Improving the transient stability-constrained optimal power flow with ...

ISSN 10683712, Russian Electrical Engineering, 2014, Vol. 85, No. 12, pp. 777–784. © Allerton Press, Inc., 2014.

Improving the Transient StabilityConstrained Optimal Power Flow with Thyristor Controlled Series Compensators1 ThanhLong Duonga, b, *, Yao JianGanga, and VietAnh Truongc a

Department of Electrical Engineering, Hunan University, Changsha, Hunan, 41082 China Department of Electrical Engineering, Industrial University of Hochiminh City, 70000 Vietnam cDepartment of Electrical Engineering, University of Technical Education Hochiminh City, 70000 Vietnam *email: [email protected] b

Received November 14, 2014

Abstract—Due to the rapid increase of electricity demand and the deregulation of electricity markets, power systems tend to operate closer to stability boundaries. The scheme of preventive control should bring in a tradeoff between economics and security, which are often the two major inconsistent requirements for the daily operation of many power systems in the world. Optimal power flow (OPF) is a powerful tool to weaken the conflict between economy and security, but the main obstacle faced is that, the complexity involved for OPF with transient stability constraints is several orders of magnitude higher than that of conventional OPF with merely static constraints. Therefore, consideration of Transient Stability Constraints in Optimal Power Flow (OPF) problems is becoming more and more imperative. FACTS devices such as Thyristor Controlled Series Compensator (TCSC) can be very effective to power system security in case of a contingency. In this paper, we concentrate on the improving transient stabilityconstrained optimal power flow to against single contingencies via the use of TCSC. Study results on IEEE 30bus system have proved the effectiveness of using TCSC to improve transient stabilityconstrained optimal power flow (TSCOPF). Keywords: power system security, TCSC, transient stability, OPF, TSCOPF DOI: 10.3103/S1068371214120165 1

1. INTRODUCTION

The present day power systems are facing many challenges in terms of system operation to obtain eco nomic benefit and security. Various factors, such as environmental, rightofway and cost constraints have limited the expansion of the transmission networks. Utilities try to maximize the utilization of the existing transmission asset that may, some times, lead to inse cure operation of the system. Hence, power system security has become one of the most important issues in the electricity market operation [1]. Better market and system operating conditions may be achieved when system security and economy are better accounted. Solution of this problem is known as Secu rity Constrained Optimal Power Flow (SCOPF). The SCOPF is an extension of the Optimal Power Flow (OPF) problem which is used to obtain an eco nomical operation of the system while considering not only normal operating limits, but also violations that would occur during contingencies. The SCOPF changes the system precontingency operating point so that the total operating cost is minimized, and at the same time no security limit is violated if contingencies occur. Although the SCOPF are still challenges related 1 The article is published in the original.

to computations, it is expected that the SCOPF will eventually become a standard tool in the electricity industry [2]. Various approaches to approximate this region in OPF models have been proposed. For example, in [3] has proposed an algorithm for solving SCOPF prob lem through the application of evolutionary program ming (EP). A new robust differential evolution algo rithm for SCOPF considering detailed generator model is presented in [4]. Florin Capitanescu and Louis Wehenkel et al. [5] has proposed a new iterative approach to the corrective SCOPF Problem. [6] has presented a approach to solve an optimal power flow problem with embedded security constraints repre sented by a mixture of continuous and discrete control variables, where the major aim is to minimize the total operating cost, taking into account both operating security constraints and system capacity requirements. In [7] has proposed of DC SCOPF approximation to improve iterative AC SCOPF Algorithms. A novel approach to pricing the system security by paralleliz ing the security constrained optimal power flow (SCOPF) based marketclearing model is presented in [8]. The use of a security constrained in OPF is increas ingly necessary in today’s stressed power system,

777

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THANHLONG DUONG et al. Bus i

Bus j

–jXs Rij + jXij

Fig. 1. Model of transmission line with TCSC.

