A Methodology for Determining Permissible Operating Region of Power Systems According to Conditions of Static Stability Limit

Van Duong Ngo 1, Dinh Duong Le 2,*, Kim Hung Le 2, Van Kien Pham 2 and Alberto Berizzi 3 1 The University of Danang, 41 Le Duan st., Danang 59000, Vietnam; nvduong@ac.udn.vn 2 Department of Electrical Engineering, The University of Danang—University of Science and Technology, 54 Nguyen Luong Bang st., Danang 59000, Vietnam; lekimhung@dut.udn.vn (K.H.L.); pvkien@dut.udn.vn (V.K.P.) 3 Department of Energy, Polytechnic University of Milan, via La Masa, 34, 20156 Milan, Italy; alberto.berizzi@polimi.it * Correspondence: ldduong@dut.udn.vn; Tel.: +84-905-320-755


Introduction
The rapid development of economies, science, and technology in the world has led to an increasing demand for energy consuming in general and electricity in particular.Power systems are growing and expanding to link power systems of regions or to connect large power plants to consumers using ultra-high-voltage (UHV) or extra-high-voltage (EHV) transmission lines.Large interconnected power systems with long UHV or EHV lines have different features to small ones.The presence of a large amount of reactive power generated by a high voltage line during its operation influences the transmission capacity of the line and the stability of the system.
The definition and classification of power system stability are given in detail in [1].Power system stability is broadly classified into three categories [1]: rotor angle stability [2][3][4], frequency stability [5,6] and voltage stability [7].Voltage stability is defined as the ability of a system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition [1,7].In voltage stability analysis, it is useful to further classify voltage stability into two subcategories, i.e., large disturbance and small disturbance voltage stability.The former refers to the ability of the system to maintain stable voltages followed by large disturbances such as system faults, loss of generation, or circuit contingencies.On the contrary, the latter refers to the capability of the system to continue steady voltages when subjected to small perturbations such as incremental changes in system load.Generally, the analysis for small disturbances is done as a steady-state stability analysis.In the present paper, static voltage stability is considered.In static stability analysis, the system is assumed to be operated in an equilibrium state.Static analysis evaluates the feasibility of the operating point to provide the system operators with a permissible region where the system can operate normally.
It is of great importance to operators of the system to identify how far the power system is from voltage collapse in order to carry out appropriate remedial actions to avoid unexpected results.
The distance towards voltage instability can be obtained by using voltage stability indices.A detailed review of the voltage stability indices can be found in [8].Voltage stability indices could generally be divided into two groups: Jacobian matrix and system variables based voltage stability indices [9].Jacobian matrix based voltage stability indices could be used to predict the voltage collapse point and determine the voltage stability margin.The main drawback of such indices is that they require high computational time; hence, they are not suitable for online voltage stability studies.On the other hand, system variables based voltage stability indices are obtained by using the elements of the admittance matrix of the system and some system variables, e.g., bus voltages, power flows, etc., so they require less computational time compared to the first group.Thanks to this advantage, the second group is appropriate for online voltage stability assessment.
So far, many techniques for assessing voltage stability have been developed in the literature.Among them, P-V and Q-V curves are still widely used both in the research and in industry [7,[10][11][12][13][14][15]; with these curves, the maximum active and reactive power at a system load bus can be determined when the voltage at that bus reaches the voltage collapse boundary.In particular, a P-V curve gives the relationship between the real power load and bus voltage.It can intuitively provide the operators of the system voltage stability criterion as the margin between the voltage collapse point and the current operating point.On the contrary, a Q-V curve shows the change of bus voltages with respect to reactive power injection or absorption.It gives the operators information on the maximum reactive power at a bus when reaching the minimum voltage limit.In general, both procedures for building P-V and Q-V curves are time-consuming since a large number of power flows need to be executed using conventional methods and models [16].Owing to this drawback, they cannot to be used for online analysis.Moreover, P-V and Q-V curves could be used to analyze voltage stability only for certain increasing modes and not for the whole view of the analysis.Commonly, those procedures focus on individual buses by stressing the bus considered independently; hence, they are not able to fully reflect the real stability condition of the system.
For determining the visual characteristics of the voltage stability limit and the operating region for the system, a P-Q curve [7,[17][18][19][20] is very useful.On a power plane, the operating region indicates both the possible active and reactive power in which the load bus of the system can operate normally.Once the boundary is determined, the distance to the voltage collape of the system can be directly assessed.In [17], the curve is assumed to follow a circle.In [7,[18][19][20], the P-Q curve is defined by a parabolic equation.Those assumptions about the shape of the curve are not realistic and may lead to the over-estimation of the reactive power limit.
In this paper, we develop an analytical method to construct a P-Q curve point-by-point, without using any assumption about the shape of the curve on a power plane, and provide a visual operating region for voltage stability studies.The technique can be applied to large-scale power systems to assess voltage stability in real-time.In addition, the P-Q curves we build are realistic since there is no assumption about the change of load used.
The remainder of this paper is organized as follows: In Section 2, a new technique for determining the permissible operating regions of power systems according to static stability limit on the power plane is represented.In Section 3, the testing of the proposed methodology on a system with long-distance UHV or EHV transmission lines is described and the results are discussed.Finally, Section 4 concludes the paper.

