Continuous Power Flow and Time Domain Analysis for Assessing Voltage Stability

Voltage instability is considered one of the primary sources of insecurity in power systems. Voltage instability is the phenomenon in which the voltage on the receiver's side is dropping far below the normal value and does not return even after establishing a voltage recovery mechanism, or continue to oscillate due to lack of attenuation. In this study, there will be a voltage stability analysis on the power system in several phases by conducting a power flow analysis, static voltage stability analysis using continuous power flow method, and dynamic voltage stability analysis using the time-domain analysis. The power system tested on this research is the IEEE 6 bus system from Wood & Wollenberg, with the software used for the simulation, which is PSAT. The results show that voltage stability analysis using Continuous Power Flow and Time Domain Analysis can provide a more comprehensive overview of a power system stability level.  The highest decrease in voltage with a value of 1.5% appears in Bus 5 which is also suffering the lowest stability level. The system has a load margin (λ) value of 1.6130 a clearing time of under 0.4 seconds


I. INTRODUCTION
Voltage stability analysis is the one effort to maintain continuity and availability of electrical energy supply to the consumer [1], [2]. It is to maintain the operating conditions of an electrical power system to be stable by fulfilling various operating criteria and must be [3]- [7] safe in case of possible situations out of range. For that, the power system should be operated far from the limits of its stability, so that at the time of disruption, the system has sufficient operational area limit to recover its state [8], [9]. However, current energy systems are often operated close to the limits of stability due to economic and environmental constraints. Maintaining this stability is a challenge in the operation of power systems [10], [11].
Voltage instability can result in the inability of the generator and transmission to provide power on demand loads. The cause may result from equipment outages and limitations of reactive power supplies or due to increased sharp loads. Therefore, the generator reactive power limit must take into account. The Phasor Measurement Unit (PMU) method can handle the Q limit.
Furthermore, load characteristics are also given a significant impact on power system stability analysis. Therefore, it is necessary to obtain an accurate load model in digital simulation. The load increment can affect the voltage stability index, especially in the transmission line. If there is a transmission contingency, the system will get closer to voltage instability [12], [13].
Voltage stability also depends on the initial condition of the system [14]- [16]. Voltage dynamic stability estimation can be done by applying voltage time-series data, known as the maximum Lyapunov exponent. This method gives a proper evaluation for dynamic voltage stability in a short time but not for a more extended period.
Several stages of analysis have done to get an overview of the voltage stability in a power system, which are; Analysis of system power flows, static voltage stability analysis, and dynamic voltage stability analysis. Simulated results on the tested power system are presented to illustrate the voltage stability issues and basic methods for analyzing the problem.
In the AC transmission system, the Unified Power Flow Controller (UPFC) has been used widely to overcome voltage instability [17]- [19]. By optimizing the UPFC location, followed by applying the Equivalent Current Injection power flow model in Power System Analysis, the voltage stability index decreased [20]- [22]. Therefore, the voltage stability margin will improve.
Besides UPFC, Power System Stability (PSS) can be applied to improve voltage stability to some extent. The use of PSS on the 11 bus system not too influential to dynamic stability.
One of the research proposed Continuous Power Flow (CPF) to analyze voltage stability in the micro-grid system [10], [23]- [26]. The relationship between Distribute Generator placement and its effect on the voltage stability can also explain using this method.
Previous studies described previously were conducted to observe voltage stability in only one state. In this study, voltage stability under normal, static, and dynamic conditions was investigated. Power flow analysis is performed to see the voltage drop on each bus for normal conditions [27]. Static voltage stability analysis is carried out to observe the voltage stability when the system experiences a periodic load increase. Meanwhile, dynamic voltage stability analysis conducted to see the voltage stability when the system is experiencing disturbance.
All analyses conducted on this research were performed using the PSAT software in MATLAB, and the systems tested

A. Transmission Line
In the electric power network, the interconnection system is common because the utilization of network interconnection will increase reliability and can enlarge the resulting power supply. The interconnection system is the development of a network or mesh system. This system distributes the electric power of some of the desired power plant centers. It works in parallel so that the transmission of electrical power can take place continuously (unbroken) despite the high load density in one or several points/areas within the network. In this interconnection system, when one of the power plant centers is damaged, electricity transmission can be transferred to another plant center. For the center of plants that have small capacity is usually used as a helper of the main plant center that has a larger power capacity. An example of an interconnection system of electrical power systems can be seen in Figure 1.

