INTRODUCTION
The problem of voltage instability is gaining more and more importance in recent
years. Because of the growth of power systems and inefficient reactive power
management. The voltage control service connected to reactive power supply is
one of the fundamental factors to guarantee stability and security of power
systems. The voltage instability is strongly associated with the lack of reactive
power support in the system caused by limitations in the generation or transmission
of the reactive power. Because of the continuous increase of power demands,
modern power systems are operated in proximity of their maximum operation limits.
Under such conditions, voltage instability of the systems likely occurs. Since,
optimization technique was applied to power system analysis, Optimal Power Flows
(OPF) have been widely used in planning and realtime operation of power systems
for active and reactive power dispatch to minimize generation costs and system
losses and to improve system VSM. Optimal reactive power dispatch (ORD) is a
subproblem of the OPF calculation. ORD determines all kinds of controllable
variables, such as reactivepower outputs of generators and static VAR (Volt
Ampere Reactive) compensators, tap ratios of transformers, outputs of shunt
capacitors/reactors and etc., achieving an adequate VSM and minimizing transmission
losses or other appropriate multi objective functions are general ORD objectives.
ORD optimization should satisfy a given set of physical and operating constraints
(Zhao et al., 2005; Esmin
et al., 2005; Yan et al., 2006; Venkatesh
et al., 2000; Hooshmand, 2008).
Voltage stability problem has received much attention in recent years and has
been a target concern during planning and operation of heavily loaded and complex
power systems. One of the most efficient and wellknown tools for assessment
of voltage stability is modal analysis technique, which introduced by Gao
et al. (1992), Kundur (1994) and Da
Silva et al. (2004). Dong et al. (2005)
proposed a Reactive Reserve Management Program (RRMP) based on an optimal power
flow to manage critical reactive power reserves. Various generators were assigned
different weights in order to maintain maximum reactive reserves within the
areas that are most vulnerable to voltage instability problem. Affonso
et al. (2004) presented a joint active/reactive power scheduling
methodology that increases voltage stability margin using active participation
factors derived from the modal analysis technique. Menezes
et al. (2003, 2004) presented a onedayahead
predispatch scheduling methodology considering improvement of the voltage stability
margin by scheduling of dynamic VAR sources. In their work modal analysis was
done and active participation factors were used to define penalty factors for
dynamic VAR sources, which are then incorporated to OPF formulation. Lin
et al. (2003) presented an ORD methodology with consideration of
the VSM as a constraint. Reactive power generation cost was minimized such that
required VSM was satisfied through a unified OPF. Hosseinpour
et al. (2008) proposed a combined operation of the Unified Power
Quality Conditioner (UPQC) with wind power generation system considering investment
cost.
This study presents a methodology for the inclusion of evaluation and improvement of voltage stability margin by optimizing the reactive power injections of generators and synchronous condensers. The objective is to maximize voltage stability margin maintaining the economical dispatch of active power by using Generator Participation Factors (GPFs) derived from the critical eigenvalue of the reduced Jacobian matrix. GPFs define penalty indices for all generators reactive injections, which are then added to the Optimal Power Flow (OPF) objective function as weighting factors, to obtain the most adequate reactive power injection for each generator or synchronous condenser. Generators reactive power injections are in proportional to their participation factors. These weighting factors cause that generators with high participations in critical mode of voltage (corresponding to critical eigenvalue of reduced Jacobian matrix) generate more reactive power support. As a result, the VSM will be improved with no negative impact on the active economical dispatch. The main contribution of this study is to use GPFs in the ORD problem. The problem formulated as an optimization problem. This optimization problem is a Non Linear Programming (NLP), which is solved using optimization toolbox of MATLAB software.
The results obtained for the IEEE 30bus test system are presented to show that the proposed methodology leads to a significant improvement in the VSM of power systems.
Voltage stability is the ability of a power system to maintain steadily acceptable bus voltage at all buses under normal operating condition, after load increase or when the system is being subjected to disturbance. Voltage stability margin represents the distance, in MW or percentage, from the base case operation point to the maximum power transfer capability point of the system (PV curve nose point). For each load increase a load flow problem is solved and the set of obtained equilibrium points defines the PV curve. In this work, PV curves are obtained by considering load increases for all load buses in a proportional way to the base case loading (keeping constant power factor). System generation level is also increased in order to match the load increases during the PV curve construction process.
MATERIALS AND METHODS
Modal analysis for voltage stability assessment: Modal analysis technique
identifies critical areas of voltage stability and provides information about
the best actions to be taken for the improvement of system stability margins.
GPFs indicate which of the dynamic VAR sources should inject more reactive power
to improve the VSM and which of them should inject less. The linearized powerflow
equations of a general power system are given by:
Where:
ΔP 
= 
Vector of bus active power injection variations 
ΔQ 
= 
Vector of bus reactive power injection variations 
Δθ 
= 
Vector of bus angle variations 
ΔV 
= 
vector of bus voltage magnitude variations 
System voltage stability is affected by both P and Q. However, at each operating
point we keep P constant and evaluate voltage stability by considering the relationship
between Q and V (Gao et al., 1992; Kundur,
1994). By assuming:
Where:
where, J_{RQV} is called the reduced Jacobian matrix of the system,
which directly relates the bus voltage magnitude and bus reactive power injection.
Voltage stability characteristics of the system can be identified by computing
the eigenvalues and eigenvectors of J_{RQV}. Let:
Where:
ξ 
= 
Right eigenvector matrix of J_{RQV} 
η 
= 
Left eigenvector matrix of J_{RQV} 
Λ 
= 
Diagonal eigenvalue matrix of J_{RQV} 
substituting Eq. 4 in Eq. 2:
Or,
where, ξ_{i} is the ith right eigenvector, η_{i} is the ith left eigenvector and λ_{i} is the ith eigenvalue of J_{RQV}.
Since ξ^{1} = η, Eq. 5 may be written
as:
Where:
v = ηxΔV 
= 
The vector of modal voltage variations 
q = ηxΔQ 
= 
The vector of modal reactive power variations 
Assuming that ith mode, the vector of modal reactive power variations (q) has
all elements equal to zero except for the ith, which equals to 1. The corresponding
vector of bus reactive power variations is:
With the vector of bus reactive power variations equal to ΔQ_{i},
the vector of bus voltage variations, ΔV_{i} is:
And the corresponding vector of bus angle variation for ith mode, is:
The relative participation of machine m in mode i is given by the generator
participation factor (Gao et al., 1992; Kundur,
1994):
where, N_{G} is the number of generator buses.
The expression for Q at any bus k, is given by:
where, the is
the kjth element of admittance matrix (Y_{bus}).
For PV buses, Eq. 12 could be linearized as below:
Where:
where, N_{PQ} is the number of PQ buses.
Using Eq. 910 and 13,
it could be written for ith mode:
Substituting Eq. 8 in 16 the participation
factor of machine m in mode i, I, GPF_{m,i}, can be:
Generator participation factors indicate, for each mode, which generator supply
the most reactive power in response to an incremental change in system reactive
loading. Generator participations provide important information regarding proper
distribution of reactive reserve among all the machines in order to maintain
an adequate voltage stability margin. Generators with high GPF_{i} are
important in maintaining stability of mode i (Gao et al.,
1992; Kundur, 1994).
Proposed approach: The proposed MVAR (Mega Volt Ampere Reactive) scheduling method is based on two main steps: The first step obtains weight (or penalty) factors using modal analysis technique and the second step is the ORD, which is used to determine the desired reactive dispatch. The proposed methodology is shown in Fig. 1 and its main steps are the following.
Determination of desired penalty factors for dynamic VAR rescheduling:
Present method maximizes the VSM directly based on GPFs. It can be defined as
a reactive power rescheduling process and is solved by adding a penalty term
on the OPF objective function. Generators with greater GPFs should inject more
reactive power support, because their impact on VSM is more than the other dynamic
VAR sources. The penalty (or weighting) factor is related with the machine reactive
power impact on VSM and it is updated from the modal analysis, as follows:
• 
At iteration k, calculate the generator participation factors
corresponding to the least stable mode from Eq. 17 
• 
Through it, build the vector d (it gives the direction for changing the
machines reactive injection) 

