A simple implementation of power mismatch ...

A simple implementation of power mismatch STATCOM model into current injection Newton–Raphson power-flow method Salah Kamel, Francisco Jurado & David Vera

Electrical Engineering Archiv für Elektrotechnik ISSN 0948-7921 Volume 96 Number 2 Electr Eng (2014) 96:135-144 DOI 10.1007/s00202-013-0288-4

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Author's personal copy Electr Eng (2014) 96:135–144 DOI 10.1007/s00202-013-0288-4

ORIGINAL PAPER

A simple implementation of power mismatch STATCOM model into current injection Newton–Raphson power-flow method Salah Kamel · Francisco Jurado · David Vera

Received: 11 June 2012 / Accepted: 21 October 2013 / Published online: 6 November 2013 © Springer-Verlag Berlin Heidelberg 2013

Abstract This paper presents a simple implementation of Static Shunt Compensator (STATCOM) into Newton– Raphson current injection load flow method. The controlled STATCOM bus in the network is represented by voltagecontrolled bus with zero active power generation at the required voltage magnitudes. The power mismatch equation of the connected STATCOM bus is included in Newton– Raphson current injection load flow algorithm, while the other PQ buses are represented by current mismatch equations. Moreover, the parameters of STATCOM can be calculated during iterative process and the final value will be updated after the convergence is achieved. This representation of generator buses reduces the number of required equations with respect to the classical and improved versions of the current injection methods. In addition of that the developed model reduces the complexities of the computer program codes and enhances the reusability by avoiding modifications in the Jacobian matrix. The performance of the developed STATCOM model has been tested using standard IEEE systems. Keywords FACTS · Newton power flow · Current injection mismatches · STATCOM

Abbreviations PV NR PQ CB FACTS STATCOM SVC VSC N NPV CM PM ICM HM GTO IGBT IGCT MCT

S. Kamel e-mail: [email protected]

P, Q V δ  I h r,m k, f sp st Z G,B

D. Vera e-mail: [email protected]

∂

S. Kamel · F. Jurado (B) · D. Vera Department of Electrical Engineering, University of Jaén, EPS Linares, Linares, Jaén, Spain e-mail: [email protected]

Voltage controlled buses Newton–Raphson method Load buses Connected STATCOM bus Flexible AC transmission systems Static synchronous compensator Static VAR compensator Voltage source convertor Total number of buses Total number of voltage controlled nodes Current mismatch Power mismatch Improved current mismatch load flow method Hybrid mismatch load flow method Gate turn-off Insulated gate bipolar transistors Integrated gate commutated thyristors Metal-oxide semiconductor controlled thyristor Active and reactive complex powers Magnitude of node voltage Angle of node voltage Refer to mismatches of variables Current injection iteration number Subscripts refer to real and imaginary parts Subscripts refer to nodes Superscript refers to specified values Subscript refer to STATCOM device Refer to impedances of devices Real and imaginary parts of admittance matrix Refers to partial derivatives

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1 Introduction Flexible alternating current transmission system devices have been increasingly introduced in electrical power systems around the world. The FACTS devices can perform control of line reactance, bus voltage, line active power-flows, and line reactive power flows. These capabilities have made FACTS continually being installed fast in power system networks [1–6]. There are four categories of these devices: series compensator, shunt compensator, combined series–series compensator, and combined series–shunt compensator. The static synchronous compensator is considered a shunt controller. This device has capability of controlling the power flow through the line by injecting appropriate reactive power to electrical power system. The STATCOM was developed as an advanced static VAR compensator [6] where the voltage source convertor (VSC) is used instead of the controllable reactors and switched capacitors. However, the voltage source convertors require self-commutated power semiconductor devices such as GTO, IGBT, IGCT and MCT. Unlike in the case of variable impedance type SVC, there are many advantages of a STATCOM over SVC. These advantages are – STATCOM has faster response than SVC – Requires less space as bulky passive components are eliminated – Inherently modular and relocatable – It can be interfaced with real power sources such as battery, fuel cell. – A STATCOM has superior performance during the condition of low voltage as the reactive current can be maintained constant (in a SVC, the current of capacitive reactive drops linearly with the voltage at the limit of capacitive susceptance). It is possible to increase the reactive current in STATCOM under transient conditions if the devices are rated for the transient overload. But, In a SVC, the maximum reactive current is determined by the rating of the passive components (reactors and capacitors). The implementation of STATCOM into load flow solution required several modifications to the usual load flow algorithm. The following are the general modifications that need to be applied in the load flow analysis in order to include this device [7]: – New terms owing to the contributions from the STATCOM devices need to be included in the existing power flow equations of the concerned buses. – STATCOM impedance should be added into the admittance matrix. – New sub-blocks of Jacobian related to the STATCOM device should be included.

