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The paper presents an optimal power flow (OPF) formulation using an AC power system model based on current nodal analysis. One of the principle advantages of the proposed formulation is all components of the OPF problem are quadratic, which results in a constant Hessian matrix. The proposed model is compared to the traditional models using interior...

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**Context 1**

... relationship between the currents defined in equa- tions (7) and (8) and the nodal voltages which will be the principle variables in the OPF formulation are easily de- fined using traditional techniques. For example, the trans- mission line branch currents (as illustrated in Figure 1) can be written as: For the proposed formulation, the relationship be- tween the complex current associated with transformers and the bus voltages is derived by modeling the trans- former as an ideal transformer in series with a series impedance as illustrated in Figure 2. To ensure that power system model remains quadratic, two additional current and voltage variables are introduced as follows: are introduced to simply the equations describing the current voltage relationship at the sending and receiving buses. ...

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## Citations

... More recent works employing current injection variables to construct the ACOPF problems include [4], wherein it is argued that the resulting structure of the Hessian matrix for the constraint equations will ensure a convex constraint set. However, that work requires all buses in the power flow formulation to be PQ buses, which may provide a poor representation of the operational characteristic of generator buses. ...

This paper examines three different formulations of AC optimal power flow problem, and compares performance of well-established, general purpose optimization algorithms for each, over different initial conditions. Polar power-voltage, rectangular power-voltage and rectangular current-voltage are formulated to evaluate ACOPF solution characteristic. The formulation here maintains line flows as explicit variables, and employs summations of these quantities to impose conservation conditions at each node. Two representations of line thermal limits are considered, one using real power (to allow comparisons to DC power flow approximations), and a more physically-based, ampacity limit using current magnitude. Nonlinear generator capability curves are represented (“D-curves”), including options to allow active and reactive limits dependent on generator voltage. A uniform objective function is used throughout, that of minimizing quadratic generator operating cost curves. Numerical performance case studies are performed for these formulations over six different classes of initial conditions, evaluating computational time and also robustness of convergence.

... PCIPA differs from IPA by equation (25), which introduces the second order nonlinear terms ∆W · ∆Z · ê and ∆W · ∆Z · ê which can be by the predictor and corrector steps in [14]. The OPF algorithm can be summarized in the flow chart of the PCIPA in fig. 2. Thus the objective function and the constraints are modeled properly with linear or quadratic functions [15] and [12] then control and state variables can be avoided in forming the Hessian matrix, else these variables would have been involved besides the Lagrangian multipliers while forming a general OPF problem. ...

... The OPF algorithm can be summarized in the flow chart of the PCIPA infig. 2. Thus the objective function and the constraints are modeled properly with linear or quadratic functions [15] and [12] then control and state variables can be avoided in forming the Hessian matrix, else these variables would have been involved besides the Lagrangian multipliers while forming a general OPF problem. Yes NoFigure 2. Flowchart of PCIPA To further enhance the performance of [11], following assumptions are made for PV buses. 1) | | cos | | sin and | | 1.0 2) Network has low R/X ratio that is 3) | | The reduced OPF problem can now be decomposed into two suboptimal problems as 1) Active sub problem with active variables ( , ) and associated constraints. ...

This paper presents the comparative analysis of conventional optimal power flow methods — Newton Raphson method, Fast Decoupled Load Flow method with the Optimal Power Flow Techniques based on Equivalent Current Injection. With the use of Predictor Corrector Interior Point Algorithm (PCIPA), an Equivalent Current Injection based OPF models — Equivalent Current Injection Optimal Power Flow, Decoupled Equivalent Current Injection Optimal Power Flow, and Fast Decoupled Equivalent Current Injection Optimal Power Flow — are proposed and analyzed in this paper. The algorithm has been tested on IEEE 9 bus, 14 bus and 30 bus networks. The minimization problem based on equivalent current injection has been compared with the conventional Optimal Power Flow techniques. The result shows that, the previous ones converge nicely compared to conventional methods.

... We can get a nearly constant Jacobi with a few generator buses still state-dependent and need to be updated at each iteration. Pioneering the rectangular-form current-based OPF, [16] did a brief test with rectangular nodal voltages and branch currents used for state variables. The generator PV problem was avoided by replacing the PV bus with real and reactive power (PQ) directly ; however, the oversimplification by replacing PV with PQ is not a common practice in handling generator buses. ...