Determining Points on Stability Boundary
For assessing the stability of a power system, stability margins are often used, i.e., K P = [(P lim − P 0 )/P 0 ] × 100% and × 100%, where K P and K Q are the stability reserve ratios, P lim and Q lim are the values of limitation, and P 0 and Q 0 are the values at the operating points for active and reactive power, respectively (see Figure 1).The determination of P lim and Q lim for buses In this section, the pragmatic criterion dQ/dV presented in [21] is used to construct the permissible operating region on a power plane under the conditions of a static stability limit.The operating states of a power system can be assessed based on dQ/dV, i.e., if dQ/dV is negative, the system is stable; otherwise, the system is unstable.Based on this criterion, we develop a mathematical formulation to determine points on the boundary (i.e., determine a curve on the power plane) between stable and unstable states of the system, and then we use this boundary to separate the operating regions into two regions, i.e., stable and unstable regions on the power plane, and assess its stability.
Suppose that a power system has n buses in which there are k generation buses (among k generation buses, Bus 1 is the slack bus of the system).The system is represented by an equivalent diagram, and then an adopted Gaussian elimination method [22] is used to simplify the diagram to a simple one that includes k generation buses and the load bus considered, as shown in Figure 2. It is worth noting that, thanks to this technique, the proposed methodology in this paper can be effectively applied for large-scale power systems.
In Figure 2, • Ei is the electromotive force of generators connected at the ith generation bus (i = 1 ÷ k); = + j is the equivalent impedance of the branch connecting generation bus i and the considered load at bus t; • Pid and Qid are the real and reactive power transferred from generation bus i to the considered load at bus t and before equivalent impedance , respectively; • Pic and Qic are the real and reactive power transferred from generation bus i to the considered load at bus t and after equivalent impedance , respectively; • Pt, Qt, and St are the real, reactive, and apparent power of the considered load at bus t; • is the equivalent admittance-to-ground at the considered load bus t, = 1/ , where = + j is the corresponding equivalent impedance; • P0, Q0, and S0 are the real, reactive, and apparent power transferred through .In this section, the pragmatic criterion dQ/dV presented in [21] is used to construct the permissible operating region on a power plane under the conditions of a static stability limit.The operating states of a power system can be assessed based on dQ/dV, i.e., if dQ/dV is negative, the system is stable; otherwise, the system is unstable.Based on this criterion, we develop a mathematical formulation to determine points on the boundary (i.e., determine a curve on the power plane) between stable and unstable states of the system, and then we use this boundary to separate the operating regions into two regions, i.e., stable and unstable regions on the power plane, and assess its stability.
Suppose that a power system has n buses in which there are k generation buses (among k generation buses, Bus 1 is the slack bus of the system).The system is represented by an equivalent diagram, and then an adopted Gaussian elimination method [22] is used to simplify the diagram to a simple one that includes k generation buses and the load bus considered, as shown in Figure 2. It is worth noting that, thanks to this technique, the proposed methodology in this paper can be effectively applied for large-scale power systems.
In Figure 2, • E i is the electromotive force of generators connected at the ith generation bus (i = 1 ÷ k); . Z 0 = R 0 + jX 0 is the corresponding equivalent impedance; • P 0 , Q 0 , and S 0 are the real, reactive, and apparent power transferred through .Y 0 .The mathematical formulation for determining points on a stability boundary is as follows.
Based on the power balance conditions at the load bus, we have: so: where: ; The power transferred from generation bus i to the considered load bus can be calculated using electromotive force and voltage at the load bus .The equations for calculating the active and reactive power transferred to the end of branch i are as follows where, Ei and Vi are magnitudes of and , respectively; δi is the angle difference between and ; and αi is the phase angle of .
From Equations (4a) and (4b), Qic is deduced as follows: where: The mathematical formulation for determining points on a stability boundary is as follows.
Based on the power balance conditions at the load bus, we have: so: where: The power transferred from generation bus i to the considered load bus can be calculated using electromotive force .E i and voltage at the load bus .V i .The equations for calculating the active and reactive power transferred to the end of branch i are as follows where, E i and V i are magnitudes of .E i and .
V i , respectively; δ i is the angle difference between .E i and .
V i ; and α i is the phase angle of .Z i .From Equations (4a) and (4b), Q ic is deduced as follows: where: Energies 2017, 10, 1163 5 of 15 From Equations ( 2), (3), and (5), the reactive power at the load bus is computed as follows: From the formula to calculate the active power at two endings of branch i, P ic , corresponding to the parameters of the operation modes, can be deduced as: We perform some trigonometric transformations as follows: From Equations (4a), (7), and (8), we can determine P ic as: As can be seen from Equations (4a) and ( 6), the reactive power at the load bus Q t is a function of the voltage at load bus V t : If there is a fluctuation of voltage at the load bus, it will lead to an imbalance of power.The amount of imbalance of reactive power can be identified as follows: where Q ΣG is the total reactive power transferred to the load bus from all generators in the system; Taking the derivative of both sides of the above equation with respect to V t , we obtain: Suppose that Q t varies very little with the variation of V t , so: According to pragmatic criterion dQ/dV in [14], the system reaches its static stability limit when: The process for determining points on a stability boundary is implemented step by step as follows.
Energies 2017, 10, 1163 6 of 15 Step 1: Suppose: in which, Q imin and Q imax are the regulating limits for generating the reactive power of source i (limits of the regulator of an excitation system) and Q id is the reactive power at the first point of branch i.The nonlinear equations in Equation ( 10) are solved for voltage at the load bus in limited state V t,lim .Q t is calculated by putting V t,lim and a given value P t into Equation (6).In order to conclude whether the point M (Q t , P t ) is located on the static stability limit of the system or not, it is needed to verify Hypothesis (11).
The reactive power at the first point of branch i can be calculated by using the following formula: If all the generation buses satisfy Hypothesis (11), point M (Q t , P t ) will be located on the static stability limit of the system.In case there are m generators within their regulating limits and (k-m) ones exceeding their regulating limits, Step 2: For source j exceeding its regulating limit: In this case, the reactive power transferred to load bus can be approximately determined: where U j,rated denotes the rated voltage at bus j.
The formula to calculate the reactive power at load bus (6) can be changed as follows: where, P ic is calculated in Equation ( 9) and P jc and Q jc are obtained from Equation (13).
Solving nonlinear Equation (10) we can obtain V t,lim and f (V t ) (see (14)); continue to check m adjustable sources if there is any source violated.If there is no source violated, M(Q t , P t ) is located on the stability limit; otherwise, we recalculate Step 2.