which operate under market rules. Therefore, Tran sient Stability Constrained OPF (TSCOPF) to guar antee an appropriate security level has become an attractive research topic recently. Transient stability is the ability of power system to maintain synchronous operation of machines when it is subject to a set of credible contingencies. Transient stability testing for the optimal solution obtained from the OPF has to be performed under all the credible disturbances to ensure the system stability performance. With TSCOPF, optimal operating state where the power system remains stable after certain contingencies can be obtained. However, TSCOPF is a nonlinear optimization problem with both algebraic and differential equa tions, which is difficult to be solved even for small power systems. There have been a number of papers proposed in the literature for incorporating transient stability constraints in the OPF in the context of eco nomic dispatch. Singh and David have proposed a transient energy function based redispatch method in [9] where the generators are redispatched if the stabil ity margin of power system, calculated is inadequate. The redispatch was also made sensitive to price sig nals so that the competing generators had an input to the redispatch. In [10–12], the authors convert the original TSCOPF into an optimization problem via a constraint transcription based on functional transfor mation techniques. This approach seems to be a promising method to solve large systems. A new approach to formulate and solve the TSCOPF prob lem in the initial operating state based on the SIngle Machine Equivalent (SIME) method has proposed in [13]. In [14], the authors convert the power system transient stability model into an algebraic set of equa tions for each time step of the timedomain simula tion. This set of algebraic equations is introduced in the OPF as transient stability constraints. The size of the resulting problem is typically large. Ref. [15] has proposed the OPF with transient stability constraints (OTS) problem using functional transformation tech niques by converting the infinite dimensional OTS into a finite dimensional optimization problem. In [16], based on SIME the transient stability is con strained by limiting the corresponding unstable OMIB trajectory no larger than a critical OMIB trajectory at one specific integration time step. In [17] the multi machine dynamics are replaced by the corresponding OMIB swing equation, and the discretization is then applied for solution.

Rapid development of selfcommutated semicon ductor devices has made it possible to design power electronic equipment. This equipment is well known as flexible ac transmission system (FACTS) [18]. FACTS devices such as Thyristor Controlled Series Compensator (TCSC) can be very effective to power system security in case of a contingency. The objective of FACTS devices is to control power flow so that it flows through the designated routes, increase trans mission line capability to its maximum thermal limit, and improve the security of transmission system with minimal infrastructure investment and environmental impact. The studies for solving such problems can be found in the literature. Ref. [19] has presented a method for the modal synthesis of the control of an electrical power system with FACTS devices. The control is intended for damping the variation of power flow. The task of minimizing the undesirable interac tion of FACTS devices in electric mains has been exam ined in [20]. In [21] has been presented the decentral ized synthesis of stabilizing controllers based on FACTS devices in an integrated electrical power system (IEPS), which secures minimum energy consumption for con trol. In this paper, we concentrate on improving tran sient stabilityconstrained optimal power flow to against single contingencies via the use of TCSC. Study results on IEEE 30bus system have proved the effec tiveness of using TCSC to improve transient stability constrained optimal power flow (TSCOPF). 2. STATIC MODELING OF TCSC The effect of TCSC on the network can be seen as a controllable reactance inserted in the related trans mission line [22]. Series capacitive compensation works by reducing the effective series impedance of the transmission line by canceling part of the inductive reactance. Hence the power transferred is increased. The model of the network with TCSC is shown in Fig. 1. The maximum compensation by TCSC is lim ited to 70% of the reactance of the uncompensated line where TCSC is located. A new line reactance (Xnew) is given as follows X new = X ij – X s , X new = ( 1 – K )X ij . Where K = Xs/Xij is the degree of series compensa tion and Xij is the line reactance between busi and busj. The power flow equations of the line with a new reactance can be derived as follows 2

P ij = V i G ij – V i V j ( G ij cos δ ij + B ij sin δ ij ),

(1)

2

Q ij = – V i B ij – V i V j ( G ij sin δ ij – B ij cos δ ij ),

(2)

2

P ji = V j G ij – V i V j ( G ij cos δ ij – B ij sin δ ij ),

(3)

2

Q ji = – V j B ij + V i V j ( G ij sin δ ij + B ij cos δ ij ).

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IMPROVING THE TRANSIENT STABILITYCONSTRAINED OPTIMAL POWER FLOW

Where δij is the voltage angle difference between bus i and bus j R ij X new and B ij = 2 . G ij = 2 2 2 R ij + X new R ij + X ij

779

3.2.1. Conventional OPF constraints —Power balance equation Pi(V, δ) + Pdi – Pgi = 0, i = 1,…, Nb,

(8)

Qi(V, δ) + Qdi – Qgi = 0, i = 1,…, Nb.

(9)

—Power generation limit min

max

(10)

min

max

(11)

3. TSCOPF PROBLEM FORMULATION

P gi ≤ P gi ≤ P gi , i = 1, …, N g ,

3.1. OPF Formulation

Q gi ≤ Q gi ≤ Q gi , i = 1, …, N g .

The OPF is a constrained optimization problem that requires minimization of an objective function. One of the possible objectives of OPF is the minimiza tion of the power generation cost subject to the satis faction of the generation and load balance in the trans mission network as well as the operational limits and constraints of the generators and the transformers [23]. The OPF is generally expressed in mathematical form as: min f(x, u).

(5)

g(x, u) = 0, h(x, u) ≥ 0.