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– The power contributed by STATCOM at the connected bus should be included into power mismatches vector. With respect to the power flow methods, the Newton– Raphson method is considered as excellent load flow technique used in industry applications. But in this method, the elements of Jacobian matrix need to update during the iteration process. A new technique based on current injections approach [8–10] has been developed to solve this drawback. The main advantage of the load flow formulation based on current injections approach is the calculation of the Jacobian matrix, where, the off-diagonal elements are constant and equal to the corresponding element in the admittance matrix. Also the method does not need at all the use of transcendental functions during the iterative process. The current injection mismatch equations are very suitable especially for a large-scale system that contains on only PQ buses and few PV buses like the distribution systems. However, in the case of the existence PV buses, this formulation has slow convergence character. Consequently, developed technique has been presented to improve the convergence character of the method (ICM). This technique has been based on additional equations to represent the voltage mismatches for all PV buses [11]. The total required number of equations has been increased according to the number of PV buses. As a result of that, Newton-Raphson current injection load flow method became unreliable for solving the transmission systems and has used widely in distribution systems. This paper presents a simple modeling of STATCOM into NR current injection load flow method. The developed model is based on power mismatch approach. The STATCOM is represented as PV bus with the desired voltage magnitude and zero generated active power. The developed model avoids the change in the Jacobian matrix. Consequently, the complexity of the load flow program is decreased. The representation of generation buses using power mismatches will reduce the number of equations to be equal with the classic power mismatches Newton–Raphson load flow method. Moreover, the elements corresponding to load buses in the Jacobian matrix are constant.

2 Static synchronous compensator (STATCOM) A static synchronous compensator is a shunt-connected type of FACTS devices. STATCOM is based on the principle of voltage source inverter, where it could generate a controllable ac voltage source behind a transformer leakage reactance when a dc source is applied [12]. The schematic diagram of STATCOM is shown in Fig. 1. The main component of STATCOM is the voltage converter that changes dc source into ac voltage source. Although

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137

Connected bus

P

also could exchange active power by adding an external dc energy source such as dc generator, wind turbines, and photovoltaic [14]. The STATCOM function is identical to the synchronous condenser. However, its response time is much faster than the synchronous condenser due to the absence of the rotating mass. It can provide smooth, continuous voltage regulation to prevent voltage collapse, to improve transmission stability, and to damp power oscillations. Although STATCOM has many applications, this paper will cover only the application for voltage control on particular bus due to its application in load flow analysis.

Q Coupling Transformer

VSC Controller

3 Power mismatch STATCOM model

Energy source (Optional)

Pref or θ ref

Qref or Vref

P

This paper presents a simple implementation of STATCOM steady-state model into current injection Newton–Raphson load flow program. The equivalent circuit of STATCOM is shown in Fig. 2. The developed model decreases the complexities of the computer load flow program codes with STATCOM device. The model is based on the representation of the connected STATCOM bus by generator (PV-type) bus with desired voltage magnitude and zero active power generation. The mismatch for this PV bus is the power mismatch and the corresponding correction is the angle. For the other load buses (PQ-type) the mismatches are real and imaginary current mismatches similar to Newton–Raphson current injection mismatches load flow formulation [8]. The load flow is carried out with the simple equivalent STATCOM model using Eq. 1.

