Continuation Power Flow Analysis of Power System Voltage Stability with Unified Power Flow Controller

Facing the rising demand for electrical energy, it is increasingly challenging for the power transmission systems to respond with high stability and good controllability. The current transmission infrastructure is already strained, trying to meet the growing power demand. Power system engineers are primarily concerned with the quality and quantity of the power supply. To ensure the quality of the delivered power, the system features a few control mechanisms that bring the temperature and stability constraints closer [1], [2].

The bus voltage may drop to a point where it cannot recover when a system is overloaded and already under stress. This can result in voltage collapse and, if ignored, system termination [3]. This viewpoint contends that improved levels of quality, dependability, and stability provided by new developments in power electronic equipment can be advantageous to the power system [4].

Over the past ten years, the power electronic has developed into an all-purpose solution to all problems and limitations. Flexible AC transmission systems (FACTS) based on power electronics are mainly converters or thyristor-controlled devices that manage the X-mission and distribution power flow as well as other system variable features like X-mission reactance, phase angle, and voltage [5]. The unified power flow controller (UPFC) is widely regarded as the most effective FACTS, thanks to its ability to control all three factors simultaneously [6].

With the development of UPFC concepts, it is important to devise new methods for power flow analysis on the power system with UPFCs. Several researchers have tried to improve voltage stability using FACTS devices. Pereira et al. [7] assessed the dynamic voltage stability for a power system, including shunt FACTS devices like static compensator (STATCOM) and static var compensation (SVC). Fei et al. [8] presented a coordinated control strategy for multi-objective voltage stability of series FACTS devices. Aydin and Gumus [9] developed a location-based algorithm to enhance the voltage stability of a power system using a shunt device.

This paper incorporates a series-shunt (hybrid) device in continuation power flow (CPF) analysis, and investigates the impact of the UPFC on voltage stability. After illustrating a CPF procedure, the authors put forward a UPFC load flow model. To measure the effect of loadability on system stability limit, a simulation was carried out using the power system analysis toolbox (PSAT) in MATLAB environment for a power system with and without the UPFC.

The UPFC, which consists of a static compensator and a static series compensation, concurrently functions as a phase-shifting device and a shunt-compensating device.

As shown in Figure 1, the UPFC is composed of a pair of converters, a shunt converter and a series converter. The two converters are connected by two voltage source converters and a common DC capacitor. Owing to the DC circuit, the active power exchange between the two converters controls the phase shift of the series voltage. This configuration assures the complete control over voltage and power flow. A thyristor bridge is required to protect the series converter. The number of real-world applications where both voltage and power flow control are required is constrained by the high cost of voltage source converters and protection, which makes a UPFC fairly expensive [10].

As shown in Figure 2 [11], the UPFC can be visualized as two ideal voltage sources: one voltage source V_se is connected in series to the transmission line, and the other voltage source V_shu injects the shunt current. Impedances in series with the voltage sources are used to mimic the coupling converter losses. Mathematically, the ideal voltage sources can be expressed as:

where, V_shu and δ_shu are the controllable magnitude and angle of the ideal voltage source of the shunt converter, respectively, which represent the shunt converter between the limits $0 \leq \delta_{s h u} \leq 2 \pi$ and $V_{\text {shumin }} \leq V_{\text {shu }} \leq V_{\text {shumax }}$ ; V_se and δ_se are the controllable magnitude and angle of the ideal voltage source of the series converter, respectively, representing the series converter between the limits $V_{\text {semin }} \leq V_{s e} \leq V_{\operatorname{semax}}$ and $0 \leq \delta_{s e} \leq 2 \pi$.

In the equivalent circuit of UPFC (Figure 2), the active and reactive powers at bus k can be respectively expressed as:

The active and reactive at bus m can be respectively expressed as:

For the series converter, we have:

For the shunt converter, we have:

where, G and B are the corresponding conductance and susceptance, respectively.