(6)

Subject to

Where f(x, u) is the objective function. The equal ity constraints (6) are the power flow equations, while the inequality constraints are due to various limita tions. The limitations include lower and upper limits on generator real and reactive powers limits on voltage magnitudes, line and transformer maximum currents, and sets of possible transformer taps position and shunt admittances. The vector of independent vari ables u is given by the active powers of the generators, the voltages of the PV nodes and transformer tap set tings. The vector of dependent variables x is given by the voltages of PQ nodes, argument of PV nodes volt ages and reactive power generation. 3.2. Objective Function Transientstability constrained OPF can be mathe matically considered as a conventional OPF with additional inequality constraints imposed by the rotor angle limits [14]. The power flow solution should not only meet the steadystate constraints imposed by the conventional OPF problem but also the dynamic con straints imposed on the rotor angles during the tran sient period under study for a given set of contingen cies. An objective function and all related constraints for TSCOPF are described as follows. min

∑ C (P i

gi ).

(7)

i ∈ Ng 2

Where Ci(Pgi) = aP gi + bPgi + c is the bid curve of ith generator; a, b and c are cost coefficients for the generator. RUSSIAN ELECTRICAL ENGINEERING

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—Bus voltage limits min

Vi

max

≤ V i ≤ V i , i = 1, …, N b .

—Apparent line flow limit S1 ≥ S1, max, 1 = 1,…, N1.

(12) (13)

Where Pgi, Qgi are the active and reactive power generation at busi: Pdi, Qdi the active and reactive power demand at bus i: Vi the voltage magnitude at bus i: Vi, min and Vi, max the minimum and maximum voltage limits; Pgi, min and Pgi, max are the minimum and maxi mum limits of real power generation: Nb the total number of buses, Ng is the total number of generation buses: Sl the apparent power flow in transmission line connecting nodes i and j, and Sl, max is its maximum limit. 3.2.2. Transient stability constraints. The transient stability problem in a power system is described by a set of differentialalgebraic equations [24], which could be solved by timedomain simulation. The swing equation set for ith generator is · δi = ωi – ω0 , (14) M i ω· i = ω 0 ( P mi – P ei – D i ω i ), i = 1, 2…NG. (15) Where δi: rotor angle of ith generator ωi: rotor speed of ith generator Di: damping constant of ith generator Pmi: mechanical input power of ith generator Pei: electrical output power of ith generator ω0: synchronous speed. For simplicity the criterion for transient stability is defined as the rotor angle deviation with respect to the centre of inertia (COI), and hence the inequality con straints of transient stability are formulated as δ i – δ COI max ≤ δ max .

(16)

Where |i – COI|max corresponds to the maximum rotor angle deviation of ith generator from COI, and δmax is the maximum allowable rotor angle deviation. The setting of δmax is often based on operational expe rience. Most utilities would have it set to 100°–120° to allow the system to have sufficient stability margin.

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3.3. Formulation of TSCOPF According to Sections 3.2, the TSCOPF problem has now been formulated: Minimize (7) Subject to (8)–(13) (15) and (16)

G6 29 28

27 25

30 26

24

23

G5 15

19

18

20

17

21 16

14 13

22

12

10

G4

11

9

G3 1

8

6

4

3

7

TCSC

G1 2

5

G2

Fig. 2. IEEE 30bus system oneline diagram.

Here, δmax is set to 100°, and the position of COI is defined as NG

∑M δ

i i

i=1 δ COI = . N

(17)

G

∑M

i

i=1

Where Mi is the moment of inertia of ith generator.

4. CASE STUDY AND DISCUSSIONS The proposed method is applied to IEEE30 bus power system. The IEEE 30bus system consists of 41 lines and six generators. The line and bus data are available in [25, 26]. The single line diagram of the sys tem is shown in Fig. 2. The total load for the operating condition considered is 283.4 MW and 126.2 MVAR. The total fuel cost obtained from the MatPower pack age [26], which is considered as the base case without transientstability constraints. A three phase to ground fault at bus 2 and cleared by tripping line 2–5 at 0.3 s. Power System Toolbox [27] is used to perform time domain transientstability simulations for determin ing the variation of the generator angles. From table (row 2) it was observed that, when tran sientstability constraints were not considered (case1), the fuel cost was obtained optimal 984.34 $/h. How ever with this generation schedule, it was found that system transient stability was lost following the fault disturbance at bus 2 as shown in Fig 3. Clearly the net work cannot be operated in this way since security of the network was violated. In order to secure operation of the power system, TSCOPF must therefore be con sidered (case2). From table (row 3) it can be seen that, to achieve transient stability (case2), there is substantial change in the generation schedule to meet the transient stability constraints as compared with the generation schedule in case1 (row 2). The active power loading of generator 2 is reduced from 80 MW (case1) to 60 MW (case2) while generator 1, 3, 4, 5

Optimization results for IEEE 30bus system Pg1 (MW)

Pg2 (MW)

Pg3 (MW)

Pg4 (MW)

Pg5 (MW)

Pg6 (MW)

Fuel cost, $/h

Case 1: generation schedule without transient stability constraints

65.00

80.00

35.53

28.74

26.11

55.00

984.34

Case 2: generation schedule with transient stability constraints

78.00

60.00

40.00

30.68

27.02

55.00

998.43

Case 3: generation schedule (MW) without tran sient stability constraints, and without TCSC installation

71.03

85.00

39.99

29.57

27.12

55.00

1066.07

Case 4: generation schedule (MW) with transient stability constraints and TCSC installation in line 2–6 (XTCSC = –0.08 pu).