Fig. 1 STATCOM schematic diagram

various topologies of dc to ac converter are available, converter using GTO thyristors is presently the most practical for high-power utility applications [12,13]. The capacitor acts as a dc source that automatically adjusted as required to serve as a voltage source for the converter. The controller is used to operate the inverter so that the phase angle between the inverter voltage and the line voltage is dynamically adjusted so that the STATCOM generates or absorbs desired VAR at the point of connection. Similar to SSSC, the STATCOM ⎡

⎡

G 11 − b1

B12

−B11 − d1 G 12

G 12

···

−B12

···

B22 − a2 G 22 − b2

−B21 .. .

G 22 − c2 .. .

···

−B22 − d2 · · · .. .

∂ Im1 ∂δPV ∂ Ir1 ∂δPV ∂ Im2 ∂δPV ∂ Ir2 ∂δPV

.. .

∂ Im1 · · · ∂δ ··· CSB

· · · B1n

G 1n

∂ Ir1 · · · ∂δ ··· Stat

· · · G 1n

−B1n

∂ Im2 · · · ∂δ ··· Stat

···

∂ Ir2 · · · ∂δ ··· Stat .. .

···

..

G 21

..

∂ PPV ∂ Vm1

∂ PPV ∂ Vr2

∂ PPV ∂ Vm2

∂ PStat ∂ Vr2

∂ PStat ∂ Vm2

.. . .. . G n1

.. . .. . Bn2

.. . .. . G n2

··· .. . .. . ···

−Bn1

G n2

−Bn2

···

∂ PStat ∂ PStat ∂δPV · · · ∂δStat · · ·

.. . .. .

∂ Imn ∂δPV · · · ∂ Irn ∂δPV · · ·

.. . .. .

∂ Imn ∂δStat ∂ Irn ∂δStat

.. .

.

∂ PStat ∂ Vm1

.. .

.. .

..

.. .

.

.. .

∂ PPV ∂ PPV ∂δPV · · · ∂δStat

... ..

.. .

.

.

Im1 ⎢ I ⎢ r1 ⎢ ⎢ Im2 ⎢ ⎢ Ir2 ⎢ ⎢. ⎢. ⎢. ⎢ ⎢ PPV ⎢ ⎢. ⎢. ⎢. ⎢ ⎢ PStat ⎢ ⎢. ⎢ .. ⎢ ⎢. ⎢. ⎢. ⎢ ⎣ Imn Irn

B11 − a1 ⎤ ⎢ ⎢G −c 1 ⎢ 11 ⎥ ⎢ ⎥ ⎢B ⎥ ⎢ 21 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ G 21 ⎥ ⎢ ⎥ ⎢. ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂ PPV ⎥ = ⎢ ∂ Vr1 ⎥ ⎢. ⎥ ⎢. ⎥ ⎢. ⎥ ⎢ ⎥ ⎢ ∂ PStat ⎥ ⎢ ∂V r1 ⎥ ⎢ ⎥ ⎢. ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢. ⎥ ⎢. ⎦ ⎢. ⎢ ⎢ Bn1 ⎣ G n1

⎤

⎥⎡ ⎥ Vr1 ⎥ ⎥ ⎢ Vm1 ⎥⎢ B2n G 2n ⎥⎢ ⎥ ⎢ Vr2 ⎥⎢ ⎥ ⎢ Vm2 G 2n −B2n ⎥⎢ ⎥⎢. .. .. ⎥⎢. . . ⎥⎢. ⎥⎢ ⎥ ⎢ δPV ∂ PPV ∂ PPV ⎥⎢ ⎥⎢ . ∂ Vrn ∂ Vmn ⎥⎢ . .. .. ⎥⎢ . ⎥⎢ . . ⎥ ⎢ δStat ⎥⎢ ∂ PStat ∂ PStat ⎥⎢. ∂ Vrn ∂ Vmn ⎥⎢. ⎥⎢. .. .. ⎥⎢ . . ⎥ ⎢ .. ⎥⎢. .. .. ⎥⎢ . . ⎥ ⎣ Vrrn ⎥ Bnn − an G nn − bn ⎥ ⎦ Vmn

