Accuracy Enhancement of Mixed Power Flow Analysis Using a Modified DC Model

The mixed power flow analysis method decreases the computational complexity and achieves a high level of simulation accuracy. The mixed approach combines the ac with the dc power flow models, depending on the area of interest. The accurate ac model is used in the study area of interest to obtain high simulation accuracy, while the approximate dc model is used in the remainder of the system to reduce the required computations. In the original mixed approach, the errors originating from the use of the dc model may propagate to the area of interest where accurate simulation outcomes are required; thus, the simulation accuracy might not be satisfactory. This paper presents a new method of enhancing the simulation accuracy of the mixed power flow analysis using available information. In the proposed approach, a modified dc model is used instead of the traditional one and is constructed from an initial base-case ac solution. The new dc model compensates for the errors originating from the neglect of the real power losses and the assumption of a flat voltage magnitude in the conventional dc model. Thus, the proposed method can improve the simulation accuracy in the area of interest. The superior computational benefits can also be preserved by maintaining linear characteristics of the dc model. Case studies with the IEEE 118-bus system are provided to validate the enhanced accuracy of the proposed method.


Introduction
Steady-state power flow analysis is a fundamental tool and the most frequently used routine in power system planning, operations, and control.It provides the bus voltages of each bus, and the power flows through a transmission network for a specified load and generator information.With this analysis, various contingencies and expansion plans can be carefully explored, and security indexes such as component overloading and power transfer capability can be evaluated.However, owing to interconnections, modern electric power systems have become increasingly complex, large-scale systems comprising numerous components.Power system equations are inherently nonlinear.The complexity issues in the analysis have grown and become even more burdensome with the recent integration of renewable energy sources.Thus, the need for the development of rapid and precise power system analysis tools has become critical [1].
The primary and the most accurate approach for power flow analysis models the electric power networks with a set of nonlinear power balance equations; this is typically referred to as the ac model.Based on Kirchhoff's law and the definition of complex power, nonlinear real and reactive power balance equations are derived for each bus.A large number of equations are required to represent the modern large power systems.The set of nonlinear equations is usually solved using iterative techniques such as Newton's method.Hence, the ac model is computationally expensive, especially for the contingency analysis of large power systems [2].
the simulation accuracy can be improved.The superior computational benefits can also be preserved with a linear formulation of the dc model.
The remainder of this paper is organized as follows.Section 2 presents a brief overview of the mixed power flow analysis.The enhanced method is proposed along with the problem definition in Section 3. Section 4 illustrates the simulation results using the IEEE 118-bus system.Various system conditions are studied to evaluate the performance of the proposed method.The conclusions are presented in Section 5.

Mixed Power Flow Analysis
In this section, the ac and dc models and the mixed approach are briefly explained.

AC Model
The ac power flow model shown in Equations ( 1) and ( 2) is the most accurate approach for modeling the steady-state behavior of balanced three-phase electric power networks.The real and reactive power balance equations are formulated for each bus.The complete set of nonlinear equations is solved with iterative numerical methods.Thus, the ac model introduces heavy computational burdens; however, it provides accurate solutions.
where P sp k , Q sp k are the real and reactive power injection at bus k; V k is the voltage magnitude at bus k; θ k is the voltage angle at bus k; G km is the real part of admittance matrix element Y km ; B km is the imaginary part of admittance matrix element Y km ; and superscript sp denotes the specified value.