B. Power Flow
The study of power flow is essential to a mathematical approach to determining the large and voltage angles on each bus, with the voltage being discovered, hence the active power (P) and the reactive power (Q) can be calculated. If P and Q on two buses are known, the power flow can be identified, including the loss-loss that is on the connected channel.
Three methods are often used in the calculation of power flow. For large-scale electric power systems, the Newton-Raphson method is a more efficient and practical method, where the number of iterations needed for fewer calculations is compared to other methods. In this method, the equation of power flow is formulated in the polar form [28]- [31].
Active and reactive power on the bus-i are: By addressing (1) to (2) will be derived equations: The real and imaginary parts of the (3) are then separated The above equation was developed from the Taylor series as in the following equation. [ The calculations for the new phase angle and bus voltage are:

C. Static Stability Analysis
The analysis of static stability (during steady-state) is usually used in the voltage instability problems caused by small disturbances, such as a load splash. The main study method used for static analysis is a simulated power flow. This method is divided into two, namely the P-V curve and the Q-V curve. Both methods can determine the load limit on the steady-state associated with voltage stability.
The P-V curve is a curve describing the relationship between the active power (P) and the voltage (V). The P-V curve is obtained by conducting a simulated power flow where the load changes. This simulation will result in magnitude values of voltage and active power on a constantly changing bus forming a curve-shaped "nose" or nose. In Figure 2, it appears that the larger the active power is supplied on a system, the system voltage will be decreased. The active power supplied by the system starts at a value of 0 PU, and the voltage is at one pu. Then, when active power is added, the voltage will begin to decline. It continues to occur at a point called a critical point, where if the addition of power passes through this point will cause tension instability.

D. Dynamic Stability Analysis
Dynamic analysis is usually done by interfered with disruptionsa significant disruption to the system, including Transient simulation is performed for various scenarios of interference, and system behavior must be observed. If the system voltage returns to an acceptable value after clearing the disorder, the system is stable, but otherwise, the system voltage is unstable. Simulation needs to be done using Time Domain Analysis as the observation time can occur very briefly.

E. Continuous Power Flow
The Jacobian matrix of the power flow equation becomes singular at the voltage stability limit. Continuous Power Flow (CPF) solves this problem. CPF discovers a load-flow solution that participates in compliance with load scenarios. This method consists of predictive steps and corrections. From a known basic solution, a tangent predictor is used to estimate the next solution for a given load-pattern lift. The correction step then determines the appropriate solution using the Newton-Raphson technique used by conventional power streams. After that, a new prediction is made for a specific load increase based on the new tangent vector. Then the correction step is applied. This process lasts until a critical point is reached. An illustration of the Predictor scheme is depicted in Figure 3. The power injected into the system for the I-Bus on the Nbus system can be written as follows: The G and D subscripts in the equation signify the generation and demand for load in-between participating for the corresponding bus. The load-λ parameter is inserted into the power request PDi and Q to simulate a load change, Di.
PDi0 and QDi0 are the initial load requests on the i-bus, and the PΔbase and QΔbase are the specific power values selected to scale the λ properly. After addressing the equation (12) to the equation (11), the new equation can be represented as follows: Where θ is a vector of bus voltage angles, and V is a vector of bus voltage magnitude. The primary solution for λ = 0 can be obtained through the power stream. Then, the CPF and parameterization processes are applied.

F. PSAT (Power System Analysis Toolbox) on MATLAB
Power System Analysis Toolbox (PSAT) is the toolbox of MATLAB for the analysis and simulation of electrical power systems. All operations can be accessed using a graphical user interface (GUI), and a SIMULINK-based library provides the tools needed to design the network. Key features of PSAT include Power Flow (PF), continuous power Flow (CPF), optimal power flow, small-signal analysis (SSA), timedomain simulation, FACTS modeling, angina turbine models, and data conversion from multiple Format. Export results from this tool can be done to Microsoft Excel files and latex files.

A. System Modeling
This type of research is an application of experimental research that is applicative so that from the scope of problems can be conducted the assessment with the method of literature studies, data collection methods (both in literature and field) followed by Design and modeling systems.
In this study, simulations were performed using PSAT software 2.1.10 (Power System Analysis Toolbox) in MATLAB R2018a. The network topology used is the IEEE 6 Bus network system. Figure 4 is an IEEE 6 Bus network system image. The system consists of 3 bus generators (PV generators on bus 2 and 3, while bus one is installed slack bus as a reference bus). There are 11 transmission lines on this network system that connects each of the buses. The network-mounted loads are on buses 4, 5, and 6.

B. System Parameters
In this research, the parameter needed in the system simulation is generator data, bus data, and transmission line data. These data can be seen in Tables 1, 2, and 3, respectively.  The network topology used in this study is the IEEE 6 Bus network system, as shown in Figure 4.

C. Research Procedure
Based on the data that has been obtained and the system modelling that has been done, power flow simulation, static voltage stability analysis, and dynamic voltage stability analysis can then be performed using PSAT 2.1.10 / MATLAB R2018a software. Figure 5 shows the stages in analyzing the voltage stability of the power system with the continuous power flow method and time-domain analysis.

A. Load Flow Analysis
Power flow simulations are performed using the Power Flow menu in PSAT 2.1.10 / MATLAB R2018a. The Newton-Raphson method is used in this simulation. With a maximum iteration of 20 times and 10 -5 tolerance. Table 4 shows the results of power flow simulations performed on the IEEE 6 bus network system.  The table shows that the most significant voltage drop occurred on bus five, which is 1.5%, but the whole bus is still in normal condition because the voltage drop is not more than 10%. The total active power (P) and reactive power (Q) generated by the system are 217.88 MW and 179.95 MVAR respectively, while the total active power (P) load and reactive power load (Q) drawn by the system respectively 210 MW and 210 MVAR. Table 5 Table 6, it can be observed that the total active and reactive power losses for each load bus are 2.59MW / 7.36MVAR, 2.67MW / 7.89MVAR, and 1.59MW / 6.69MVAR for buses 4, 5, and 6, respectively. Bus 5 has the highest power losses compared to other buses because this bus has the highest line impedance (as shown in Table 3), which congruent with the fact that bus 5 has the highest voltage drop.