.. . .. . ···

··· .. . .. . ···

···

· · · G nn − cn −Bnn − dn

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(1)

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The terms a f , b f , c f and d f depend on the specified load and generation at bus f and also on the load model; these terms are presented in the Appendix. The complex current mismatch at PQ bus f is given by

Vk Zst

sp

Ist

I f =

sp

Pf − j Q f E ∗f

−

n 

Y f i Ei ,

(5)

i=1

where sp

(6)

sp

(7)

P f = PG( f ) − PL( f )

Vst

Q f = Q G( f ) − Q L( f )

Equation 5 can be expanded into its real and imaginary components as

Fig. 2 Equivalent circuit of STATCOM device

sp

During the iterative process, the parameters of STATCOM can be calculated using simple Eq. 2. Vst = Vk + Z st (Sst /Vk )∗ ,

Ir f =

Vr2f + Vm2 f −

(3)

i=1 sp sp P f Vm f − Q f Vr f Vr2f + Vm2 f n 

Im f =

Psh equal to zero (there is no active power generation from STATCOM side) and Qsh can be calculated each iterative process using Eq. 4. Q st = Q = k

n 

n 

(2)

where Sst = Pst + j Q st

sp

P f Vr f + Q f Vm f

|Vk ||Vi |(G k,i sin δk,i − Bk,i cos δk,i ) + Q load k

i=1

−

(8)

sp

(9)

(G f i Vmi − B f i Vri ) = Im f − Imcalf

i=1

The current mismatches in Eqs. 8 and 9 can be expressed only in terms of power mismatches and voltages at bus f as in Eqs. 10 and 11:

(4) After load flow convergence has happened, the final value of STATCOM parameters can be updated using the same equations. The developed STATCOM model enhances the reusability for load flow code without Jacobian modification. The number of iterations is similar to hybrid mismatches load flow without control.

sp

(G f i Vri − B f i Vmi ) = Ir f − Ircal f

Ir f = Im f =

P f Vr f + Q f Vm f V f2

(10)

P f Vm f − Q f Vr f , V f2

(11)

where V f2 = Vm2 f + Vr2f

(12)

The active power mismatches are used for PV buses is 4 Hybrid mismatches load flow method In the hybrid power-flow formulation, both current and power mismatches are used. Equation (1) shows the Jacobian matrix for the Hybrid power and current mismatches load flow method [15]. The mismatches for PQ buses are real and imaginary current mismatches similar to current injection mismatches load flow method [8]. The corrections of real and imaginary voltages are updated every iteration. The mismatches of STATCOM bus and any other PV bus are power mismatches and the corresponding correction is the angles of the PV buses. The angles are updated every iteration similar to the standard Newton–Raphson method.

123

sp

Pk = Pk −

n 

|Vk ||Vi |(G ki cos δki + Bki sin δki )

i=1

(13) Derivation of Jacobian matrix elements in case of PV buses 1) Diagonal element n  ∂ Pk Vi (G ki sin δki − Bki cos δki ) = −Vk ∂δk i =1 i = k

(14)

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2) Off diagonal elements

Step 2: Calculate the current mismatches for PQ buses and power mismatches for STATCOM and PV buses Step 3: The hybrid mismatches load flow method is solved without any changes in the Jacobian matrix. Step 4: Compute the STATCOM parameters during the iterative process using Eq. 1 and the final values after load flow convergence.

