其他智慧財產權，■一年□二年後可公開查詢

執行單位：國立臺灣科技大學電機工程系

中 華 民 國 98 年 10 月 27 日

行政院國家科學委員會專題研究計畫成果報告

使用快速審視法分析大型電力系統動態穩定度問題

Dynamic Stability Analysis of Large-scale Power Systems Using a Fast Investigation Method

計畫編號：NSC 97-2221-E-011-144 執行期限：97 年 8 月 1 日至 98 年 9 月 30 日

主持人：郭明哲 臺灣科技大學電機工程系助理教授

中文摘要

隨著電力市場的自由化與標準市場設 計的提出，位於電力網邊界的電力傳輸交 易活動顯著增加，傳統電力公司的穩定性 分析已經無法滿足現今的操作環境，然而 模擬整個電力互連系統已超過現有的商業 穩定性分析軟體的能力，很幸運地大多數 穩定性問題是從區域性的故障開始，所以 可以借由在初期就發現並減輕這些故障的 影響來避免造成大系統的穩定度問題，基 於雅可比特徵向量的分解，這個計畫將提 出一個新的演算法來解決大型互連電力系 統 的 動 態 穩 定 度 問 題 ， 使 用 Python 和 MATLAB 程式語言在商用電力軟體 PSS/E 周 遭建造綜合的模擬環境不僅克服大型電力 系統無法模擬的問題而且縮短模擬時間，

將提出的方法會使用擁有接近三萬個匯流 排的美國東南電力系統與台灣電力系統來 進行驗證。

關鍵詞：大型電力系統、暫態穩定度、譜 圖分割、模擬時間

Abstract

Following power market deregulation and the proposed standard market design, the activities of cross boundary power transactions are increased significantly.

Traditional company based stability analysis can not satisfy today’s operating environment.

However, simulating the entire interconnection power system may exceed the capacity of existing commercial stability analysis software. Fortunately, most of the stability problems are started from the local level. One will be able to avoid most of the

system wide stability issues by mitigating the problems in the early stage. Based on eigenvector decompositions of the Jacobian, this plan proposes a new algorithm for performing the dynamic stability problem for a large interconnected power system. Using Python and MATLAB to build an integrated simulation environment around PSS/E not only overcomes the large-scale power system problem but also shortens the simulation time.

The proposed method will be verified on the Taiwan Power Company and Southwest Power Pool (SPP) system, which has close to 30,000 buses.

Keywords: Large-Scale Power System、

Transient Stability、Spectral Graph Partition、Simulation Time

1. Introduction

As social advancement and industry development have increased power consumption, the demand for power on the North America continent has been doubling every ten years since the invention of the induction motor by Nikola in 1888 [1]. Due to the utility deregulation, the complexity of the power system connecting generators and users across the US and Canada has increased dramatically as independent system operators transfer increasing amounts of power to each other. Controlling and managing this huge and complex power system has therefore become an engineering challenge. Moreover, even though many software packages are available to assist engineers with these tasks, no available commercial software enables to simulate the entire eastern interconnection system.

2

Therefore, developing tools and/or algorithms to handle such simulation is an urgent topic.

The northeast blackout on August 14th, 2003, the largest in US history, not only caused a huge pecuniary loss but also reminded engineers of the importance of power system stability. Because of increased computing ability, personal computers can now simulate a larger system with increasingly accurate results. On the other hand, the dynamic model also grows ever more complex. The huge systems and complex models exponentially increase the computational work, thereby hindering the building of realtime simulation. To solve this problem of stability analysis in a large-scale power system, two types of strategies were proposed: applying a parallel computing algorithm to accelerate the speed of simulation or developing dynamic equivalency techniques to simplify the circuit model.

Parallel software for cluster systems on high-performance computers has been widely used in every field. Moreover, research on parallel algorithms and their application in transient stability analysis has been well developed over the last 16 years [2]. The focus of this development lies in how to divide a large power system grid into several smaller blocks whose data are sent to different computers and computed separately at the same time. Thus, the computational load of each block must be the same, while the system partitioning and communication between blocks must be minimal.

Dynamic equivalent approach is based on the observation that oscillations in local areas seldom affect other areas for a traditional monopolistic electric power system. Therefore, one will be able to obtain reasonable simulation results by keeping full circuit model of the local power system while using dynamic equivalents for the system outside the local area to reduce the computational burden. Constructing a dynamic equivalent for a power system requires partitioning the system into coherent areas, aggregating the coherent areas, and aggregating the coherent generators and their

control devices [3].

