Keywords

1 Introduction

The introduction of the FACTS devices into the power system offered great opportunities for the power engineer in the area of operation and control of modern power systems. For example, FACTS devices are often planned for power flow regulation in the steady state thus enhancing the power transfer capability of existing transmission lines. Various types of FACTS devices and their location at different places have varying advantages [1, 2]. The FACTS devices like Static VAR Compensator (SVC), Thyristor Controlled Series Capacitor (TCSC), Static Synchronous Series Compensator (SSSC) and Static Synchronous Compensator (STATCOM) are the first generation FACTS devices available in the literature for control of power flow in transmission systems [3].

The introductory FACTS devices were able to regulate either the flow of active or reactive power along a single transmission line. A breakthrough was made with the introduction of the UPFC [4], which is one of the most versatile FACTS devices and also capable of simultaneously controlling the flow of both active and reactive power in the transmission line. Another newly developed FACTS device, namely IPFC, further extends the capability of independently influencing the active and reactive power flows to simultaneous compensation of multiple transmission lines. These significant functions are made possible by the combination of multiple compensators coupled via a common dc link. Thus, both the UPFC and IPFC are defined as the combined compensators [5].

The IPFC is an advanced FACTS device aimed at controlling the power flow in multiline systems in a substation [6]. IPFC employs Voltage Source Inverter (VSI) as basic building block. Generally, it composes of two VSIs which are capable of transferring real power from one line to any other line and thereby facilitating transfer of real power among the lines, and also achieving independent control of series compensation of each individual line.

IPFC is presented as a power injection model and is implemented to study the effect of IPFC parameters on bus voltages, active and reactive power flows in the lines [7]. The applications of IPFC to improve damping of the system have been reported by few researches and they have applied IPFC to improve transient stability of power system [8]. It can also be utilized to compensate against reactive voltage drops and the corresponding reactive line power and thereby increase the effectiveness of the compensating system against dynamic disturbances [9]. The minimization of generation cost, transmission losses and maximization of the loadability of the transmission system can be achieved by optimally placing IPFC. Different operating conditions of the power system must be considered while determining the optimal size and location of the power flow controller.

Contingency analysis deals with the study of the impact and performance of the system during the outage of the power system components such as transmission lines, transformers and generators. Among these contingencies referring to major disturbances like loss of a transmission line or a generator may create sudden and large changes in both the configuration and the operating state of the power system. Contingencies sometimes may also result in severe violations of the operating constraints. Consequently, to have a secure operating evaluation and planning for contingencies forms an important aspect. [10, 11].

This paper proposes an algorithm for optimal location of the IPFC to improve voltage stability under the over loaded line outage contingency in a power system network. This paper also analyses the performance of the IPFC for various combinations of voltage magnitudes and angles at best IPFC location.

2 Interline Power Flow Controller (IPFC)

In general the IPFC utilizes a number of DC to AC converters each providing a dedicated series compensation for a given line as shown in Fig. 1 and equivalent circuit shown in Fig. 2. The series compensation is achieved by employing two or more independently controllable static synchronous series compensators (SSSC) which are solid state voltage source converters (VSC). By maintaining DC link voltage at the desired level the combination of the series connected VSC can inject a voltage at fundamental frequency with controllable magnitude and phase angle. In practice the DC link is represented as a bidirectional link for exchange of active power among the converters. SSSC is employed for increasing real power transfer on a given line by directly compensating for the voltage drop due to inductive loading of a transmission network. In addition, active power can also be exchanged through these series converters via the common DC link in IPFC. It is noted that the sum of the active powers resulting from VSCs to transmission lines should be zero when the losses in the converter circuits are ignored.

Fig. 1
figure 1

Simple model of IPFC

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Fig. 2
figure 2

Equivalent circuit of IPFC

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3 Modeling of IPFC

The modeling for IPFC which will be referred to as power injection model is presented here. This model is helpful in understanding the impact of the IPFC on the power system in the steady state. Furthermore, the IPFC model can easily be incorporated in the power flow model. For steady state analysis of power systems the normal practice is to represent VSC as a synchronous voltage source injecting an almost sinusoidal voltage with controllable magnitude and angle. On this basis, the equivalent circuit of IPFC has been modified and is represented as shown in Fig. 3.