∂ Im f = Vk (G f k cos δk − B f k sin δk ) ∂δk ∂ Ir f = −Vk (G f k sin δk + B f k cos δk ) ∂δk ∂ Pk = Vk (G k f sin δk − Bk f cos δk ) ∂ Vm f ∂ Pk = Vk (G k f cos δk + Bk f sin δk ) ∂ Vr f

(15) (16) (17) (18)

Bus Voltage Updating Case (1) PQ Buses During the iterative load flow process and after obtaining bus voltage corrections using Eq. 1, the voltage values can be updated in rectangular coordinates as following: (h+1) Vr f (h+1) Vm f

=

Vrhf

+ Vr f

h

= Vmh f + Vmh f V f = Vr f + Vm f

 Vm f δ f = tan−1 Vr f

(19) (20) (21) (22)

Case (2) PV Buses The angle values can be calculated during the iterative load flow process using Eq. 23: (h+1)

δk

= δkh + δkh

(23)

The main advantage of the power mismatches representation for connected STATCOM and PV buses in NR current injection formulation is the reduction of power-flow equations to be solved in comparison with the other Newton– Raphson current injection load flow methods [8,11,15]. Table 1 shows a comparison for the required number of equations for the Newton–Raphson based on power mismatch method, Newton–Raphson based on current mismatch method, Newton–Raphson based on improved current mismatch method, and finally the hybrid mismatch method. By assuming that the number of connected STATCOM buses (CB) is equal to 3.

5 Load flow solution process with power mismatch STATCOM model The following steps are required for a simple implementation of STATCOM model into Newton–Raphson current injection load flow method: Step 1: Modify the bus data of the network by converting the connected STATCOM bus to PV bus with desired control voltage

Figure 3 shows the developed algorithm of Newton–Raphson hybrid mismatches load flow method with power mismatch STATCOM model without any change in Jacobian

6 Results and discussion The IEEE 14 bus, the IEEE 30 bus, the IEEE 57 bus, and the IEEE 118 bus are used for validating the developed power mismatch STATCOM model in Newton–Raphson based on current mismatch for PQ bus and power mismatch for PV bus. 6.1 IEEE 14 bus test system The objective of this test is to validate the result obtained from power mismatch STATCOM model in current injection method and comprehensive STATCOM model in standard NR load flow method [16], The IEEE 14-Bus system is shown in Fig. 4. The STATCOM is connected at bus 11 to maintain voltage magnitude to be 1.01 p.u. The convergence tolerance for power mismatch is set to 10−4 . The final values of STATCOM parameters are computed after the convergence and presented in Table 2. The obtained results are similar for both of two methods. The bus voltages and line flows of test system without and with STATCOM are shown in Tables 2 and 3. From Table 2, it is clear that the voltages at slack bus and generator buses are same without and with STATCOM and there is a significant change in load bus voltages. From Table 3, it can be seen that there is significant change line flows. Also, From Table 4, it is observed that the total system active power and reactive power losses decrease with STATCOM. 6.2 Performance characteristics The convergence characteristic for load flow method with the developed STATCOM model is tested using IEEE 14 bus. In this test the voltage at bus 11 is controlled to be 1.01 p.u. The active and reactive power mismatches against the number of iterations are drawn as shown in Fig. 5. From this figure, it can be observed that the developed model has better convergence characteristics than comprehensive STATCOM model in standard NR load flow.

