# Voltage stability

# Voltage stability

### 1 Introduction

Voltage stability has become one of the most important research areas in electrical power system. It has received special attention by every power system analyst and researcher. With the growing size of the power system network, along with economic and environmental constraints, voltage instability currently provides a serious threat. Therefore, lot of works have been conducted by all researchers, R&D organizations and power utilities throughout the world to better understand the problems, detect it faster and evolve a newer and more efficient strategies to cope up with the menace of voltage instability or voltage collapse.

To this purpose, an accurate knowledge of how close the actual system's operating point is from the voltage stability limit is very crucial to operators. There are several works proposed to predict the proximity to voltage collapse based on conventional approach, for example, P-V / Q-V curves , sensitivity based indices, and continuation method . Gao et al.proposed a method using modal analysis that computes the smallest eigenvalue and associated eigenvectors of the reduced Jacobian matrix of the power system. Other methods, such as bifurcation theory , energy function , and singular value decomposition have also been reported in the literature. These methods provide complete and accurate results. However, longer computation time is needed because of the requirement for repetitive power flow calculations.

Therefore, due to the ability to provide a fast, reliable, and easily interpretable indication for proximity to the voltage collapse, voltage stability indices have been extensively used in power system planning. These indices are derived by either referring to a bus or a line.

Majority of the discussed voltage stability indicators in the literature are line stability indices [line comparison]. Line stability index, LQP [x], Lmn and Fast Voltage Stability Index (FVSI) are obtained by the same concept which based on the power transmission concept in a single line. Voltage Collapse Proximity Index (VCPI) , proposed by M.Moghavvemi investigates the stability of each line in the system and they are based on the concept of maximum power transferred through a line. An on-line voltage stability index referred as LVSI is developed from the relationship between line active power and bus voltage with the line [x]. In , the stability index is deduced based on the equilibrium solution region of branch power flow.

This paper addresses two new techniques for voltage stability analysis:

- A new index for predicting voltage collapse, RVSI
- A new weak area identification method, WACM

Section 2 briefly presents the mathematical formulation of some bus voltage stability indices. The indices are grouped according to their types. Section 3 describes the derivation of RVSI while section 4 presents the new technique for weak area identification. Section 5 is the simulation test results and discussion. A comparison between the proposed methods and other methods developed are made and discussed. Finally, a conclusion is drawn in Section 6.

### 2 Review of bus voltage stability indices

The use of voltage stability indices in voltage stability analysis is becoming popular these days due to their simpleness and practicability. In general, the indices can be derived by referring to a bus or line of a power system network. Bus voltage stability index is the index that is derived by referring to a bus system and also uses the magnitude value at the busbar to calculate the index. Most bus voltage stability indices use Thevenin's equivalent circuit in deriving the index. These indices can reveal the critical or weakest bus, or the stability of each line connected between two busses in an interconnected network as well as evaluate the voltage stability margin of a power system. According to [versatile], bus stability indices are expected to be used in large industrials centers. In this section, several bus stability indices are discussed briefly and grouped into the concept of the indices.

### 2.1 Power Flow in a Single Line based Indices

### A. Z-Index

Sterling et al. [x] established the criterion of the stability by using the Maximum Power Transfer theorem which states that maximum power is transferred to a load if the load impedance, ZL is equal to the conjugate of the source impedance, ZS, i.e, ZL=Z S*. By using the Thevenin equivalent circuit, the index that referred to a load bus is derived by applying the condition at which maximum power is transferred to a load, which is when (where PL is the active power at the load) and this is correspond to,

### C. Power Transfer Stability Index (PTSI)

PTSI, derived from [x] is similar like VCPI[x]. The different is that this index is referred to a bus for dynamic analysis and use apparent power, S instead of active and reactive power, P and Q, while VCPI is referred to a line and is used for static analysis. It is derived by using the Maximum Power Transfer theorem for a simple 2-bus system.

The complex current drawn by the load is given by [x],

The complex load power, SL can be written as,

By considering that and, and substituting equation (9) to (10), the simplified equation will be,

where α and β are the phase angle of the load and Thevenin impedance.