Fig. 3
figure 3

Equivalent circuit of IPFC

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In Fig. 3, Vi, Vj and Vk are the bus voltages at the buses i, j and k respectively, Vx = Vx∠θ (x = i, j and k). In Vse it is the controllable voltage source injected by connecting in series, Vse = Vse ∠θse (n = j, k) and in Zse (n = j, k) is the transformer impedance. The complex power injected into any bus can be determined by modeling IPFC as a current source. The line and the series coupling transformer’s resistances are neglected for making the calculations simpler. The injected power at buses are summarized and The Power flow equations for IPFC can written as below,

$$\begin{aligned} P_{i} = & V_{i}^{2} g_{{ii}} - \sum\limits_{{j = 1,j \ne i}}^{n} {V_{i} } V_{j} \left( {g_{{ij}} \cos \left( {\theta _{j} - \theta _{i} } \right) + b_{{ij}} \sin \left( {\theta _{j} - \theta _{i} } \right)} \right) \\ & - \sum\limits_{{j = 1,j \ne i}}^{n} {V_{i} } Vse_{{ij}} \left( {(g_{{ij}} \cos \left( {\theta _{i} - \theta se_{{ij}} } \right) + b_{{ij}} \sin \left( {\theta _{i} - \theta se_{{ij}} } \right)} \right) \\ \end{aligned}$$
(1)
$$\begin{aligned} Q_{i} = & V_{i}^{2} b_{{ii}} - \sum\limits_{{j = 1,j \ne i}}^{n} {V_{i} } V_{j} (g_{{ij}} \sin \left( {\theta _{j} - \theta _{i} } \right) + b_{{ij}} \sin \left( {\theta _{j} - \theta _{i} } \right)) \\ & - \sum\limits_{{j = 1,j \ne i}}^{n} {V_{i} } Vse_{{ij}} \left( {g_{{ij}} \sin \left( {\theta _{i} - \theta se_{{ij}} } \right) + b_{{ij}} \sin \left( {\theta _{i} - \theta se_{{ij}} } \right)} \right) \\ \end{aligned}$$
(2)

where: V = Bus voltage magnitude, θ = Voltage angle, Vse = magnitude of injected voltage, θse = Angle of injected voltage.

4 Voltage Stability Index Formulation

In this study the Voltage Stability Index [12] abbreviated by “Lij” and referred to a line is formulated as the measuring unit in predicting the voltage stability condition in the system. The mathematical formulation presented here is very simple and also achieves faster computation. By using the second order linear voltage equation at the receiving bus on a two bus system the Lij is obtained (Figs. 4, 5 and 6).

Fig. 4
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Plot between bus number and voltage mangitude witout IPFC and with IPFC during various conditions

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Fig. 5
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Plot between line number and apparent power witout IPFC and with IPFC during various conditions

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Fig. 6
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Plot of power losses witout IPFC and with IPFC during various conditions

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From Fig. 3, the voltage quadratic equation at the receiving bus is written as

$$\left\lfloor {V_{j}^{2} - \left( {\frac{R}{X}\sin \delta + \cos \delta } \right)V_{i} V_{j} + \left( {X + \frac{{R^{2} }}{X}} \right)Q_{j} = 0} \right\rfloor$$
(3)

Setting the discriminate of the equation to be greater than or equal to zero:

$$\left\lfloor {\left( {\frac{R}{X}\sin \delta + \cos \delta } \right)V_{i}^{2} } \right\rfloor - 4\left( {X + \frac{{R^{2} }}{X}} \right)Q_{j} \ge 0$$
(4)

Rearranging above equation, Voltage Stability Index “Lij” is

$$L_{ij} = \frac{{4Z^{2} Q_{j} X}}{{V_{i}^{2} \left( {Rsin\delta + Xcos\delta } \right)^{2} }}$$
(5)

where: Z and X are line impedance and reactance respectively,

Qj and is the reactive power at the receiving end, Vi and Vj are sending end and receiving end voltages.

5 Contingency Analysis

Contingency analysis aims at studying the effect of the outage of components of power system like transmission lines, transformers and generators on the power system network. Contingencies referring to disturbances such as transmission line outages or generator outages may cause large amount of load may stay connected or removed and thus resulting a change in either the state or configuration of the power system. Contingencies may result in severe changes of the operating parameters. Consequently, planning for contingencies forms an important aspect of secure operation of the power system network. Contingency analysis helps the power system engineer at many stages like network design, programmed maintenance, network expansion and also in the identification of network weaknesses and thus serve as an important tool for estimating security of the power system during operation and planning. Contingency analysis allows the power system to be operated defensively. Majority of the faults occurring in the power system network can cause serious troubles within a small time if the operator could not take fast remedial action. Keeping in view of these, modern computers are equipped with contingency analysis programs which model the power system network and are used to know outage events and give alert to the operators of potential overloads and voltage violations.