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Author's personal copy 140 Table 1 Required equations for different NR load flow methods with STATCOM model

Fig. 3 Flow chart for NR based on current mismatches with power mismatches STATCOM model

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IEEE data N

NPV

CB

NR based on PM

NR based on CM

NR based on ICM

NR based HM

2N − NPV − CB−2

2N − 2

2N + NPV + CB−2

2N − NPV −CB-2

14

4

3

19

26

33

19

30

5

3

50

58

66

50

57

6

3

103

112

121

103

118

53

3

178

234

290

178

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141

Fig. 4 A standard IEEE 14 bus system with STATCOM device

G 13

1

14

VSC + VSh −

Vdc

12 11 10

9

6

G

c 7 5

8

4

2

3

c G Table 2 Bus voltage with and without STATCOM model

Bus number

Magnitude of voltage (p.u)

Angle of voltage (degree)

Without STATCOM

With STATCOM

Without STATCOM

1

1.0600

1.0600

0.0000

0.0000

2

1.0450

1.0450

−4.9511

−4.9699

3

1.0100

1.0100

−12.6141

−12.6510

4

1.0295

1.0270

−10.4188

−10.4123

With STATCOM

5

1.0349

1.0332

−8.9546

−8.9807

6

1.0700

1.0700

−14.6700

−14.9146

7

1.0559

1.0481

−13.5516

−13.5239

8

1.0900

1.0900

−13.5516

−13.5239

9

1.0497

1.0342

−15.1674

−15.1488

10

1.0458

1.0214

−15.3638

−15.1431

11

1.0543

1.0100

−15.1436

−14.3606

12

1.0547

1.0535

−15.5140

−15.7526

13

1.0495

1.0472

−15.5734

−15.7799

14

1.0315

1.0216

−16.3510

−16.4485

STATCOM parameters

–

0.9776

–

−14.3607

6.3 Robustness of the developed STATCOM model in many IEEE data systems The IEEE 30 bus, the IEEE 57 bus, and the IEEE 118 bus are tested with power mismatch STATCOM model at different case studies. Table 5 shows the ability of STATCOM model to solve large-scale system, when one and more STATCOM is implemented in the network to control the voltage at different places.

6.4 Execution time and numbers of iterations This test has been carried out using the personal computer (Pentium IV, 2 GHz processor and 2 GB RAM). In this test, the average execution time for HM load flow method is compared with the CM load flow method in case of including the developed power mismatch STATCOM model. The STATCOM is connected to bus 11 to maintain the voltage magnitude at 1.01 p.u for both of IEEE 14-bus system and

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Author's personal copy 142 Table 3 Line flow with and without STATCOM model

Table 4 Line losses with and without STATCOM model

Electr Eng (2014) 96:135–144

Bus number

Active power flow (MW)

Reactive power flow (MVAR)

Without STATCOM

Without STATCOM

With STATCOM

With STATCOM

1–2

155.95

156.52

−20.19

−20.32

1–5

76.42

76.71

−3.34

−2.57

2–3

72.53

72.69

3.63

3.61

2–4

56.01

56.07

−8.35

−6.93

2–5

41.47

41.78

−7.87

−6.94

3–4

−23.95

−23.80

−2.33

−0.91

4–5

−62.31

−61.30

7.40

5.23

4–7

28.41

27.94

−12.21

−9.59

4–9

16.09

15.77

−3.07

−0.68

5–6

43.76

45.35

−12.24

−12.75

6–11

7.11

8.28

5.07

28.35 3.13

6–12

7.80

7.95

2.70

6–13

17.65

17.92

7.99

9.71

7–8

0.00

0.00

−20.45

−24.94

7–9

28.41

27.94

6.35

13.62

9–10

5.48

5.07

2.73

13.78

9–14

9.52

9.15

2.65

0.60

10–11

−3.53

−4.00

−3.10

7.81

12–13

1.63

1.77

0.95

1.36

13–14

5.55

5.94

2.71

4.79

Total active power losses (MW)

Total reactive power losses (MVAR)

Without STATCOM

With STATCOM

Without STATCOM

With STATCOM

13.3752

13.3534

9.8764

5.2785

IEEE 118-bus system. The results are presented in Table 6. It can be observed that in the case of IEEE 14-bus system the saving time is very small and cannot be measured. But in the case of IEEE 118-bus system the time saving is more than 50 %. This result is similar to the result obtained from the load flow methods in uncontrolled case [2]. And also the reductions in number of iterations of theses load flow methods have been saved with and without the STATCOM model. The numbers of iterations are presented in Table 7.

