Post-Contingency Loading Margin through Plane Change in the Continuation Power Flow

Abstract

This paper introduces a parameterization technique to obtain complete P-V curves for electrical power systems under various conditions, including contingencies (N-1, N-1 severe, and N-2) as well as normal operating conditions N-0 (pre-contingency). The distinct feature of this methodology lies in the utilization of the new (θ_k, V_k) plane for constructing the P-V curve. In this new framework, the solution trajectory exhibits a linear characteristic in the vicinity of the maximum loading point (MLP), effectively mitigating any numerical challenges. Another crucial aspect is the point denoted as the center of the set of lines, point O (λ⁰, V_k⁰). This point can be adjusted along the ordinate axis, further enhancing the method’s ability to achieve a smoother ‘flat start’. The ‘flat start’ approach employed in the proposed methodology enables the determination of the loading margin for any system subjected to both simple and severe contingencies. Additionally, it facilitates the use of larger steps in acquiring complete P-V curves with a reduced number of iterations. A procedure presented is to not update the Jacobian matrix at each iteration but only when the system undergoes significant changes. This technique reduced the CPU Ratio by 29.74% in obtaining the P-V curve for all scenarios.

1. Introduction

Electricity is an essential benefit worldwide, and continuity and quality of its supply are fundamental to our quality of life [1]. Nowadays, with the increasing demand, along with the deregulation of the power sector and policies restricting the construction of new transmission lines and hydroelectric plants, electric power systems operate close to their operational limits, near the maximum loading point (MLP). Not to mention the lack of rainfall, which often leads to energy rationing, especially in countries where the majority of their energy comes from hydropower, such as Brazil, where 51% of the demand comes from hydroelectric power [2].

Systems operating close to their operational limits are subject to a higher number of contingencies. In this context, security analysis is essential to identify contingencies that can affect the system. An electrical system is exposed to a large number of contingencies, but few are severe enough to cause instability [3].

Voltage stability analysis is a crucial component in the management and operation of electrical power systems. This refers to the ability of an electrical system to maintain voltage stability within acceptable limits during operational variations or disturbances. Voltage stability is vital to ensure reliable system operation, preventing the occurrence of failures, blackouts or damage to equipment [4]. Modern electrical systems are subject to a number of challenges, including the introduction of intermittent power sources such as wind and solar power, the transmission of power over long distances, variation in power demand, and the increasing complexity of interconnections between different regions. Voltage stability analysis, which involves P-V (real power-voltage) and Q-V (reactive power-voltage) curves to determine the loading margin, is the main tool in contingency studies. The continuation power flow (CPF) with parameterization techniques [5,6,7,8,9] is the method used to obtain these curves, as they can generate the entire P-V curve by using the appropriate parameter.

Ref [5] introduces efficient correction techniques based on Newton-like methods to expedite continuation power flow, showcasing their superiority over the conventional Newton– Raphson method in terms of computational time. A parameterization to continuation power flow is presented in [6], proposing a coordinated combination of methods to enhance convergence. Ref [7] applies the continuation power flow method to voltage stability analysis, comparing results between Matlab and the Power System Analysis Toolbox and discussing the effectiveness of three-dimensional visualization. Ref [8] addresses precision issues in the holomorphic embedding method and suggests a multi-stage scheme to refine the P-V curve. Lastly, [9] presents an alternative methodology for shunt reactive power compensation allocation, aiming to reduce total reactive power losses, improve voltage profiles, and increase the loading margin. In the literature, several studies have been conducted on contingency analysis [10,11,12,13]. It is known that obtaining all possible contingencies of the system is impractical, so the idea is to have a fast method that simulates as many contingencies as possible.