Development of Algorithm for Drawing Characteristics of Stability Limit
From the calculation for determining points located on the stability boundary in Section 2.1, we can build an algorithm for drawing the characteristics of the stability limit, as in Figure 3. adjustable sources if there is any source violated.If there is no source violated, M(Qt, Pt) is located on the stability limit; otherwise, we recalculate Step 2.

Development of Algorithm for Drawing Characteristics of Stability Limit
From the calculation for determining points located on the stability boundary in Section 2.1, we can build an algorithm for drawing the characteristics of the stability limit, as in Figure 3.

Application to Building a Program for Determining Permissible Operating Region of Power Systems on Power Plane
Based on the algorithm developed in Section 2, we build a procedure and then a computer program to determine the operating region of power systems using Delphi programming language.For the illustration, we run it for six-bus power system, as shown in Figure 4, in which Power System 2 (PS2) is connected to Power System 1 (PS1) with an unlimited capacity via a long-distance 500 kV UHV transmission line, in the middle of the line at Bus 6, to which a power plant (G2) is connected.In addition, there are two regional electricity networks with low capacity that are represented as two loads connected to Bus 5 and Bus 6.There is also a load connected to Bus 3.

Application to Building a Program for Determining Permissible Operating Region of Power Systems on Power Plane
Based on the algorithm developed in Section 2, we build a procedure and then a computer program to determine the operating region of power systems using Delphi programming language.For the illustration, we run it for six-bus power system, as shown in Figure 4, in which Power System 2 (PS2) is connected to Power System 1 (PS1) with an unlimited capacity via a long-distance 500 kV UHV transmission line, in the middle of the line at Bus 6, to which a power plant (G2) is connected.In addition, there are two regional electricity networks with low capacity that are represented as two loads connected to Bus 5 and Bus 6.There is also a load connected to Bus 3.  PS1 has an unlimited capacity so the voltage at Bus 1 (U1) can be considered constant, and Bus 1 is considered the slack bus in the system.PS2 is represented by an electromotive force Eeq and an equivalent impedence Zeq.Therefore, the system considered can be represented by an equivalent diagram, as shown in Figure 5. From this equivalent diagram, the algorithm in Section 2 is applied to construct the characteristics of the stability limit on the power plane.The program is run for the test PS1 has an unlimited capacity so the voltage at Bus 1 (U1) can be considered constant, and Bus 1 is considered the slack bus in the system.PS2 is represented by an electromotive force E eq and an equivalent impedence Z eq .Therefore, the system considered can be represented by an equivalent diagram, as shown in Figure 5. From this equivalent diagram, the algorithm in Section 2 is applied to construct the characteristics of the stability limit on the power plane.The program is run for the test system with its initial interface, as in Figure 6.PS1 has an unlimited capacity so the voltage at Bus 1 (U1) can be considered constant, and Bus 1 is considered the slack bus in the system.PS2 is represented by an electromotive force Eeq and an equivalent impedence Zeq.Therefore, the system considered can be represented by an equivalent diagram, as shown in Figure 5. From this equivalent diagram, the algorithm in Section 2 is applied to construct the characteristics of the stability limit on the power plane.The program is run for the test system with its initial interface, as in Figure 6.For example, when clicking on Bus 3, the resulting P-Q curve is obtained, as shown in Figure 7 in which the circular point represents the operating point.In this case, the operating point belongs to the stable region (the region covered by two axes and the P-Q curve), indicating that the system is stable.If the load at Bus 3 is increased, exceeding P-Q curve (stability boundary), a warning message will appear to notify the user that the system is unstable, as in Figure 8.  PS1 has an unlimited capacity so the voltage at Bus 1 (U1) can be considered constant, and Bus 1 is considered the slack bus in the system.PS2 is represented by an electromotive force Eeq and an equivalent impedence Zeq.Therefore, the system considered can be represented by an equivalent diagram, as shown in Figure 5. From this equivalent diagram, the algorithm in Section 2 is applied to construct the characteristics of the stability limit on the power plane.The program is run for the test system with its initial interface, as in Figure 6.For example, when clicking on Bus 3, the resulting P-Q curve is obtained, as shown in Figure 7 in which the circular point represents the operating point.In this case, the operating point belongs to the stable region (the region covered by two axes and the P-Q curve), indicating that the system is stable.If the load at Bus 3 is increased, exceeding P-Q curve (stability boundary), a warning message will appear to notify the user that the system is unstable, as in Figure 8.For example, when clicking on Bus 3, the resulting P-Q curve is obtained, as shown in Figure 7 in which the circular point represents the operating point.In this case, the operating point belongs to the stable region (the region covered by two axes and the P-Q curve), indicating that the system is stable.