Although parallel computing can accelerate the speed of simulation, there are some disadvantages. To apply the parallel software for cluster systems in dynamic stability is a complex procedure both in software development and hardware architecture. Therefore, it is difficult to apply parallel computing widely in power system simulation [12]. For dynamic equivalency approaches, although the simplified equivalent circuit can speed up the simulation, it faces some challengings after deregulation. Because the boundary of local areas disappears after power market deregulation, studying area may have to cover multiple traditional boundaries. It needs to develop a more appropriate algorithm to define the local area.

Because deregulation, the demarcation boundary among areas has become increasingly fuzzier and simulating the entire eastern interconnection of the North America power system exceeds the capacity of existing commercial software, the key point of the proposed method is how to properly define and aggregate a local area from a large-scale power system for dynamic stability simulation. The ability of the proposed method to overcome the software limitation to solve the dynamic stability of a large power system is verified through the studying of a large-scale power system. The proposed method is simpler and faster than the dynamic equivalency or parallel computing.

2. Partitioning Algorithm for Power Systems

When electric utilities or areas have a very large power flow exchange, simulating the entire power system is either impossible or extremely time consuming because of the maximum bus limitation in the commercial software. Therefore, the first step is to appropriately subdivide the large-scale power system by converting it into a graph partition problem. The object of this transformation is to solve the dynamic stability problem by finding the deviation of power flow. First, the power flow Jacobian must be calculated from the initial condition using the following

matrix:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

= ∂

∂

∂

δ real S δ P^N

(3)

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

∂

−

∂

∂

∂

∂

∂

∂

∂

=

L I N

N L

N N

V Q Q δ Q

V P δ

P

J (1) Once the eigenvalue for the Jacobian

matrix is solved, the variation in every branch can be used to calculate the power variation matrix ^δ

P^N

∂

∂

and produce a nondirectional graph. No matter whether the branch absorbs or provides energy, it is the absolute of variation that is of interest.

Therefore, the diagonal term of ^δ

P^N

∂

∂

is deleted and the sums of each column are inserted into the diagonal elements.

In addition to the traditional Jacobian matrix solution [4], another technique can be used to account for the power system losses.

For the purposes of this proposed application, this derivation [5] is deduced and modified during the final phase of the development.

The overbar of the form“ ” is used to emphasize a complex valued quality. is a vector with n real numbers. Similarly, is a vector with n complex numbers and is a matrix with complex numbers.

Let

ℜn

Cn n

Cn^×

n n×

The block

δ P^N

∂

∂ obtained from this derivation, which still preserves the properties of the weighted Laplacian matrix [6], is then combined with the sensitivity area so spectral graph partitioning can be applied in the next stage.

= ℜ

∈ ,δ:

δ ⁿ vector of bus voltage phase angles relative to an arbitrary synchronous reference frame of frequency

(no reference angle is deleted);

ω0

= ℜ

∈ ,V:

V ⁿ vector of variable bus voltage magnitudes;

=

∈C ,V:

V ⁿ vector of complex bus voltages; note that ^{V =V.*exp(jδ)}

=

∈C ^× ,Y:

Y ⁿⁿ full bus admittance matrix (reference bus rows and column not eliminated);

Solving the sensitivity area requires injection of 1MW power into the fault bus, followed by calculation of the power flow difference in each branch to identify those branches with larger variations and find the buses in their two ends. For this method, the power flow must first be solved; however, if the system is too large, an alternative method can be used to find the nearest n layers of buses from the fault bus, which then become the sensitivity area.

=

∈C , I:

I ⁿ vector of currents absorbed into network at each bus; note that ^{I =Y*V}

=

∈C , S:

S ⁿ vector of complex powers absorbed into network at each bus;

Once the sensitivity area is identified, the original system can be combined with the sensitivity area to produce the following equation:

wL J

J_final = + (4)

Where

note that

) (

* ).

( ) (

*

. conjI conj I V V

S= = J^final: Final Jocobian matrix

δ

∂

=∂P^N

J : Part of Jocobian matrix of the original system

} { diag

* )}

exp(

* .

{V j j V

V =

∂

=∂

∂

∂

δ δ δ

} { diag

* )} *

exp(

* .