Ref [10] provides an overview of methods for steady-state security analysis in power systems, discussing full AC power flow, approximate methods, and contingency ranking, while emphasizing techniques to improve computational efficiency. An efficient method for selecting multiple contingencies using genetic algorithms with high accuracy is presented in [11]. Ref [12] focuses on computational efficiency in contingency analysis, proposing a novel approach that significantly reduces computation time by reusing symbolic factorization, yielding impressive results in power system security assessment. In [13], a method via artificial neural networks was proposed to obtain the complete P-V curves of electrical power systems subjected to double contingencies (N-2). Results similar to those desired were obtained by the network. Methods aimed at reducing computational time have also been applied to improve electrical systems [14,15,16,17].

The Western Electricity Coordinating Council (WECC) [18] requires its companies to maintain a safe loading margin of 5% for real power in any single contingency (N-1), and for double contingencies (N-2), the loading margin is 2.5% for real power.

In this context, this work presents a new parametrization technique for contingency analysis with slight modifications to Newton’s method. The technique involves a change in the plane for plotting the P-V curve, resulting in a linearized solution trajectory around the MLP and thereby avoiding the Jacobian matrix singularity problems. The performance of the method using the parameterization techniques is assessed by tracing the P-V curves of the IEEE 14, 57 and 300-bus system, of a 638-bus system corresponding to part of South-Southeast Brazilian system, and of a 904-bus Southwestern American system.

2. Proposed Methodology

Commonly, the (λ-V) plane (with λ representing the loading factor) or the (P, V) plane is used for plotting the P-V curve. In this work, we will use the (θ-V) plane, which represents the angle versus magnitude of nodal voltages (Figure 1). The equations for the proposed technique are presented as follows:

$$\begin{array}{l}\mathbf{H}(\mathbf{\theta },\mathbf{V},\mathsf{\lambda })=\mathbf{0}\\ {\mathbf{T}}_{\mathbf{1}}(\mathbf{\theta },\mathbf{V},\mathsf{\lambda },\mathrm{m})=\mathrm{m}(\mathsf{\lambda }-{\mathsf{\lambda }}_{}^0)-({\mathrm{V}}_{\mathrm{k}}-{\mathrm{V}}_{\mathrm{k}}^0)=0\end{array}$$

(1)

and

$$\begin{array}{l}\mathbf{H}(\mathbf{\theta },\mathbf{V},\mathsf{\lambda })=\mathbf{0}\\ {\mathbf{T}}_{\mathbf{2}}(\mathbf{\theta },\mathbf{V},\mathsf{\lambda },\mathrm{m})={\mathrm{m}(\mathsf{\theta }}_{\mathrm{k}}-{\mathsf{\theta }}_{\mathrm{k}}^0)-({\mathrm{V}}_{\mathrm{k}}-{\mathrm{V}}_{\mathrm{k}}^0)=0\end{array}$$

(2)

wherein: H is the basic equation of the continuation power flow, T₁ and T₂ are the linear equations for the (λ-V) and (θ-V) planes, and m is the angular coefficient of the straight-line equation.

Expanding in Taylor series, the following equations present the matrix form for solving systems of Equations (1) and (2), as shown in Figure 2:

$$\left[\begin{array}{ccc}{G}_{\theta }& G_V& G_{\mathsf{\lambda }}\\ [\dots ,0,\dots ,0,\dots ]& [\dots ,0,\dots ,-1,\dots ,0,\dots ]& \mathrm{m}\end{array}\right]\left[\begin{array}{c}\mathsf{\Delta }\theta \\ \mathsf{\Delta }V\\ \mathsf{\Delta }\mathsf{\lambda }\end{array}\right]=J_m\left[\begin{array}{c}\mathsf{\Delta }\theta \\ \mathsf{\Delta }V\\ \mathsf{\Delta }\mathsf{\lambda }\end{array}\right]=\left[\begin{array}{c}\mathsf{\Delta }P\\ \mathsf{\Delta }Q\\ \mathsf{\Delta }W\end{array}\right]$$