SL is said to be maximum when ZL=ZThev. Thus, by substituting ZL=ZThev in equation (11),

SL=SLmax shows that no more power can be transferred at this point and proximity to voltage collapse is likely to take place. Hence, to avoid voltage collapse, the power margin has to be greater than zero which also means that, the ratio of SL to SLmax has to be less than 1. Therefore, the proposed index is defined as follow:

### D. Voltage Collapse Indicator Index (VCI)

VCI[x] is derived from a simple observation from equation below:

This equation indicates that when the load apparent power, Si is changed, both the load voltage, Vi and current, Iichange as well.

By using the Taylor's theorem and neglecting the higher order terms, the incremental change of the relationship above can be written as:

When the load of a bus increases, the load current increases and the load voltage decreases. However, the increase of load at a bus may not increase the load of apparent power Si when the load of the bus approaches the critical value or the voltage collapse point. This is due to a rapid decrease of voltage compared to the increase in current. Therefore, at this point ∆Si of (15) will approach 0. Hence, by dividing equation (15) with Vi∆Ii, the stability limit is given by:

where α >1 is used to linearize the trend of the index itself especially. The value of the index varies between 0-1 where at no load, Ii =0, and thus the index becomes unity and zero when the system collapses.

### 2.2 Local Measurements based Indices

### A. Wide Area Voltage Stability Index

With the recent development of synchronized measurement technology, some researchers have proposed phasor measurement algorithms using phasor measurement units (PMU) to determine the closeness to voltage collapse. The local measurement value of a two bus equivalents can be estimated by using synchronized voltage phasors and current phasors. The stability of the load bus at the current operating point is based on the assessment of the voltage stability margin from the two-bus equivalents [static,16]. Several indices use this concept, for instance, WAVI (Wide Area Voltage Index) [WAVI] and WASI (Wide Area Stability Index [WASI].

Both, WAVI and WASI are developed from the established Z-Index by Sterling [x] as discussed in A. WAVI is exactly as Z-Index given in equation (2). The differences from other methods lie on the methods of the determination of local measurement value. It is suitable for monitoring voltage stability of KEPCO system. It uses only voltage and current phasor data obtained from PMUs without coupling with EMS data.

### 2.3 Long Transmission Lines based Indices

### A. Ls-Index

Dike et al. derived an index called, Ls-Index from the pi-equivalent structure of medium and long transmission lines as shown in figure 2.

3 Proposed Voltage Stability Index, RVSI

RVSI is derived from the current equation in a complex form of a simple 2-bus Thevenin equivalent system as shown in figure 3 above. Looking at the load bus concerned, the equation of current, I in complex form can be written as,

Equation (26) above will only have real root if the discriminant is greater or equal to 0. Hence,

which can be simplified to,

Therefore, RVSI is given by

In order to maintain the stability of the system, the RVSI value must be less than 1 as in equation (27). Any bus that has RVSI value near to 1 implies that the bus is approaching its instability point.

### 3.1 Algorithm for Obtaining RVSI

To calculate RVSI value, the following steps are employed:

- Create a contingency condition such as step increase either in active power, reactive power or both or line outage.
- Run a load flow to obtain the system's power flow and voltage profile.
- Obtain and by running the load flow with the load connected at the bus concerned removed.
- By using the data obtained from step 2 and 3, calculate the Thevenin equivalent impedance and its phase angle, as follows:
- Calculate RVSI value.
- Repeat steps 1-5 by considering another contingency.

### 3.2 Critical Bus Identification

Weak buses identification in a power system is very important since it recognizes which load buses are most likely to collapse first under a certain loading condition. Therefore, several actions can be taken in advance and precautions can be made in order to avoid voltage collapse.

- Obtain Thevenin equivalent circuit for each load bus.
- Evaluate RVSI value for each load bus.
- Rank RVSI value in descending order.
- Buses with the highest RVSI value are identified as the weak buses.
- Identify the most critical buses for the whole system by integrating weak buses under each contingency.

### 3.3 Line Outage Contingency Ranking

### 4 Weak Area Identification Techniques

Voltage collapse is the result of a series of voltage instability which can eventually lead to a power blackout. It has been agreed that the voltage instability problems start locally at the weakest bus and then spread to neighbouring buses, resulting in a cascading event [WA1]. Therefore, identification of weak area in a power system for hierarchical control is very useful since it can reduce the computational burden of voltage collapse analysis.