78.28

77.00

40.00

30.50

27.14

55.00

1070.08

Case


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Phase angle difference (fault cleared at 0.3 s) 150

100

Delta, deg

50

0

–50

–100 0

0.2

0.4

0.6

1.0 t, s

0.8

1.2

1.6

1.4

1.8

2.0

Fig. 3. Relative rotor angles for fault at bus 2 (case 1).

Phase angle difference (fault cleared at 0.3 s)

100 80 60

Delta, deg

40 20 0 –20 –40 –60 –80 0

0.2

0.4

0.6

1.0 t, s

0.8

1.2

1.4

1.6

1.8

2.0


increases from 65, 35.53, 28.74 and 26.11 MW (case1) to 78, 40, 30.68, and 27.02 MW (case2) respectively. A consequence of satisfying the transient stability con straints is that of increasing fuel cost from 984.34 $/h (case1) to 998.43 $/h (case2) as shown in table (col umn 8) and Fig. 4. RUSSIAN ELECTRICAL ENGINEERING

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The study in case 2 shows that, when the load demand is specified, the system without TCSC can operate with transient stability being maintained for the fault disturbances considered but fuel cost is increased. In order to investigate the contribution of the TCSC for minimum generation cost and remain

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THANHLONG DUONG et al. Phase angle difference (fault cleared at 0.3 s)

150

100

Delta, deg

50

0

–50

–100 0

0.2

0.4

0.6

0.8

1.0 t, s

1.2

1.4

1.6

1.8

2.0

1.8

2.0


Phase angle difference (fault cleared at 0.3 s) 100 80 60

Delta, deg

40 20 0 –20 –40 –60 –80 –100 0

0.2

0.4

0.6

0.8

1.0 t, s

1.2

1.4

1.6


transiently stable of power system following the fault disturbances considered, the load demand used in cases 1 and 2 is modified, and now has the total value of 299.2 MW (active power at bus 5 is increased from 94.2 MW to 110 MW). The description of the two fur ther cases 3 and 4 together with the dispatch solutions

and system transient stability assessments is given in table (row 4, 5). It can be seen from table (row 4) and Fig. 5 that, at this point in load demand, system transient stability is lost following the fault disturbance at bus 2. An attempt was made to reschedule the generation based


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on solving an OPF with transient stability constraints for system without TCSC. However, this solution was not still obtained. In this case, in order to maintain the transient stability, a TCSC is installed near bus 2 and in series with the transmission line between bus 2 and bus 6. In the initial, the level (K) of series compensa tion will be set equal to 1%. If the transient stability violations are not eliminated under the optimal power flow scenario, the compensation level of TCSC is increased as K = K + 1% and perform optimal power flow solution till all the violations are eliminated. Simulation results base on OPF with transient sta bility constraints and TCSC installation in line 2–6 is given in table (row 5) and Fig. 6. From Fig. 6 it can be observed that, system transient stability is obtained following the fault disturbance at bus 2. These simula tion results demonstrate the effectiveness of TCSC in improving system operation. Proper use of TCSC gives better results in terms of OPF solution and also ensures system transient stability following the fault disturbance, therefore enhancing the system dynamic security. 5. CONCLUSIONS The development of the modern power system has led to an increasing complexity in the study of power systems, and also presents new challenges to power system stability, and in particular, to the aspects of transient stability. Transient stability control plays a significant role in ensuring the stable operation of power systems in the event of large disturbances and faults, and is thus a significant area of research. Although the base OPF provides a dispatch that respects the physical and operational limitations, it did not guarantee the transient stability of the system after the fault has been cleared. Incorporating the transient stability constraints into the OPF to limit the value of the rotor angle allowed us to ensure that the system would be transiently stable following the fault. This paper investigates the improvement of tran sient stability with OPF using TCSC which is an effec tive FACTS device capable of controlling the active power flows in a transmission line by controlling appropriately its series parameters. Simulations are carried out in Matlab environment with TCSC to ana lyze the effects of TCSC on transient stability perfor mance of the system. The transient stability of the sys tem is compared with and without the presence of TCSC in the system. The simulation results demon strate the effectiveness of the proposed TCSC on tran sient stability improvement with OPF. ACKNOWLEDGMENTS The support for this research under Chinese National Science Foundation Grant (no. 51277059) is gratefully acknowledged. RUSSIAN ELECTRICAL ENGINEERING

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