(3)

and

$$\left[\begin{array}{ccc}{G}_{\theta }& G_V& G_{\mathsf{\lambda }}\\ [\dots ,0,\dots {\mathrm{m}}_{\mathrm{k}}\dots ,0,\dots ]& [\dots ,0,\dots ,-1,\dots ,0,\dots ]& 0\end{array}\right]\left[\begin{array}{c}\mathsf{\Delta }\theta \\ \mathsf{\Delta }V\\ \mathsf{\Delta }\mathsf{\lambda }\end{array}\right]=J_m\left[\begin{array}{c}\mathsf{\Delta }\theta \\ \mathsf{\Delta }V\\ \mathsf{\Delta }\mathsf{\lambda }\end{array}\right]=\left[\begin{array}{c}\mathsf{\Delta }P\\ \mathsf{\Delta }Q\\ \mathsf{\Delta }W\end{array}\right]$$

(4)

Figure 1 displays the geometry of the method, while Figure 2 show the flowchart for the (λ-V) and (θ-V) planes, respectively. The novelty of this method lies in the new plane for plotting the P-V curve, where the new solution trajectory has a linear aspect around the MLP. In Figure 1, in the λ-V plane, it is observed that the maximum loading point (MLP) or critical point (point T) is located at the vertex of the λ-V curve, which presents an issue, as the Jacobian matrix is singular at this point. However, when switching to the θ-V plane, this problem no longer occurs, as the MLP (point U), corresponding to the T point in the λ-V plane, is situated in a linearized region of the θ-V curve, demonstrating the robustness of the proposed method. Another key factor is the point represented by the center of the set of lines, point O (λ⁰, V⁰), which can be adjusted along the ordinate axis. In Figure 1a, point O assumes the value (0, 0), but another value can be chosen, such as (0, 0.5), as shown in Figure 1b. This value was chosen because the MLP of the critical bus of all studied systems has a voltage magnitude of around 0.5 pu.

The methodology could obtain the entire P-V curve through the presented flowchart without any problems related to the Jacobian matrix.

The initial bus selected for plotting the P-V curve was bus 14 of the IEEE 57 system (chosen arbitrarily). The technique starts from a flat-start with the parameter value (the angular coefficient) set to 1.0 (45°), and with a step size Δα = 0.15 (tripled), the complete P-V curve was obtained. The step size was tripled to demonstrate the robustness of the method in plotting the P-V curve, but the commonly adopted step size is 0.05. After five calculated points (which can be fewer depending on the trajectory of the initially chosen curve) in the (λ-V) plane, the plane is switched. The new plane (θ-V) is represented by the variables θ_k and V_k of the bus that showed the lowest voltage magnitude value in the last calculated point of the first set of lines, the (λ-V) plane. The identified bus was bus 31, meaning that the switch was made to the (θ₃₁, V₃₁) plane.

Another distinctive feature of the proposed methodology is that when choosing the point O (0, 0.5) for plotting the P-V curves, there is an automatic reduction in the initial value for plotting these curves. This can be observed in Figure 1a, where the flat-start with m = 1.0 found a solution at λ = 1.0 for the loading factor (point a). In Figure 1b, this value of point a was reduced to almost half, i.e., λ = 0.5 for the flat-start with m = 1.0, resulting in exceptionally superior performance for obtaining a post-contingency loading margin, especially for systems with N-2 contingencies. For example, in the case of the contingency between buses 1 and 2 in the IEEE 14-bus system, a negative loading margin (−0.0191) was obtained, as shown in Figure 3. The results were obtained using Matlab^® software [19].

3. Test Results

For all studied systems, the threshold value for the power mismatch criterion was set to 10⁻⁵ pu. The control of limits from the reactive power (Q) at PV generation buses follows the same procedure as the conventional power flow method.

In Figure 3a, the initial step size and the point representing the center of the set of lines were 0.05 and (0, 0.5), respectively. Bus 2 was chosen arbitrarily for plotting the P-V curve. It can be noted that after five points, a switch was made to the (θ₁₄, V₁₄) plane, obtaining the rest of the P-V curve without any Jacobian matrix-related problems. This demonstrates that a technique that combines robustness with simplicity can be developed with only minor changes in Newton’s method. The number of iterations is shown in Figure 3b, and the MLP was obtained in two iterations.