* {

V j j Y V

I Y =

∂

= ∂

∂

∂

δ δ

δ L: Laplacian matrix of the sensitivity

area

}) { diag

*

* conj(

)}

( conj { diag

* )

* conj(

)}

( conj

{ I = j Y V = j Y V

∂

∂

δ w : The weighted parameter

If w is higher, the branch in the sensitivity area is stronger and therefore not easily cut off in the program.

Thus,

)}

( conj {

* ) ( conj

* } { diag

* }

* ).

I diag{conj(

* V j V Y diag V

S=j −

∂

∂ δ

(2) The next step is the spectral graph

partition of the matrix. In this Laplacian matrix, the second smallest eigenvalue , referred to as the algebraic connectivity of graph [7], and its corresponding eigenvector are called the Fiedler value and Fiedler vector, respectively.

The Fiedler value indicates a relative coupling between the resulting 2 subareas – i.e., a higher value represents higher coupling.

The eigenvector is then sorted to separate the system. The following flow chart describes this partition process.

final

J )

2(Jfinal

λ

4

Compute the eigenvector v²

corresponding to λ² of

final

J

for each node n of ^Jfinal if v₂(n) < 0

put node n in partition N- else

put node n in partition N+

endif endfor [8]

In accordance with theorem [9], G is assumed to be connected. If N- and N+ are defined by the above algorithm, N- is connected. If no component of the second eigenvector is zero, then N+ is also connected. Because the two subareas after cutting are still connected, further recursive partition is possible.

)

2(n v v2

The recursive spectral graph partitioning method terminates recursion given the following two conditions:

N

N_bus< or λ ^<C

(5) Where

Nbus: The bus number of an oncoming re-cutting area

N: A reasonable bus number

λ :The second smallest eigenvalue of an oncoming recutting area. The second smallest eigenvalue means that the two subnetworks display a higher or lower degree of connectivity.

C: A reasonable connective parameter.

If the value of λ is high, the connection

inside this area is so strong that continuous separation of the area is impossible.

The connected property of this partition method guarantees no island in the power system after partitioning.

3. Building an Integrated Test Environment

Despite the broad body of research on accelerating dynamic stability simulation, few applications exist for industrial systems because methods developed in the laboratory do not cooperate with industrial commercial software. Since the power system simulator for engineering (PSS/E) – commonly used for simulating transient stability in industry – provides an embedded Python interpreter, this situation can be improved. The Python program not only executes the PSS/E functions but also can easily be extended using code written in other languages, for example MATLAB syntax in our application [10]. Therefore, it can reduce the time for research and development. However, the dynamic functions for Python syntax do not work in version 30 of PSS/E, the user has to control the process with batch commands.

Figure 1 outlines the proposed approach in which the Python program is in control from the beginning until the choosing of the swing bus, and then, once the dynamic models are added, the batch commands control the remaining processes.

4. Test Results

To assess whether the algorithm can be practically applied to a contemporary large-scale power system, data for the complete 2005 summer peak Southwest Power Pool (SPP) system – including 29,657 buses, 40,080 branches, 5,208 generators, and 19450 loads – were tested in simulation.

It should be pointed out, however, that in operation only 29,528 buses are active and the system contains only one swing bus.

These tests were performed to compare the execution time and accuracy of results for the subsystem simulations against those for the full size system simulations [11].

4.1 Preparing the power flow data

Running the transient stability requires

advance preparation of the power flow data, the first step in which is partitioning the SPP system. Some of the corresponding partitioning processes and results are shown in figure 2.

The gray area of figure 2 shows the final subareas, the tip elements of the binary tree.

The final result is shown in figure 3.

Once the sensitivity area is calculated, the final cutting areas are the subareas that cover the sensitivity area. For example, if the fault bus is bus 25044, we start from the fault bus and expand out 15 layers of buses. We then choose both the fault bus and 15 layers of buses for the sensitivity area. If we do not weight the sensitivity area in the original system, we obtain the final cutting areas shown in the black blocks in figure 3.

We also deduce another partition process by first calculating the Laplacian matrix of the sensitivity area and then directly weighting the Laplacian matrix to the original system. The final Jacobian matrix can now apply to any partitioning method. As long as the weighting is not large, both methods are feasible and their results are almost identical. After partitioning, we must first run the power flow, after which our program will assign a swing bus if the separated area does not include the swing bus from the original system.