DC Model
The dc model is the simplified version of the ac model, assuming a lossless network, one-per-unit voltage magnitudes, and small voltage angle variations, while completely ignoring the reactive power equations.With these assumptions, the real power equation in (1) can be modified to the simple linear equation in (3).
Since the equation is linear, the solution can be calculated directly; this means that the dc model always has a single solution.The linearity and the reduced number of equations allow the dc power flow model to compute the solutions much faster than the ac model.For locational marginal price (LMP) calculations in practical power systems, the dc power flow model was shown to be about 60 times faster than the ac model [7].Because of the advantages in speed and robustness, the dc model has been an attractive tool for real-time market analyses and future expansion planning.However, the accuracy of the dc model solution is highly variable depending on the power system.An excellent simulation result can be obtained only if the underlying assumptions are satisfied.However, these assumptions cannot always be justified and sometimes might be unrealistic for practical power systems.
Many studies have been conducted to quantify the simulation accuracy of the dc model and to increase its performance [8,11,16].It was observed in [8] that the main factors affecting the accuracy of the dc model are the flat voltage profile and the ignorance of real power losses.A satisfactory accuracy has been reported with hot-start models, which take the branch losses and bus voltages into account using the base-case ac power flow results or the historical data of the system [11,16].These are state-dependent models.The hot-start model could maintain the same amount of MW losses in the entire system; thus, the total system power balance could be preserved without the need for generating additional power from a slack generator.A more advanced version even modifies the slope of the dc model linear equation corresponding to B km in Equation (3).

Mixed Approach
The mixed approach was proposed to take advantage of the best of both ac and dc models in terms of computational benefits and simulation accuracy.The approach first partitions the electric network into three subsystems, which are the internal system, the external system, and the boundary buses.The division is dependent on the areas of interest under study, and those are mutually exclusive.Then, it formulates the power flow problem by combining the ac model with the dc model.The detailed ac model in Equations ( 1) and ( 2) is used in the study area of interest and is called the internal system.The less-detailed dc model in Equation ( 3) is used in the external system, which is the remainder of the system.Thus, the approach aims to obtain accurate simulation outcomes in the internal system and to decrease computational demands by the use of the simplified dc model in the external system.The set of formulated equations can be solved using the well-known Newton-Raphson method.Figure 1 shows the overall procedure of the mixed approach.
Energies 2016, 9, 776 4 of 12 the entire system; thus, the total system power balance could be preserved without the need for generating additional power from a slack generator.A more advanced version even modifies the slope of the dc model linear equation corresponding to Bkm in Equation (3).

Mixed Approach
The mixed approach was proposed to take advantage of the best of both ac and dc models in terms of computational benefits and simulation accuracy.The approach first partitions the electric network into three subsystems, which are the internal system, the external system, and the boundary buses.The division is dependent on the areas of interest under study, and those are mutually exclusive.Then, it formulates the power flow problem by combining the ac model with the dc model.The detailed ac model in Equations ( 1) and ( 2) is used in the study area of interest and is called the internal system.The less-detailed dc model in Equation ( 3) is used in the external system, which is the remainder of the system.Thus, the approach aims to obtain accurate simulation outcomes in the internal system and to decrease computational demands by the use of the simplified dc model in the external system.The set of formulated equations can be solved using the well-known Newton-Raphson method.Figure 1 shows the overall procedure of the mixed approach.The computational benefits of the mixed approach come from the reduced number of equations in the external system and the reduction of operations required for LU factorization and forward/backward substitution to solve the linear matrix equations.Simulation time is dependent on the ratio of the number of internal buses to external buses.With the IEEE 118-bus system in [17], the mixed approach, using a ratio in which the external system has about 1.5 times more buses than the internal system, can reduce the simulation time by half.More computational gain can be expected with larger power systems and larger dimensions of the external system compared to the internal one.Comprehensive details including speed, accuracy, and examples can be found in [13].

The Accuracy Problem
The mixed approach might have problems in the accuracy of the simulation outcomes, although an exact ac model is used in the internal system.Errors originating from the simplified dc model may be transferred to the internal system.In the original mixed approach, careful considerations were made to minimize the error propagation by formulating the ac power flow equations at the boundary The computational benefits of the mixed approach come from the reduced number of equations in the external system and the reduction of operations required for LU factorization and forward/backward substitution to solve the linear matrix equations.Simulation time is dependent on the ratio of the number of internal buses to external buses.With the IEEE 118-bus system in [17], the mixed approach, using a ratio in which the external system has about 1.5 times more buses than the internal system, can reduce the simulation time by half.More computational gain can be expected with larger power systems and larger dimensions of the external system compared to the internal one.Comprehensive details including speed, accuracy, and examples can be found in [13].