Figure 4 shows the results for the IEEE 57-bus system under a severe N-1 contingency stage (an outage of the transmission line between buses 25 and 30) with a negative post-contingency loading margin (−0.0237). It means that the loading factor λ for the post-contingency MLP (0.9763) is lower than the value for the base case (λ = 1.0). The step size adopted was Δα = 0.15 (tripled), demonstrating the effectiveness of the method for larger step sizes. Again, the methodology proved to be efficient in plotting the P-V curves for the IEEE 57-bus system. The new plane used to complete the rest of the P-V curve was (θ₃₀, V₃₀), and a very linear behavior can be observed throughout its trajectory, especially around the MLP, eliminating any numerical issues related to the Jacobian matrix singularity in the conventional Newton’s method.

Flat Start Applied to the Proposed Methodology

The method starts with the flat-start values, where all load bus voltages are initialized to unity magnitude and all phase angles are set to zero. Figure 5a presents multiple P-V curves for different N-1 (an outage of the transmission line between buses 2 and 3) and N-2 (an outage of the transmission lines between buses 1 and 2) contingencies in the IEEE 14-bus system. It can be noted that for the first curve, using the proposed method with a flat-start and α1 = 1.0, there were no issues in obtaining the P-V curves for any applied contingency, even for cases where the curve showed a reduction compared to the base case, resulting in a negative loading margin (post-contingency P-V curve (V_pos) for N-2), as shown in Figure 5b.

Figure 5c,d show the same behavior for the IEEE 57-bus system, where the proposed method successfully obtained all complete P-V curves for any applied contingency, even with negative loading margins, including severe N-1 contingencies. Many methods proposed in the literature [20] would fail for these severe contingencies. For this work, N-0 represents a normal operating system, i.e., without contingencies.

Figure 6a below shows the performance of the proposed method for the flat-start with a different point O (0, 0.5). It can be observed that for any applied contingency, whether simple or severe, the methodology had no issues in obtaining all P-V curves for the IEEE 14-bus system. Another value of m = 0.4 was also presented for the initial line’s angular coefficient (flat-start), and the method would find a solution regardless of the applied contingency. The same applies to the IEEE 57-bus system, where all complete P-V curves were obtained without any singularity issues, regardless of the applied contingency, as shown in Figure 6b.

In this section, the performance of the proposed methodology is compared with the local parameterization technique proposed by [20] for the tripled step size and the number of iterations under normal operating conditions (N-0). In the local parameterization method, the parameter switch occurs for the V_k variable that shows the highest variation in the tangent vector of prediction. As a tripled step size of m was used, i.e., m = m + Δm, where Δm = −0.15, it can be seen that the local parameterization method failed, as shown in Figure 7a for the IEEE 14-bus system and Figure 7e for the IEEE 57-bus system. At point “a”, the λ variable still shows the highest variation in the tangent vector of prediction, and in this case, as the correction process is vertical, there is no intersection of this vertical trajectory with the desired P-V curve, making it impossible to use for larger step sizes. The number of iterations is shown in Figure 7b for the IEEE 14-bus system and Figure 7f for the IEEE 57-bus system.

It can be noted that for the local parameterization technique, only four points (IEEE 14-bus system) and three points (IEEE 57-bus system) were calculated, respectively, before reaching the MLP. All P-V curves of the IEEE 14-bus and 57-bus systems subjected to N-1 contingencies are shown in Figure 7c,g for the proposed methodology and Figure 7d,h for the local parameterization technique, which failed to obtain the curves.

Figure 8a presents the results for the IEEE-14 system with tripled step size (Δα = −0.15) and subjected to N-1 contingency (an outage of the transmission line between buses 2 and 3). It can be observed that the local parameterization method also fails (Figure 8d) to obtain the loading margin of the system under the contingency. However, for the proposed technique, there were no issues in obtaining the P-V curves, as shown in Figure 8c. The number of iterations used is presented in Figure 8b, where only two points were obtained by the local parameterization method.