7 Conclusions Fig. 5 Absolute power mismatches graphs for IEEE 14 bus system with STATCOM models

123

This paper has presented the implementation of power mismatch Static Synchronous Compensator (STATCOM) model

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Table 5 Different cases for many IEEE systems with power mismatch STATCOM model Case

STATCOM bus

IEEE 30

IEEE 57

IEEE118

Vst (p.u)

δst (degree)

Vst (p.u)

δst (degree)

Vst (p.u)

δst (degree)

A

7

1.0195

−12.9452

1.0812

−8.0970

1.1699

62.7356

B

7

1.0199

−12.9522

1.0789

−7.9439

1.1699

62.6784

14

1.0199

−16.0358

1.1028

−9.8241

1.0617

59.9252

7

1.0203

−12.9569

1.0789

−7.9497

1.1699

62.6249

14

1.0072

−16.1656

1.1028

−9.8296

1.0617

59.8790

16

1.0034

−15.7917

1.0041

−8.7404

1.0597

59.4116

C

D

E

7

1.0201

−12.9566

1.0781

−7.9360

1.1699

62.5752

14

1.0065

−16.1179

1.0991

−9.8371

1.0617

59.8258

16

1.0025

−15.7648

1.0041

−8.7405

1.0597

59.3608

20

1.0133

−17.0829

1.0219

−15.0650

1.0674

53.9331

7

1.0201

−12.9566

1.0710

−7.8844

1.1699

62.5698

14

1.0065

−16.1192

1.0691

−9.8240

1.0617

59.8204

16

1.0025

−15.7657

1.0041

−8.7125

1.0597

59.3554

20

1.0133

−17.0845

1.0157

−15.6908

1.0617

53.9262

22

1.0098

−16.5331

1.0737

−15.0503

1.0213

52.3937

Table 6 Execution time in seconds for NR methods with STATCOM

Hybrid mismatch method Improved current mismatch method Reduction ratio

IEEE 14

IEEE 118

>0.001 >0.001 –

0.047 0.094 53.19

Acknowledgments The authors gratefully acknowledge the contribution of the Spanish Ministry of Foreign Affairs and Cooperation (MAEC) and the Spanish Agency for International Cooperation (AECI), for providing research funding to the work reported in this research.

8 Appendix

0.0001 p.u. mismatch Table 7 Comparative convergence characteristics for NR methods with STATCOM Test system

Mismatch tolerance 0.01

0.001 CM

CM



Q f (Vr f 2 − Vm f 2) − 2P f Vr f Vm f

af =

0.0001 HM

The parameters (a f , b f , c f and d f ) in Eq. (1) can be given as following:

CM

HM

HM

IEEE 14-Bus

3

2

4

3

4

3

IEEE 118-Bus

9

4

10

6

10

6

+ bf =

into Newton–Raphson based on current injection mismatches formulation, which is capable of solving large power networks reliably. In this model, the power mismatch approach has been used to represent the connected STATCOM and PV buses; hence, the required number of equations is reduced with respect to the classical and improved versions of the current injection methods, the complexities of the computer program codes has reduced, and the reusability is enhanced by avoiding modifications in the Jacobian matrix. The model is considered more suitable for shunt FACTS devices. Many test case studies have been carried using IEEE system data to validate the power mismatch STATCOM model in load flow algorithm.