The Accuracy Problem
The mixed approach might have problems in the accuracy of the simulation outcomes, although an exact ac model is used in the internal system.Errors originating from the simplified dc model may Energies 2016, 9, 776 5 of 12 be transferred to the internal system.In the original mixed approach, careful considerations were made to minimize the error propagation by formulating the ac power flow equations at the boundary buses composed of the boundaries between the two systems.However, the errors from the use of a dc model cannot be completely avoided under certain conditions, and the mixed approach fails to provide accurate simulation outcomes.The accuracy problem arises from the excessive approximations made in the dc model, especially from the neglect of the real power losses.Static power flow analysis assumes that the real and reactive powers are balanced in the entire system, and this means that the total generation is equal to the total consumption plus losses.Ignorance of the real power losses reduces the total amount of generation.Although the real power losses on a transmission line are relatively small, the cumulative sum of these losses might be significant, particularly when a large power system is under study.For example, if the external area does not have enough real power injections, the real powers required to maintain the system balance must be provided from the internal system.The use of a dc model reduces the amount of real power injections from the internal system, and this brings about a reduction in the accuracy of the simulation outcomes in the internal system.
The mixed approach was originally developed with a cold-start dc model assuming that the initial ac solutions were not available.However, in practical power system operations, various applications are investigated with prior knowledge about the present power system conditions, which can be obtained from a state estimator.This useful information can be positively utilized to sort out the accuracy problems in the original mixed approach.

The Proposed Approach
The proposed method aims to improve the simulation accuracy of the mixed approach, while retaining its computational benefits.Simulation accuracy can be improved by preserving the real power balances, and the computational advantages can be sustained by using a linear model.The proposed approach modifies the traditional dc model to a new model assuming that the base-case ac solutions of the entire system are available.

Loss Modeling of the DC Model
The real power losses are simply modeled by calculating the differences between the net power injections in the ac and the dc models, and the loss model is provided in Equation ( 4).The loss information is calculated using the initial ac power flow solution.The superscript b denotes the information from the base-case ac solutions.

Incorporation of Voltage Magnitudes into the DC Model
The voltage magnitude information from the available base-case ac solutions is utilized to improve the dc power flow model, instead of assuming a per-unit voltage magnitude.Voltage magnitudes in power systems are closely related to the reactive powers, which are not considered in the external system.Therefore, this utilization produces a positive effect while dealing with reactive powers; thus, it helps the proposed method in enhancing the accuracy.

The New DC Model
The new dc model shown in Equation ( 6) incorporates the voltage magnitude information and loss modeling into the conventional dc model.This new model is constructed with the base-case information, which is assumed to be available.
3.2.4.The Procedure of the Proposed Approach The procedure of the proposed approach is almost the same as that of the original mixed approach.The difference is that the new dc model in Equation ( 6) is applied for the external system.Overall procedure of the proposed approach is shown in Figure 2. The proposed approach retains the linear formulation of the dc model.The construction of the new dc model requires minor additional computations to incorporate the base-case ac solutions and is made once before running the simulation.More accurate simulation outcomes can be obtained with little extra computational effort.
Energies 2016, 9, 776 6 of 12 3.2.4.The Procedure of the Proposed Approach The procedure of the proposed approach is almost the same as that of the original mixed approach.The difference is that the new dc model in Equation ( 6) is applied for the external system.Overall procedure of the proposed approach is shown in Figure 2. The proposed approach retains the linear formulation of the dc model.The construction of the new dc model requires minor additional computations to incorporate the base-case ac solutions and is made once before running the simulation.More accurate simulation outcomes can be obtained with little extra computational effort.