In power stability studies, the main expected characteristics of the power flow method used are robustness and computational efficiency. In this regard, continuation power flow methods using the Newton–Raphson algorithm have proven to be the most robust. In these algorithms, a set of nonlinear equations is linearized around the current solution at each iteration, and the state update is obtained by solving these equations. Therefore, in this process, the elements belonging to the Jacobian matrix are updated at each iteration. However, after several studies, it was concluded that the Jacobian matrix is important for the convergence of the process but does not influence the final solution. Thus, in terms of computational efficiency, a commonly used procedure in power flow methods is to not update the Jacobian matrix at each iteration but only when the system undergoes significant changes (e.g., when a bus changes from PV to PQ due to limit violations) or after the number of iterations exceeds a predefined iteration limit, which often leads to considerable processing time reduction. Regarding this context, two procedures are considered: in the first procedure (P1), the Jacobian matrix is updated at each iteration, and in the second procedure (P2), it is only updated when the system undergoes significant changes. The results of this comparison can be seen in

Table 1. Performance of the parameterization technique considering the angular coefficient (m) of the line located on planes (λ-V) and (θ-V) for procedures P1 and P2.

System	P1		P2			CPU Ratio (%)
	IC	CPU Time (p.u.)	IC	ACo	CPU Time (p.u.)
14 (1)	40	1.000	63	23	0.779	24.5
57 (2)	27	1.000	54	14	0.769	25.1
300	62	1.000	77	28	0.623	37.8
638	123	1.000	254	37	0.676	32.2
904	84	1.000	94	38	0.715	29.1

⁽¹⁾ N-2 contingency (outage of the transmission line between buses 1 and 2), ⁽²⁾ N-1 contingency (outage of the transmission line between buses 25 and 30). ACo—Actualization count.

below. For both procedures, the total number of iterations (IC) required for the complete plotting of the P-V curve is presented, and for P2, the total number of iterations (ACo) at which the matrix is updated is also shown. The computational time required by the proposed parametrization technique, considering the P2 procedure, is presented in the sixth column. The values were normalized by the respective times required considering the P1 procedure, which is shown in the third column of table.

The results show that although the total number of iterations is higher for the P2 procedure, it is possible to achieve a reduction in computational time, meaning an improvement in the efficiency of the proposed techniques without losing robustness. This is achieved with a simple change in the procedure, which is not updating the Jacobian matrix at each iteration but only when the system undergoes significant changes. The Jacobian matrix is updated at each iteration in the P1 procedure, while in the P2 procedure, it is only updated when the number of iterations exceeds eight.

4. Conclusions

This work presented a geometric parametrization technique for obtaining complete P-V curves of power systems subjected to contingencies (N-1, severe N-1, and N-2) and normal operating conditions (N-0). The novelty of this method lies in the new plane (θ_k, V_k) used for plotting the P-V curve, where the new solution trajectory has a linear aspect around the MLP, ruling out any numerical issues around it. Another crucial factor is the point represented by the center of the set of lines, point O (λ⁰, V⁰), which can be adjusted along the y-axis to further enhance the flat-start of the method. The flat-start used in the proposed methodology allowed us to obtain the loading margin for any system subjected to a simple or severe contingency and also the possibility of using larger steps in obtaining complete P-V curves. All P-V curves were obtained with a reduced number of iterations.

Another important aspect was not updating the Jacobian matrix at each iteration but only when the system undergoes significant changes (limit violations or reaching the maximum number of iterations). It can be observed that there was up to a 40% reduction in the plotting of the P-V curve compared to conventional continuation power flow techniques, which update the Jacobian matrix at each iteration.

The results also showed that a reduction in computational time and efficiency improvement are possible without losing robustness. This is achieved by updating the Jacobian matrix only when the system undergoes a significant change, achieving an average reduction (CPU Ratio) of 29.74% compared to the P1 procedure.