4.2 Single fault scenarios

To verify the proposed algorithm, six independent faults at different locations were studied. Because of the security issue, the map will not be shown but the relative locations of the faults are shown in Fig. 4.

The adjustable parameters chosen for the simulation are explained below:

w = 1 (which is similar to no weighting).

We then use the original Jacobian matrix to partition the power system.

N = 100, because the total number of buses in the SPP system is 29,528. Every subarea will then be around 100 buses.

C = 0.5. If the coupling of the next two subareas is tight, this parameter can avoid forward recursive partitioning.

During data validation, the first two seconds of stability simulation are a no-fault

test. We create a fault in two seconds and keep it 0.0833 seconds (5 cycles), after which we allow the simulation to run to 15 seconds.

After all stability cases have been simulated, we compare all the simulation graphs and list the results in Table I.

As shown in Figures 5 and 8, 6 and 9, and 7 and 10, respectively, some simulation results for the separated system are almost the same as those for the original system, some are similar, and some are different.

Because the four simulation results on fault bus 25044 are the same as those for the original system and fault bus 87456 is near the boundary, we choose both cases to compare simulation time. Our simulation environment is a laptop with 2.4G Hz CPU, 512MB RAM and Microsoft Windows XP Home Edition. The comparative results for the simulation time of dynamic stability are listed in Table II.

The simulation results indicate that the safest method is to choose 30 layers of buses near the fault in the SPP system. Their simulations results are almost the same as the original system. However, only partitioning eight layers of buses would produce a similar simulation result as for the whole system if the fault is far away from boundary.

4.3 Multiple faults scenarios

As described below, three multiple faults cases are created to study two distant faults and two near-by faults scenarios:

Case 1: Two faults occur simultaneously in load bus 54121 and generator bus 87456.

Case 2: Two faults occur simultaneously in load bus 64895 and load bus 83904.

Case 3: Two faults occur simultaneously in load bus 31426 and generator bus 25044.

The simulation results are shown in Table III. A comparison of the single and double fault test results shows that the double faults correspond to the single fault. Even though the fit is not 100%, the subsystem simulation is similar to the original whole system.

5. Graphical User Interface and Double Check Boundary Method

In order to allow the proposed algorithm

be smoothly applied to the other power system by other users, all programming codes have been integrated with the graphical user interface in the Python program. The user only needs to choose his/her wanted function, as shown in figure 11. The procedure will automatically give the commands to PSS/E or link to Matlab for complex computation, and then produce the simulation results of the dynamic stability.

6

After dynamic simulation, in order to guarantee the accuracy of simulation results, the user can use the verification procedure to double check the results. This test procedure will start from an area of eight layers of sensitivity, and then increase the layers to 15.

After comparing the differences in terms of both simulation results, if the difference is huge, the program will increase the layers again, and vice versa, as shown in figure 12.

The program will suggest whether the simulation result is accurate, and also draw the error between the suggestion layers and the compared nearby layers, as shown in figure 13 and 14. Then users can decide whether this is accurate enough.

6. CONCLUSIONS

After the deregulation, the demarcation boundary among areas has become increasingly fuzzier due to high volume cross boundary transactions. Traditional company based dynamic study is not suitable for this operation paradigm. The northeast blackout on August 14th, 2003 signifies the importance of overall system evaluation tools.

However, study the entire interconnected system can be either time consuming or technically infeasible due to size limitations of commercially software. The usage of dynamic equivalency or parallel processing is also difficult due to complicated mathematical algorithms and software/hardware requirements. This paper

proposes approaches to partition the system and aggregate pre-divided subareas to perform dynamic stability study of a large-scale interconnected power system like those in North America. Testing on SPP system indicates that this proposed partition method can achieve its goal and overcome

the size limitations of commercial software.

N N

or C

bus<

λ<

Fig. 1. The flowchart of the proposed method.

Fig. 2. The separation of the original SPP system.

Fig. 3. The final area chosen for the fault on bus 25044.

Fig. 4. The six fault positions.

Fig. 5. The voltage stability result on the Bus 82293.

Fig. 6. The voltage stability result on the Bus 82293.

Fig. 7. The voltage stability result on the Bus 82293.

Fig. 8. The voltage stability result on the Bus 87230.