Test Systems and Performance Metrics
The proposed method was implemented using Matlab.Simulations were performed on the IEEE 118-bus system shown in Figure 3.In this system, there are 180 transmission lines, 19 generators, and 99 loads.The detailed system information including generation, load, and transmission line parameters can be found in [17].

Test Systems and Performance Metrics
The proposed method was implemented using Matlab.Simulations were performed on the IEEE 118-bus system shown in Figure 3.In this system, there are 180 transmission lines, 19 generators, and 99 loads.The detailed system information including generation, load, and transmission line parameters can be found in [17].The proposed approach was tested based on the system division presented in Table 1.In the test system, there are 47 buses in the internal system, 5 boundary buses, and 66 buses in the external system.The internal system is presented in Figure 2. Simulation comparisons are made among the ac model, the original mixed model, and the proposed approach.Four metrics are used to quantify the simulation accuracy, and they are calculated considering the ac solution as a reference.They include the Euclidean norm in Equation ( 7), a sum of the Euclidean norm considering only the internal system buses in Equation ( 8), absolute values of the differences between the real power flows in corresponding branches of the three approaches in Equation ( 9), relative values of the difference between the real power flows in Equation (10), and a sum of the real power errors in Equation (11).
% 100 where ENi is the Euclidean norm of voltage difference at bus i; SEN is the sum of the Euclidean norm; EPFi is the error in the real power flow through transmission line i; REPFi is the relative error (%) in the real power flow through transmission line i; PFi is the real power flow through the transmission line i; SEPF is the sum of real power errors.The proposed approach was tested based on the system division presented in Table 1.In the test system, there are 47 buses in the internal system, 5 boundary buses, and 66 buses in the external system.The internal system is presented in Figure 2. Simulation comparisons are made among the ac model, the original mixed model, and the proposed approach.Four metrics are used to quantify the simulation accuracy, and they are calculated considering the ac solution as a reference.They include the Euclidean norm in Equation ( 7), a sum of the Euclidean norm considering only the internal system buses in Equation ( 8), absolute values of the differences between the real power flows in corresponding branches of the three approaches in Equation ( 9), relative values of the difference between the real power flows in Equation (10), and a sum of the real power errors in Equation (11).
where EN i is the Euclidean norm of voltage difference at bus i; SEN is the sum of the Euclidean norm; EPF i is the error in the real power flow through transmission line i; REPF i is the relative error (%) in the real power flow through transmission line i; PF i is the real power flow through the transmission line i; SEPF is the sum of real power errors.

Case Studies
The proposed approach is constructed assuming that base-case ac solutions are given.Performance evaluation of the accuracy is conducted under different operating conditions that deviate from the base-case operating point.In this study, three different types of contingencies are applied: (1) a loss in the generation to cause an imbalance in the real power, (2) a loss in the transmission line for which the system balance is maintained, and (3) both contingencies at the same time.The first simulations are performed with a generator outage (450 MW) at bus 10.The largest generator in the internal system is selected to change the system operating point substantially.The power required to keep the system balanced, after the loss of generation, is picked up by the system slack bus at bus 69.Additional powers from the slack generator are distributed through the network.
Figure 4 shows the Euclidean norm calculated with the voltage vectors at all system buses.All line power flows in both directions on a transmission line are compared in Figure 5. Relative errors in Figure 6 are calculated only with the line flows larger than 5 MW.As shown in the simulation comparisons, the difference between the ac solution and the proposed approach is much smaller than that between the ac solution and the original mixed approach, for the entire system.In Table 2, the sum of Euclidean norms for the internal system buses is 0.0111 for the proposed approach, while it is 0.2412 for the original mixed approach.The active power flow differences for the entire system are reduced from 844.79 MW with the original approach to 99.05 MW with the proposed approach.The overall simulation accuracy shows an obvious improvement.Moreover, the simulation comparisons show that the proposed method provides better matching in the external system.The improved performance can be understood from the fact that the real power imbalance from the generator outage can be correctly represented with the proposed approach.The original mixed approach has a weakness in the simulation accuracy, and ignoring the real power losses in the conventional dc model has a significant impact on the simulation accuracy.