B. Static Stability Analysis
Static voltage stability analysis can be done by observing the PV curve for each load bus in the system. The simulation carried out to obtain this PV curve is by carrying out continuous power flow (CPF), which is a continuous power flow analysis continuously on a system with an increasing load changing value. In this study, CPF simulations are performed with the CPF menu in PSAT 2.1.10 / MATLAB R2018a software, with the basic load used is the load when performing a previous power flow analysis. Figure PV curve results from the simulation can be seen in Figure 5. The load margin (λ) value obtained from the simulation is 1.6130. A load margin is a measure or distance of the fundamental closeness of a system to voltage collapse. λ max = 1.6130 means the system can bear the maximum load increase of 160.13% of the initial load. It can be said that if the system has a larger load margin means that the system has good voltage stability against small disturbances (increased load).
From equation (12), the maximum load that can be accommodated by the system (for each load) before a voltage collapses or a voltage drop outside the allowable limit is: Based on the PV curve in Figure 5, it can be noted that the bus that has the lowest level of voltage stability is bus 5. As explained earlier, this is because each line leading to bus 5 has the highest impedance when compared to the line to the other load bus, so the power losses become highest. With significant line losses, the voltage drop is even greater.

C. Dynamic Stability Analysis
Dynamic voltage stability simulation is performed using the Time Domain menu on PSAT 2.1.10 / MATLAB R2018a. The simulation is carried out with two types of interference: (1) line termination; (2) disturbances or faults in the load bus. The simulation is done by terminating line 1 to line 11 on the system one by one in a row. Changes in voltage values for each load bus after line termination can be seen in Table 7.   Table 7, it can be seen that from all the scenarios carried out, there is no voltage drop, which is below 10% of its normal value. Therefore, indicates that the system voltage is still stable against major disruptions in the form of line termination. However, it can be observed, the greatest voltage drops of all line disconnection scenarios carried out is in termination of lines 5 and 9, which results in a voltage drop of 9% on bus 4 and 8% on bus 6.
In the case where line 5 is disconnected, there is a large voltage drop on bus 4 (9%) due to the largest reactive power supply line injected into bus four is cut off. From the power flow analysis conducted previously (table 4.6), it is known that line 5 supplies reactive power of 46.05 MVAR, or at 66% of the total bus reactive power requirements 4 (70 MVAR). This loss of reactive power supply is compensated as much as possible by the supply from generator one and other generators through the line that passes bus 5, causing a higher voltage drop.
The same thing occurred when line 9 is disconnected of which caused a voltage drop on bus 6 of 8%. In this case, the interrupted reactive power supply is 60.72 MVAR or 86% of the reactive power requirements on bus 6. Loss of this reactive power supply is compensated as much as possible by the supply from generator two and other generators through the channel passing bus 5, causing a greater voltage drop. The voltage dynamic stability profile for load bus when line 5 interrupted is shown in Figure 6. Observation of dynamic voltage stability behavior is carried out by simulating the giving of faults or disturbances to the load bus one by one and not simultaneously. Termination of interruption on the bus is assumed to occur at t = 1 second and fault clearing time at 0.2 seconds and 0.4 seconds. Transient stability analysis for each bus is observed. For bus disturbances that occur on buses 4, 5, and 6 with fault clearing time within 0.2 seconds, a nominal voltage can be seen in Table 8, with a representation of the dynamic voltage stability behavior depicted in Figure 7. For bus interruptions that occur on bus 4 with a fault clearing time of 0.4 seconds, the nominal voltage can be seen in Table 9, with a description of its dynamic voltage stability behavior that can be seen in Figure 8.
The result shows that after experiencing a disruption with a longer clearing time, the system cannot return to its stable state. With bus 1 experiencing a voltage collapse and other buses experiencing a very significant voltage drop, which indicates that the transient stability of the system is at a critical breaker time below 0.4 seconds. For disturbance on Bus 5, assuming a fault clearing time of 0.4 seconds, the voltage behavior for each bus is almost the same as if there is a disturbance on bus 4. However, for interference on Bus 6 assuming a fault clearing time of 0.4 seconds, the simulation experiences a singularity due to the system voltage collapses on each bus so that the system cannot be returned to its operating state. The results showed the value of the load margin (λ) is 1.6130, means that the loading value that can be applied to a load bus before the voltage on the bus experiences a voltage collapse is 160.13% of its initial load, i.e., 181.91 MW with a power factor of 0.707.
Simulating system interference performed the dynamic evaluation of voltage stability. Also, a study of the time domain is carried out because the measurement process will take place very briefly. The results of the simulation show that the tested power system has a clearing time of under 0.4 seconds