If the load at Bus 3 is increased, exceeding P-Q curve (stability boundary), a warning message will appear to notify the user that the system is unstable, as in Figure 8. Plot stability limited curver for BUS 3 P-Q curve for BUS 3 [Mvar] The system is stable Plot stability limited curver for BUS 3 The system is unstable Using the program, the permissible operating region at Buses 3, 5, and 6 can be investigated.Moreover, factors that affect the operating region at a load bus can be examined such as the power change at other load buses in the system, the change in power generated (e.g., P2f), the number of working units (change in Q2max) of the power plant connected to Bus 6, the change in power generated (P3f), and the limit of adjusting the reactive power Q3max of PS2.In order to identify the critical bus and factors affecting the stability of the system, the operating parameters are assumed to be set in the program, as in Table 1.Plot stability limited curver for BUS 3 P-Q curve for BUS 3 [Mvar] The system is stable Plot stability limited curver for BUS 3 The system is unstable Using the program, the permissible operating region at Buses 3, 5, and 6 can be investigated.Moreover, factors that affect the operating region at a load bus can be examined such as the power change at other load buses in the system, the change in power generated (e.g., P2f), the number of working units (change in Q2max) of the power plant connected to Bus 6, the change in power generated (P3f), and the limit of adjusting the reactive power Q3max of PS2.In order to identify the critical bus and factors affecting the stability of the system, the operating parameters are assumed to be set in the program, as in Table 1.Using the program, the permissible operating region at Buses 3, 5, and 6 can be investigated.Moreover, factors that affect the operating region at a load bus can be examined such as the power change at other load buses in the system, the change in power generated (e.g., P 2f ), the number of working units (change in Q 2max ) of the power plant connected to Bus 6, the change in power generated (P 3f ), and the limit of adjusting the reactive power Q 3max of PS2.In order to identify the critical bus and factors affecting the stability of the system, the operating parameters are assumed to be set in the program, as in Table 1.
The operating regions at Buses 5, 6, and 3 are plotted in Figures 9-11, respectively.By comparing the distances from the operating points to the P-Q curves for all load buses in the system, the most dangerous bus can be identified.In this system, Bus 3 is found to be the most dangerous one that needs to be carefully supervised during the operation of the system.The operating regions at Buses 5, 6, and 3 are plotted in Figures 9-11, respectively.By comparing the distances from the operating points to the P-Q curves for all load buses in the system, the most dangerous bus can be identified.In this system, Bus 3 is found to be the most dangerous one that needs to be carefully supervised during the operation of the system.P-Q curve for BUS 5 Table 1.Operating parameters for the six-bus test system.The operating regions at Buses 5, 6, and 3 are plotted in Figures 9-11, respectively.By comparing the distances from the operating points to the P-Q curves for all load buses in the system, the most dangerous bus can be identified.In this system, Bus 3 is found to be the most dangerous one that needs to be carefully supervised during the operation of the system.P-Q curve for BUS 5  In particular, while surveying the operating region at Bus 3, as in Figure 11, we decrease Q3max from 1150 Mvar to 1000 Mvar; the characteristics of stability limit will be changed, as shown in Figure 12.In other cases, we adjust P2f from 360 MW to 240 MW, for which the characteristics of stability limit are shown in Figure 13, or we adjust the load at Bus 6 from P6 + jQ6 = 400 + j240 MVA to P6 + jQ6 = 500 + j360 MVA, so that the curve will be obtained as in Figure 14.Thus, by adjusting the operating parameters and observing the change in the P-Q curve, we can find effective solutions to improve the stability of the system.In particular, while surveying the operating region at Bus 3, as in Figure 11, we decrease Q 3max from 1150 Mvar to 1000 Mvar; the characteristics of stability limit will be changed, as shown in Figure 12.In other cases, we adjust P 2f from 360 MW to 240 MW, for which the characteristics of stability limit are shown in Figure 