V f4

cf =

df =

Vr f b p P0 f Vm f + b p Q 0 f Vm2 f V f3

+ cq Q 0 f

(24)

(P f (Vr f 2 − Vm f 2) + 2Vr f Vm f Q f ) V f4

(b p Q 0 f Vm f Vr f + b p P0 f Vr f 2) − − c p P0 f V f3  P f ((Vm f 2 − Vr f 2) − 2Q f Vr f Vm f ) V f4 b p Q 0 f Vm f Vr f − b p P0 f Vm f 2) + − c p P0 f V f3  Q f (Vr f 2 − Vm f 2) − 2P f Vr f Vm f

(25)

(26)

V f4

(Vm f b p P0 f Vr f − b p Q 0 f Vr f 2) + − cq Q 0 f , (27) V f3

123

Author's personal copy 144

Electr Eng (2014) 96:135–144

where P f = PG( f ) − PL( f ) = PG( f ) − P0 f a p

(28)

Q f

(29)

= Q G( f ) − Q L( f ) = Q G( f ) − Q 0 f aq

When the bus has only a constant power load, the parameters are simplified to af = df =

Q f (Vr2f − Vm2 f ) − 2P f Vr f Vm f

b f = −c f =

V f4 P f (Vr2f − Vm2 f ) + 2Vr f Vm f Q f V f4

(30)

(31)

These parameters are updated during the iterative load flow process.

References 1. Gotham DJ, Heydt GT (1998) Power flow control and power flow studies for systems with FACTS devices. IEEE Trans Power Syst 13:60–66 2. Kamel S, Abdel-Akher M, Jurado F, Ruiz-Rodríguez FJ (2013) Modeling and analysis of voltage and power control devices in current injections load flow method. Electr Power Compon Syst 41:324–344 3. Radman G, Raje SR (2007) Power flow model/calculation for power systems with multiple FACTS controllers. Electr Power Syst Res 77:1521–1531 4. Canizares CA, Pozzi M, Corsi S, Uzunovic E (2003) STATCOM modeling for Voltage and angle stability studies. Int J Electr Power Energy Syst 6:421–422 5. Yankui Z, Yan Z, Bei W, Jian Z (2006) Power injection model of STATCOM with control and operating limit for power flow and voltage stability analysis. Electr Power Syst Res 76:1003–1010

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6. Kamel S, Abdel-Akher M, Song MK (2012) Analysis and modeling of a unified power flow controller in Newton–Raphson load flow. Adv Sci Lett 8:884–889 7. Bhowmick S, Das B, Kumar N (2011) An advanced static synchronous compensator model to reuse Newton and decoupled power flow codes. Electr Power Compon Syst 39:1647–1666 8. da Cost VM, Martins N, Pereira JLR (1999) Developments in the Newton–Raphson power flow formulation based on current injections. IEEE Trans Power Syst 14:1320–1326 9. da Cost VM, Martins N, Pereira JLR (2001) An augmented Newton–Raphson power flow formulation based on current injections. Electr Power Energy Syst 23:305–312 10. Garcia PAN, Pereira JLR, Carneiro S, da Cost VM, Martins N (2000) Three-phase power flow calculations using the current injection method. IEEE Trans Power Syst 15:508–514 11. Garcia PAN, Pereira JLR, Carneiro S, Vinagre MP, Gomes FV (2004) Improvements in the representation of PV buses on threephase distribution power flow. IEEE Trans Power Deliv 19:894– 896 12. Gyugyi L (1994) Dynamic compensation of AC transmission lines by solid-state synchronous voltage source. IEEE Trans Power Deliv 9:904–911 13. Prenesh R, Crow ML, Yang Z (2000) STATCOM control for power system voltage control applications. IEEE Trans Power Deliv 15:1311–1317 14. Zhiping Y, Crow ML, Chen S, Lingli Z (2000) The steady-state characteristics of a STATCOM with energy storage. Power Eng Soc Summer Meet 2:669–674 15. Kamel S, Abdel-Akher M, Jurado F (2013) Improved NR current injection load flow using power mismatch representation of PV Bus. Int J Electr Power Energy Syst. doi:10.1016/j.ijepes.2013.03. 039 16. Acha E, Fuerte-Esquivel CR, Ambriz-Perez H (2004) FACTS modeling and simulation in power networks. Wiley, England