The most difficult practical problem to manage within contingency analysis is the correctness of the method and the speed of solution of the model used. The operator should have an idea of the performance of the existing network which is instable condition and also he should possess the knowledge of the effect of a particularly contingency like outage of a particular generator or transmission line.

Recently, due to the problems such as the congestion management, the minimization of the operational cost and the overall generating cost, the additional degree of freedom possessed by the FACTS devices have aroused great interest in the application of the FACTS devices, especially the UPFC, the IPFC and the generalized Unified Power Flow Controller (GUPFC), in the OPF control. However, very few publications have focused on the comparison between the performance of the UPFC and the IPFC in the OPF control. This paper proposes an algorithm for optimal location of the IPFC to improve voltage stability under the over loaded line outage contingency in a power system network.

6 Performance Index

The contingency analysis process gives an idea about the effect of individual contingency cases, hence the above process take large time to evaluate the contingency in the power system network. The contingency analysis is selected by calculating a kind of severity indices known as Performance Indices PI [13]. These indices values are calculated using the conventional power flow algorithms for individual contingencies. Based on the line flow limit in overloaded lines, contingencies are ranked in a manner where the highest value of PI is ranked first. This will continues till the no severe contingencies are found.

There are two kinds of performance indices used in power system networks, one is active power performance index (PIP) and other one is reactive power performance index (PIV). The active power performance index (PIP) reflects the violation of line active power flows and is given as

$$PI_{P} = \mathop \sum \limits_{i = 1}^{L} (\frac{{P_{i} }}{{P_{{i^{max} }} }})^{2n}$$
(6)

where: Pi = active power flow in line I, P maxi  = maximum active power flow in line i

n = specified exponent, L = number of transmission lines in the power system

The maximum power flow in each line will be calculated as

$$P_{i}^{max} = \frac{{V_{i} *V_{j} }}{X}$$
(7)

And other performance index parameter which is used in reactive performance index corresponding to the bus voltage magnitude violations. The value can be evaluated as below

$$PI_{V} = \mathop \sum \limits_{i = 1}^{{N_{pq} }} \left[ {\frac{{2(V_{i} - V_{inon} )}}{{V_{imax} - V_{imin} }}} \right]^{2}$$
(8)

where Vi = voltage at bus I, Vimax and Vimin max. and min. values voltage limits,

Vinon = average value of Vimax and Vimin, Npq = total number of voltage buses.

7 Results and Conclusions

The proposed method is implemented in MATLAB working platform. The performance of proposed method is tested with IEEE 30 bus system. Initially severity indices known as Performance Indices are calculated and are ranked in a manner where the highest value of PI is ranked first. Based on the line flows (MVA) outage lines (lines which are overloaded) and contingency rank have been determined and are indicated in Tables 1 and 2 respectively. Also line flows under rank 1 contingency criterion are provided in Table 3.

Table 1 Over loaded lines of IEEE-30 bus system during contingency analysis
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Table 2 Contingency ranking
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Table 3 Line flows under rank-1 contingency
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Base on the results of line flows it can be concluded that the best location of IPFC will be such that interline power flow takes place between lines 2−6 and 6−7. In short it is read as 2−6−7 (Tables 4, 5 and 6).

Table 4 Voltage magnitudes
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Table 5 Line apparent power flows
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Table 6 Total power losses
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IPFC LOCATION: 6−2−7

Voltage values

Case-1:

Vseij = 0.02; Thseij = 072; Vseik = 0.10; Thseik = 360

Case-2:

Vseij = 0.04; Thseij = 144; Vseik = 0.08; Thseik = 288

Case-3:

Vseij = 0.06; Thseij = 216; Vseik = 0.06; Thseik = 216

Case-4:

Vseij = 0.08; Thseij = 288; Vseik = 0.04; Thseik = 144

Case-5:

Vseij = 0.10; Thseij = 360; Vseik = 0.02; Thseik = 072.