13, or we adjust the load at Bus 6 from P 6 + jQ 6 = 400 + j240 MVA to P 6 + jQ 6 = 500 + j360 MVA, so that the curve will be obtained as in Figure 14.Thus, by adjusting the operating parameters and observing the change in the P-Q curve, we can find effective solutions to improve the stability of the system.In particular, while surveying the operating region at Bus 3, as in Figure 11, we decrease Q3max from 1150 Mvar to 1000 Mvar; the characteristics of stability limit will be changed, as shown in Figure 12.In other cases, we adjust P2f from 360 MW to 240 MW, for which the characteristics of stability limit are shown in Figure 13, or we adjust the load at Bus 6 from P6 + jQ6 = 400 + j240 MVA to P6 + jQ6 = 500 + j360 MVA, so that the curve will be obtained as in Figure 14.Thus, by adjusting the operating parameters and observing the change in the P-Q curve, we can find effective solutions to improve the stability of the system.Extensive testing on the above six-bus system indicates the good performance of the proposed approach in comparison with the results obtained by other commercial software that includes functions for power system stability studies.For example, Figure 15 shows the operating region at Bus 3 and the determination of values for calculating the stability reserve ratios, while Table 2 gives the results obtained by the proposed method compared with the results from the Power System Simulator for Engineering (PSS/E) software.It should be noted that the curve we build and the stability reserve ratios that are computed are realistic because we construct the curve point-by-point without using any assumption about the shape of the curve or assumption about the change of load.As illustrated in Figure 15, when surveying the operating region at a certain load bus (e.g., Bus 3), from the operating point of the load and the characteristics curve under the static stable limit built, the stability reserve ratios can be computed as follows.At the operating point, P0 and Q0 are known; then S0 is calculated as = + .To determine the limitation of Q (Qlim in Figure 15), P is kept Extensive testing on the above six-bus system indicates the good performance of the proposed approach in comparison with the results obtained by other commercial software that includes functions for power system stability studies.For example, Figure 15 shows the operating region at Bus 3 and the determination of values for calculating the stability reserve ratios, while Table 2 gives the results obtained by the proposed method compared with the results from the Power System Simulator for Engineering (PSS/E) software.It should be noted that the curve we build and the stability reserve ratios that are computed are realistic because we construct the curve point-by-point without using any assumption about the shape of the curve or assumption about the change of load.As illustrated in Figure 15, when surveying the operating region at a certain load bus (e.g., Bus 3), from the operating point of the load and the characteristics curve under the static stable limit built, the stability reserve ratios can be computed as follows.At the operating point, P0 and Q0 are known; then S0 is calculated as = + .To determine the limitation of Q (Qlim in Figure 15), P is kept Extensive testing on the above six-bus system indicates the good performance of the proposed approach in comparison with the results obtained by other commercial software that includes functions for power system stability studies.For example, Figure 15 shows the operating region at Bus 3 and the determination of values for calculating the stability reserve ratios, while Table 2 gives the results obtained by the proposed method compared with the results from the Power System Simulator for Engineering (PSS/E) software.It should be noted that the curve we build and the stability reserve ratios that are computed are realistic because we construct the curve point-by-point without using any assumption about the shape of the curve or assumption about the change of load.As illustrated in Figure 15, when surveying the operating region at a certain load bus (e.g., Bus 3), from the operating point of the load and the characteristics curve under the static stable limit built, the stability reserve ratios can be computed as follows.At the operating point, P 0 and Q 0 are known; then S 0 is calculated as S 0 = P 2 0 + Q 2 0 .To determine the limitation of Q (Q lim in Figure 15), P is kept constant (P = P 0 ), while Q is increased until it reaches the curve.Analogously, P lim is determined by increasing P until it reaches, the curve whilst Q is held constant (Q = Q 0 ).Maintaining constant cosϕ and increasing S (i.e., increasing both P and Q along the line that crosses the origin point and the operating point in Figure 15) until it reaches the curve, P i and Q i are found out and S i is calculated as S i = P 2 i + Q 2 i .Eventually, stability reserve ratios K P , K Q , and K S are computed: Using the above formulas, results are obtained as shown in Table 2.