T.K.A Rahman et al. [x] and Zambroni et al. [103] both proposed a network portioning by using inverse Jacobian matrix of the load flow. Zambroni used the determinant of the inverse Jacobian, to identify the weak area while T.K.A Rahman employed the voltage stability variation at the critical load bus with respect to load variations at other load buses, , where i is the critical bus and j is the other load buses. In both methods, weak area in a power system is identified as the subnetwork which includes the critical load bus. There are only two areas, stable and unstable areas.

Several methods have been reported in the literatures that form few weak sub-networks in power systems, for instance, Voltage Control Area (VCA) technique. VCA proposed in [WA1] uses the coherency of the voltage dependence among the buses of the electrical grid as the indicator. The algorithm is disturbance-independent and requires only a limited number of basic arithmetic operations on the reduced Jacobian matrix of the load flow.

On the other hand, in [x], bus clusters are also formed by using line stability index, FVSI. A line outage contingency ranking is performed in order to identify the cluster, which includes the most critical line outages in a system. FVSI values for each line outage are then calculated. The highest FVSI values from each line outage are ranked in descending order. The ranking of these critical line outages are then grouped on the system in order to form the weak area clustering. This technique also has two weak area clusters, stable and non-stable areas. In this paper, a new method to cluster weak area called WACM is proposed to reduce the computational burden associated with the analysis of voltage instability in large power system networks. The details of the proposed method are described in the next section.

### 4.1 Weak Area Clustering Margin (WACM)

WACM is formulated based on the concept of voltage stability index margin at the critical load bus between a specified critical point and identified collapse point. This method studies the relation between the load increase at other load buses and the voltage stability at the critical bus, analysed by using RVSI. WACM is given as below:

where, RVSI value in the equation (29) above is the RVSI value at the critical bus, i, and the base and final value are the specified value that is considered to be critical and the identified collapse point (when RVSI very close to 1).

It is believed that the increased load buses that highly affect the voltage stability at the critical buses are electrically close and strongly connected. Thus, they are belonged to the same cluster. Besides that, it has also been confirmed that buses in a cluster behave in a coherent manner [WA1,prof, dr ismail]. The change in voltage and angle at these buses are about the same for any control action or contingency [prof]. Therefore, the change in voltage and angle are used as a coherency indicator or the cut off value to determine the weak area cluster boundary. All generator buses connected to these buses are considered to belong to the clustered weak area.

The algorithms of WACM are outlined as follows:

- Create a contingency condition such as step increase either in active power, reactive power or both or line outage.
- Run a load flow programming to compute the voltage profile and power flow in the system
- Obtain and for each load bus by running the load flow with the load connected at the bus concerned removed.
- Calculate RVSI value for each load bus to determine the critical bus in the network for the given operating point. This value is referred to as base value,
- Increase load at other each load buses until the system reaches its collapse point. The final value is the value just before the system collapses,
- Evaluate WACM value by using equation (29) and normalise the value by using equation below:
- where WACMn is the value at any load buses and WACMmax is the maximum value of WACM.
- Rank the normalised WACM value in descending order.
- The clusters are grouped based on the coherency in the change of voltage and phase angle of the load buses between the base case and the final case.

[OPF] The weakest bus/area in the system is identified by an indicator achieved with these marginal costs via the Kuhn-Tucker theorem.

[103] - weak area is defined as the area that contains the critical bus of the power network.

[103] - A network portioning is important because voltage collapse analysis is faster after portioning.

If the system is divided into several areas for hierarchical control….

[WA1]One of the basic features of the voltage collapse is that in its primary stages the problem arises in a reduced area of the power system, and only affects the buses closest to the bus under voltage strength. This generally suggests identifying reduced areas of the power system with coherent bus voltage behaviour, network clustering and analysis of the voltage instability of the reduced area.

[WA1] .. the voltage dependence among buses of the electrical grid can be used as a coherency indicator to obtain reduced areas of the network (approaches based on physical considerations)

All generator buses connected to these buses are considered belonged to the clustered weak area.

A group of load buses that have similar voltages and angles are considered as electrically close and belonged to the same cluster. On the other hand, for load buses that have voltages and angles not affected by the load variations are considered as electrically distanced or weakly connected and belonged to a different cluster.