Case Studies
The proposed approach is constructed assuming that base-case ac solutions are given.Performance evaluation of the accuracy is conducted under different operating conditions that deviate from the base-case operating point.In this study, three different types of contingencies are applied: (1) a loss in the generation to cause an imbalance in the real power, (2) a loss in the transmission line for which the system balance is maintained, and (3) both contingencies at the same time.

Generator Outage (450 MW) at Bus 10
The first simulations are performed with a generator outage (450 MW) at bus 10.The largest generator in the internal system is selected to change the system operating point substantially.The power required to keep the system balanced, after the loss of generation, is picked up by the system slack bus at bus 69.Additional powers from the slack generator are distributed through the network.
Figure 4 shows the Euclidean norm calculated with the voltage vectors at all system buses.All line power flows in both directions on a transmission line are compared in Figure 5. Relative errors in Figure 6 are calculated only with the line flows larger than 5 MW.As shown in the simulation comparisons, the difference between the ac solution and the proposed approach is much smaller than that between the ac solution and the original mixed approach, for the entire system.In Table 2, the sum of Euclidean norms for the internal system buses is 0.0111 for the proposed approach, while it is 0.2412 for the original mixed approach.The active power flow differences for the entire system are reduced from 844.79 MW with the original approach to 99.05 MW with the proposed approach.The overall simulation accuracy shows an obvious improvement.Moreover, the simulation comparisons show that the proposed method provides better matching in the external system.The improved performance can be understood from the fact that the real power imbalance from the generator outage can be correctly represented with the proposed approach.The original mixed approach has a weakness in the simulation accuracy, and ignoring the real power losses in the conventional dc model has a significant impact on the simulation accuracy.

Case Studies
The proposed approach is constructed assuming that base-case ac solutions are given.Performance evaluation of the accuracy is conducted under different operating conditions that deviate from the base-case operating point.In this study, three different types of contingencies are applied: (1) a loss in the generation to cause an imbalance in the real power, (2) a loss in the transmission line for which the system balance is maintained, and (3) both contingencies at the same time.