Fig. 1: 
Reactive power rescheduling methodology 
• 
Compute penalty factors as below: 
where, α is a controlled step size (TleloCuautle
et al., 2007). Generators with grater GPFs have smaller penalty factors,
due to their more effectiveness on VSM. Therefore, these generators will inject
more reactive power support and then, the VSM of the system will improve.
Optimal reactivepower dispatch (ORD): By adding a penalty factor for
each machine reactive injection at the OPF objective function, it is possible
to find the most adequate MVAR distribution to improve the VSM. Generators with
high GPFs are important in maintaining stability of the least stable mode. Thus
at each iteration it is necessary to identify the least stable mode (through
calculation of the smallest eigenvalue of J_{RQV}) and then calculate
the corresponding penalty factors from Eq. 18 and 19.
The OPF model, which is used in this study, is as follows.
Objective function: In this ORD problem it is assumed that active
power generations are fixed and rescheduling is carried out on reactive
power generations. Thus the objective function could be written, as following:
where, QG_{j} and VG_{j} is reactive power generation and voltage of jth reactive power source.
Constraints: The details of the OPF constraints are discussed here (Zhang
et al., 2007):
• 
Power flow constraints: 
Where:
Pg_{i} 
= 
Generator active power output 
PL_{i} 
= 
Load active power 
QG_{i} 
= 
Generator reactive power output 
QL_{i} 
= 
Load reactive power 
VG_{i} 
= 
PV bus voltage 
VL_{i} 
= 
Bus voltage 
θ_{i} 
= 
Bus voltage angle 