Considering the losses of the converter valve, the active power P_shu supplied to the shunt converter is equivalent to 1.02 of the active power P_se required by the series converter:

The linearized UPFC power equations are coupled with the equations of the AC network. Suppose bus m is a PQ bus. The UPFC involves the following variables: (1) voltage magnitude at the terminal of the shunt converter (bus k); (2) active power flow from bus m to bus k; (3) reactive power injected at bus m. Then, the linearized system of equations can be expressed as [12], [13]:

The most widely accepted assessment method for voltage collapse is the bifurcation theory. It is a suitable mathematical hypothesis capable of classifying stability issues. The theory can evaluate the system performance near failure or unstable points, providing quantifiable data on countermeasures to mitigate critical situations [14]. The bifurcation theory states that system formulas depend on several factors as well as model state parameters [15]:

where, $\rho$ is a state variable of the power system; $\lambda$ is the loading parameter.

Properties of stability or instability can be evaluated by gradually changing the settings. The loading parameter $\lambda$ is introducedo assess if a system is in danger of voltage collapse. The load values are changed as follows:

where, P_L0i and Q_L0i are the active and reactive powers at fundamental operating point of the bus, respectively; k_ip and k_iq are the load distribution factors.

Voltages are typically depicted as functions of, or the measures of system loadability. In bifurcation diagrams, they are illustrated by P-V or nose curves. Eqns. (14) and (15) can be applied to the CPF analysis [14].

The nose slopes of electrical transmission lines can be determined using CPF techniques, which are also useful for estimating scenarios with high loading and "critical" requirements (for example, saddle-node and points of bifurcation caused by limits). The CPF method used in this study can distinguish between a P-V graph's stable and unstable areas. It is unaffected by numerical instabilities. Furthermore, it can provide information like the sensitivity of the current resolution to important parameters [14].

In static and dynamic voltage stability investigations, the CPF, which from a mathematical standpoint is a homotopy technique [14], can examine the stability of power system formulas, when a system variable is changed. It is commonly the loading parameter.

Typically, the CPF consists of two stages: the predictor stage, which is attained through the calculation of the tangent vector, and the calibration stage, which can be attained through either a local characterization or a perpendicular crossing.

A typical CPF approach with predictor and corrector phases is shown in Figure 3 as an iterative process.

By means of a step that employs a tangent predictor, an estimation of the resolution point B for a given load route denoted by is carried out starting from a recognized initial position A. After that, using the corrector step, a power flow and an additional formula are used to establish the precise solution C.

This procedure is repeated until the desired bifurcation graph or P-V curve is obtained.

Our simulation targets the IEEE 14-, 30-, and 57-bus systems. Specifically, the 14-bus system has 20 lines and 5 generators, the 30-bus system has 41 lines and 6 generators, and the 57-bus system has 80 lines and 7 generators [16]. The UPFC location was taken from earlier investigations [17], [18].

The CPF was performed on the test systems with and without the UPFC to see the effect of such device on voltage stability (Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9).

From Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, it can be seen clearly that the loading parameter λ is greater for the test systems with UPFC than those without UPFC. The reason is that the maximum loading point is pushed further by using the UPFC, which thus enhances the voltage stability. Table 1 lists the loading parameters values with and without UPFC for the test systems.

The maximum loading parameter before the system loses stability is shown in Table 1. The higher the value, the more voltage stability margin the system has. It is evident that the loading parameter value is higher in all test systems with the UPFC, and that as test system size is increased, the difference in values becomes less pronounced. The load is multiplied by the loading parameter each time, and the greater the load, the greater the system's capacity to deliver additional load without losing stability.

The amount of additional load that can be added to the 14-bus test system without it losing stability is increased by 16% utilizing the UPFC, the amount of power is improved by 15% for the 30-bus test system, and the amount is reduced by 7% for the 57-bus test system.

This study carries out the CPF analysis to evaluate the impact of UPFC on the voltage stability of power systems. Firstly, the modeling technique is presented through the modified power flow equations and Jacobian matrix. To determine the point of voltage collapse, the model is implanted through the CPF analysis, which multiplies the loading parameter by the load active and reactive powers. Next, the efficiency of the UPFC in improving voltage stability is validated through a simulation on numerous IEEE test systems. The simulation shows that the UPFC pushes further the point of collapse, allowing the power system to supply more power and bear greater loads.