Generator Outage (450 MW) at Bus 10
The first simulations are performed with a generator outage (450 MW) at bus 10.The largest generator in the internal system is selected to change the system operating point substantially.The power required to keep the system balanced, after the loss of generation, is picked up by the system slack bus at bus 69.Additional powers from the slack generator are distributed through the network.
Figure 4 shows the Euclidean norm calculated with the voltage vectors at all system buses.All line power flows in both directions on a transmission line are compared in Figure 5. Relative errors in Figure 6 are calculated only with the line flows larger than 5 MW.As shown in the simulation comparisons, the difference between the ac solution and the proposed approach is much smaller than that between the ac solution and the original mixed approach, for the entire system.In Table 2, the sum of Euclidean norms for the internal system buses is 0.0111 for the proposed approach, while it is 0.2412 for the original mixed approach.The active power flow differences for the entire system are reduced from 844.79 MW with the original approach to 99.05 MW with the proposed approach.The overall simulation accuracy shows an obvious improvement.Moreover, the simulation comparisons show that the proposed method provides better matching in the external system.The improved performance can be understood from the fact that the real power imbalance from the generator outage can be correctly represented with the proposed approach.The original mixed approach has a weakness in the simulation accuracy, and ignoring the real power losses in the conventional dc model has a significant impact on the simulation accuracy.Next, comparisons were made with a transmission line outage in the line connecting buses 23 and 25.For the contingency, one of transmission lines carrying a large amount of real power in the internal system was chosen as the opening line.The transmission line carried approximately 170 MW power.Figures 7-9 show the Euclidean distances, errors in the real power flows, and percentage errors of the line flows, respectively.The results in Table 3 show that the errors were reduced from 0.2368 to 0.0035 in the sum of the Euclidean norms, and from 729.99 to 10.07 MW in the real power flows.Although the contingency does not introduce an imbalance in the real powers in the system, the original mixed approach fails to provide good solutions.Next, comparisons were made with a transmission line outage in the line connecting buses 23 and 25.For the contingency, one of transmission lines carrying a large amount of real power in the internal system was chosen as the opening line.The transmission line carried approximately 170 MW power.Figures 7-9 show the Euclidean distances, errors in the real power flows, and percentage errors of the line flows, respectively.The results in Table 3 show that the errors were reduced from 0.2368 to 0.0035 in the sum of the Euclidean norms, and from 729.99 to 10.07 MW in the real power flows.Although the contingency does not introduce an imbalance in the real powers in the system, the original mixed approach fails to provide good solutions.Next, comparisons were made with a transmission line outage in the line connecting buses 23 and 25.For the contingency, one of transmission lines carrying a large amount of real power in the internal system was chosen as the opening line.The transmission line carried approximately 170 MW power.Figures 7-9 show the Euclidean distances, errors in the real power flows, and percentage errors of the line flows, respectively.The results in Table 3 show that the errors were reduced from 0.2368 to 0.0035 in the sum of the Euclidean norms, and from 729.99 to 10.07 MW in the real power flows.Although the contingency does not introduce an imbalance in the real powers in the system, the original mixed approach fails to provide good solutions.

Both Contingencies at the Same Time
More severe conditions are studied considering the generator and line outages at the same time.Simulation comparisons are shown in Figures 10-12.Similar to the previous experiments, the proposed approach provides better simulation accuracy.The simulation results are quite similar to those from the generator outage simulation.It can be inferred that the generator outage introduces more state transitions in the system than the line outage.10-12.Similar to the previous experiments, the proposed approach provides better simulation accuracy.The simulation results are quite similar to those from the generator outage simulation.It can be inferred that the generator outage introduces more state transitions in the system than the line outage.

Both Contingencies at the Same Time
More severe conditions are studied considering the generator and line outages at the same time.Simulation comparisons are shown in Figures 10-12.Similar to the previous experiments, the proposed approach provides better simulation accuracy.The simulation results are quite similar to those from the generator outage simulation.It can be inferred that the generator outage introduces more state transitions in the system than the line outage.The quantified errors depending on the contingency applied are shown in Tables 2 and 3.Only the internal system buses are considered for the calculation of the sum of the Euclidean norms in Table 2.In Table 3, all real power errors for the entire system are summed up.They prove that the proposed method shows better matching with the ac model solution when compared to the original mixed method.
Comparison of the computation time among three different approaches was made with an Intel Core processor of 3.3 GHz, and the results are listed in Table 4.An average execution time was obtained from 10 simulations.The proposed approach shows 32.5% reduction compared to the full ac method and provides a small increase less than 1% compared with the original method.For the case study, the ratio of the internal to external buses is about 1 to 1.27.More computational benefits are expected with practical systems that have a larger external system.