Figure 1 .
Figure 1.P-Q curve on power plane.

Figure 1 .
Figure 1.P-Q curve on power plane.

Figure 3 .
Figure 3. Algorithm for drawing the characteristics of the stability limit.

Figure 3 .
Figure 3. Algorithm for drawing the characteristics of the stability limit.

Figure 5 .
Figure 5. Equivalent diagram of the six bus power system.

Figure 6 .
Figure 6.Diagram of the six-bus power system in the program.

Figure 5 .
Figure 5. Equivalent diagram of the six bus power system.

Figure 5 .
Figure 5. Equivalent diagram of the six bus power system.

Figure 6 .
Figure 6.Diagram of the six-bus power system in the program.

Figure 6 .
Figure 6.Diagram of the six-bus power system in the program.

Figure 7 .
Figure 7. P-Q curve plotted for Bus 3 in a stable state.

Figure 8 .
Figure 8. P-Q curve plotted for Bus 3 in an unstable state.

Figure 7 .
Figure 7. P-Q curve plotted for Bus 3 in a stable state.

Figure 7 .
Figure 7. P-Q curve plotted for Bus 3 in a stable state.

Figure 8 .
Figure 8. P-Q curve plotted for Bus 3 in an unstable state.

Figure 8 .
Figure 8. P-Q curve plotted for Bus 3 in an unstable state.

Figure 12 .
Figure 12.Operating region at Bus 3 when adjusting Q3max.Figure 12. Operating region at Bus 3 when adjusting Q 3max .

Figure 14 .
Figure 14.Operating region at Bus 3 when adjusting the load at Bus 6.

Table 1 .
Operating parameters for the six-bus test system.

Table 1 .
Operating parameters for the six-bus test system.