8 Fig. 9. The voltage stability result on the Bus 87230.

.

Fig. 10. The voltage stability result on the Bus 87230.

Fig. 11. The graphical user interface to sumulate transient stability in a large-scale power system.

Fig. 12. The verification procedure to double check the simulation results.

Fig. 13. The example of error in one fault case.

Fig. 14. The example of error in double fault case.

TABLEI

COMPARING THE STABILITY SIMULATION RESULTS FOR SINGLE FAULT

How many layer of buses near the fault? 30 15 8 5 The fault in load bus 54121 O O V V The fault in generator bus 87456 O V X X The fault in load bus 64895 O O V V The fault in load bus 31426 O O O O The fault in generator bus 25044 O O O O The fault in load bus 83904 O V V X O:THE SIMULATION RESULT IS ALMOST THE SAME AS THE ORIGINAL SYSTEM.

V:THE SIMULATION RESULT IS SIMILAR TO THE ORIGINAL SYSTEM. X: THE SIMULATION RESULT IS DIFFERENT FROM THE ORIGINAL SYSTEM.

TABLEII

COMPARING THE SIMULATION TIME IN THE FAULT ON BUS 25044

AND 87456

Simulation time (sec) The fault location Bus 25044 Bus 87456

Whole System 338 654

30 layers 337 251

15 layers 71 61

8 layers 24 15

5 layers 14 14

TABLEIII

COMPARING THE STABILITY SIMULATION RESULTS FOR DOUBLE FAULTS

How many layer of buses near the fault?

Bus 54121 Bus 87456

Scenarios

Compare with the original Whole system

5 15 distant V

8 15 distant V

15 15 distant V

30 30 near-by O

Bus 64895 Bus 83904

5 8 distant V

8 8 distant V

15 15 distant V

30 30 near-by O

Bus 31426 Bus 25044

5 5 near-by O

8 8 near-by O

15 15 near-by O

30 30 near-by O

7. References

1. P. M. Anderson, and A. A. Fouad, Power System Control and Stability, 2nd ed., Wiley Interscience (IEEE Series on Power Engineering), New Jersey, 2003. p 3.

2. Shu, J.; Wei Xue; Weimin Zheng, "A parallel transient stability simulation for power systems," IEEE Trans. Power Systems, Volume 20, Issue 4, pp. 1709 – 1717, Nov. 2005 3. Galarza, R.J.; Chow, J.H.; Price, W.W.; Hargrave, A.W.;

Hirsch, P.M., "Aggregation of exciter models for constructing power system dynamic equivalents, " IEEE Trans. Power Systems, on Volume 13, Issue 3, pp. 782 – 788, Aug. 1998

4. J. D. Glover, and M. S. Sarma, Power System Analysis and Design, 3nd ed., Brooks/Cole, California, 2002, pp.

285-286.

5. C.L. DeMarco, T. Yong, J. Meng, and F.L. Alvarado,

"Efficient Computation of Higher Order Derivatives of

Power Flow and Line Flow Quantities, " 13th Power System Computation Conference, 1998.

6. Supun Tiptipakorn, "A Spectral Bisection Partitioning Method for Electric Power Network Applications," Master thesis, The University of Wisconsin – Madison, Dept. ECE., December 2001.

7. M. Fiedler, "Algebraic Connectivity of Graphs", Czechoslovak Mathematical Journal, 23, pp. 298-305, 1973 8. Demmel, J. "CS 267: Notes for Lecture 23, April 9, 1999.

Graph Partitioning, Part 2." Available:

http://www.cs.berkeley.edu/~demmel/cs267/lecture20/lectur e20.html

9. M. Fiedler, "A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory", Czechoslovak Mathematical Journal, 25, pp. 619--633, 1975

10. PSS/E-30.1 Users Manual, Siemens, April 2005

11. W.W.; Hargrave, A.W.; Hurysz, B.J.; Chow, J.H.; Hirsch, P.M, "Large-scale system testing of a power system dynamic equivalencing program, " IEEE Trans. Power Systems, on Volume 13, Issue 3, pp. 768 – 774, Aug. 1998

12. Fang Hualiang; Mao Chengxiong; Zhang Buhan; Lu Jiming,

"The Grid Computing Model—A New Computing Model For The Analysis and Computation Of Large-Scale Power System," Transmission and Distribution Conference, pp. 1 – 6, Aug. 2005