Conclusions
The mixed approach is an extremely attractive power flow analysis model that can speed up the simulation with a simple dc model as well as maintain a high level of simulation accuracy with an accurate ac model.However, the use of the conventional dc model developed with a large amount of simplifications inevitably introduces errors in simulation results.In this paper, an enhanced method is proposed to improve the simulation accuracy of the mixed approach with minor extra computations.The proposed method utilizes the base-case information, which is commonly available, for power system planning, operations, and control.With the available ac solutions, the proposed method modifies the conventional dc model used in the original mixed approach into an advanced model.The The quantified errors depending on the contingency applied are shown in Tables 2 and 3.Only the internal system buses are considered for the calculation of the sum of the Euclidean norms in Table 2.In Table 3, all real power errors for the entire system are summed up.They prove that the proposed method shows better matching with the ac model solution when compared to the original mixed method.
Comparison of the computation time among three different approaches was made with an Intel Core processor of 3.3 GHz, and the results are listed in Table 4.An average execution time was obtained from 10 simulations.The proposed approach shows 32.5% reduction compared to the full ac method and provides a small increase less than 1% compared with the original method.For the case study, the ratio of the internal to external buses is about 1 to 1.27.More computational benefits are expected with practical systems that have a larger external system.

Conclusions
The mixed approach is an extremely attractive power flow analysis model that can speed up the simulation with a simple dc model as well as maintain a high level of simulation accuracy with an accurate ac model.However, the use of the conventional dc model developed with a large amount of simplifications inevitably introduces errors in simulation results.In this paper, an enhanced method is proposed to improve the simulation accuracy of the mixed approach with minor extra computations.The proposed method utilizes the base-case information, which is commonly available, for power system planning, operations, and control.With the available ac solutions, the proposed method modifies the conventional dc model used in the original mixed approach into an advanced model.The new dc model incorporates the real power losses and the voltage magnitude information, which are the most influential factors affecting the accuracy of the dc model.The compensation of real power losses is helpful in preserving the real power balance of the system.The incorporation of voltage magnitude information has an impact on considering the reactive power flows, which are totally neglected in the conventional dc model.Thus, more accurate simulation outcomes can be expected by minimizing the loss of accuracy in the conventional dc model.The superior computational benefits can be retained with a linear formulation of the dc model.The test simulation with the IEEE 118-bus system proved that the proposed approach achieved better simulation accuracy than the original mixed approach, while preserving the advanced computational benefits.The proposed approach can be very useful for applications where accurate and fast simulation outcomes are of great importance.

Figure 1 .
Figure 1.Procedure of the mixed power flow method.

Figure 1 .
Figure 1.Procedure of the mixed power flow method.

Figure 2 .
Figure 2. Procedure of the proposed approach.

Figure 2 .
Figure 2. Procedure of the proposed approach.

Figure 5 .
Figure 5. Errors in the real power with the generator outage (450 MW) at bus 10.

Figure 5 .
Figure 5. Errors in the real power with the generator outage (450 MW) at bus 10.

Figure 5 .
Figure 5. Errors in the real power with the generator outage (450 MW) at bus 10.

Figure 6 .
Figure 6.Relative errors in the real power with the generator outage (450 MW) at bus 10.

Figure 8 .
Errors in the real power with the line outage.

Figure 9 .
Figure 9. Relative errors in the real power with the line outage.

Figure 8 .
Figure 8. Errors in the real power with the line outage.

Figure 9 .
Figure 9. Relative errors in the real power with the line outage.

Figure 11 .
Figure 11.Errors in the real power with both contingencies.

Figure 12 .
Figure 12.Relative errors in the real power with both contingencies.

Figure 12 .
Figure 12.Relative errors in the real power with both contingencies.

Table 1 .
Test system configuration.

Table 1 .
Test system configuration.

Table 2 .
Sum of Euclidean norms with the contingencies.

Table 3 .
Sum of real power errors (MW) with the contingencies.

Table 2 .
Sum of Euclidean norms with the contingencies.

Table 2 .
Sum of Euclidean norms with the contingencies.

Table 3 .
Sum of real power errors (MW) with the contingencies.

Table 3 .
Sum of real power errors (MW) with the contingencies.Both Contingencies at the Same Time More severe conditions are studied considering the generator and line outages at the same time.Simulation comparisons are shown in Figures

Table 4 .
Computation time comparison of the IEEE 118-bus system.

Table 4 .
Computation time comparison of the IEEE 118-bus system.