= 
Transmission line flow 
In order to evaluate the suitability of the proposed methodology, VSM is calculated
using Fig. 2, as follows:
where, P_{initial} and P_{max} represent the active power loads
at current operating point and PV curve nose point, respectively and:

Fig. 2: 
PV curves 
Also, the loading factor is defined as
RESULTS AND DISCUSSION
The proposed ORD methodology is tested on the IEEE 30bus test system, which has 6 generators and 41 transmission lines. The configuration of the power system is shown in Fig. 3. For this test system α = 10, is suitable. As mentioned previously, the problem formulated as an optimization problem, which is a Non Linear Programming (NLP). Using optimization toolbox of MATLAB software and some programming, this NLP is solved. BFGS QuasiNewton algorithm is employed for the optimization. Present optimization problem is a problem with continues (non integer) variables and this algorithm is suitable for NLP with continuous variables.
Active power injections of the generators are fixed for the initial solution,
except for the slack bus. The relationship between VAR source 4 (at bus 8) output
VAR and its GPF, is shown in Fig. 4. Figure
4 shows that the scheduling of VAR source is in direction of its GPF. The
initial and final (after optimization) PV curves for buses 20 and 30 are shown
in Fig. 5. From Fig. 5, it can be found
that the voltage profile has improved for any loading factor and eventually,
voltage stability margin has been increased significantly.

Fig. 3: 
IEEE 30bus test system 

Fig. 4: 
VAR rescheduling and GPF for source4 (at bus 8) 
The effect is rather
significant since the ORD is applied without adding new VAR sources and the
active power dispatch is not changed. Figure 6 shows the total active and reactive losses of the
system, during rescheduling process. It could be seen that the active and reactive
losses are decreased.
In the voltage stability problem, the reactive reserve margin is extremely
important at generators, because it gives an advance indication of how close
the generator to its operation limits. Figure 7 shows the
modified reactive power injections of VAR sources at each iteration of the proposed
algorithm. It can be observed that unless VAR source 6 (at bus 13), reactive
power injections of all other sources decreased.

Fig. 5: 
Initial and final PV curves (a)for bus 20 and (b) for bus
30 

Fig. 6: 
Initial and final PV curves (a)for bus 20 and (b) for bus
30 

Fig. 7: 
Reactive power injections of generators 
Therefore, the reactive reserve of system is increased. Figure
8 justifies this conclusion and illustrates that the total reactive power
generation of the system is decreased. The increase of reactive reserve of system
indirectly increases the VSM.

Fig. 8: 
Total reactive power generation 
The numerical values of P_{initial}, P_{max} and VSM corresponding
to the figures are given in Table 1.
Table 1: 
Numerical values for voltage stability analysis 

As it is observed, the
VSM is increased 6.90%, which is significant, because no modification is done
on generators activepower dispatch.
CONCLUSION
This study discusses the management of dynamic reactive power generation in
order to improve voltage stability margin. The method is based on optimal power
flow. The management of the VAR generation is processed as an optimization problem.
Generator participation factors is introduced and then incorporated to ORD problem. This study has shown that the generator participation factors are adequate for the indication of the direction of change of reactive power injection of dynamic VAR sources, in order to increase the voltage stability margin. The method studied has proved efficient in improving voltage stability margins by modifying the reactive generation of dynamic VAR sources. By applying the proposed ORD approach, the reactive power reserves in the system increase and active and reactive power losses decrease. The voltage stability margin has been improved without adding new VAR sources and changing the active power dispatch and consequently the cost of generation.