... The generator PV problem was avoided by replacing the PV bus with real and reactive power (PQ) directly ; however, the oversimplification by replacing PV with PQ is not a common practice in handling generator buses. Besides, using KCL in [16], it was not even mentioned how load and generator power injections are handled for each iteration, which are the key factors affecting convergent behaviors in developing a current-based model. Reference [17] developed a rectangular voltage OPF, but the power flow equations are still PQ based, not current. ...

... The OPF algorithm can be summarized in the flowchart of the PCIPA inFig. 2. Numerical Advantage: Taylor series expansion of a quadratic function terminates at the second-order term with no truncation error, that is (28) From (28), it can be seen that if the objective function and constraints can be modeled properly with linear or quadratic functions [16], [17], control and state variables can be avoided in forming the Hessian; otherwise, a general OPF problem I E E E P r o o f could have these variables involved besides the Lagrangian multipliers. Note that from (17) and the Jacobi in (4) and (10), the proposed method has a nearly constant Jacobi , except for a few elements of the generator PV buses which need to be updated, while the traditional Newton–Raphson OPF has a state-dependent , which needs to modify all elements at each iteration and is time-consuming. ...

An equivalent current injection (ECI) based hybrid current-power optimal power flow (OPF) model is proposed in this paper and the predictor-corrector interior point algorithm (PCIPA) is tailored to fit the OPF for solving nonlinear programming (NLP) problems. The proposed method can further decompose into two sub-problems. The computational results of IEEE 9 to 300 buses have shown that the proposed algorithms can enhance the performance in terms of the number of iterations, memory storages and CPU times.

... I N RECENT years, the predictor corrector primal dual interior point method (PCPDIPM) has been extensively applied to solve large-sized optimal reactive power flow (ORPF) prob- lems [1]–[7], [10]–[12] due to its faster calculation speed and robustness, etc. The conventional ORPF model in polar coordinates is a higher order problem [5] . ...

... Although the voltages in rectangular coordinates are used in [3], the optimal power flow (OPF) formulation is not completely quadratic because of the presence of tap ratio variables in the load tap changing (LTC) branch power equations. A fully quadratic formulation of OPF is proposed in [12]. In that paper, the authors used the current and voltage equations to establish the OPF model in a rectangular form. ...

... Thus, a new quadratic model for the ORPF problem in a rectangular coordinate is developed. Although the introduction of the dummy nodes will still result in an increase in the number of constraints and variables of the ORPF, this increase is much less in comparison to that in [12]. The test results demonstrate that the emergence of a constant Hessian in the proposed ORPF model greatly reduces the total execution time of the PCPDIPM solution. ...

A new optimal reactive power flow (ORPF) model in rectangular form is proposed in this paper. In this model, the load tap changing (LTC) transformer branch is represented by an ideal transformer and its series impedance with a dummy node located between them. The voltages of the two sides of the ideal transformer are then used to replace the turn ratio of the LTC so that the ORPF model becomes quadratic. The Hessian matrices in this model are constants and need to be calculated only once in the entire optimal process, which speed up the calculation greatly. The solution of the ORPF problem by the predictor corrector primal dual interior point method is described in this paper. Two separate prototypes for the new and the conventional methods are developed in MATLAB in order to compare the performances. The results obtained from the implemented seven test systems ranging from 14 to 1338 buses indicate that the proposed method achieves a superior performance than the conventional rectangular coordinate-based ORPF.

The electric power industries worldwide have undergone considerable changes especially from vertical structure to full deregulated entities. These changes are now introducing new problems in terms of operations, controls and planning of the entire grid systems. This calls for a more reliable analytical tool ever than before. One feasible solution is to perform the Optimal Economic Dispatch (OED) paradigm on this restructured power system so as to provide fairness to all operators. In this paper, the economic dispatch problem with voltage and line flow constraints has been formulated for the hydro-thermal generating units feeding the Nigerian power system. In order to solve the arising power flow problem a MATLAB based simulation package, MATPOWER version 3.0 has been suitably modified to obtain feasible solutions for different loading system scenarios. The results obtained showed that the OED offered a better optimal power schedules, power loss minimization and reduced total fuel cost than earlier work based on Micro-Genetic Algorithm, (MGA) and Conventional Genetic Algorithm (CGA).

A new procedure focused on reliability analysis of subtransmission systems supported by the state enumeration technique is presented. This new methodology is conducted in three stages. First, a classical state enumeration reliability assessment is performed for the branch-node model of a subtransmission system, assuming that substations are perfectly reliable. Second, a detailed model of the subtransmission system is considered and the reliability of each substation is assessed by considering them in a 'one-by-one' process, supposing perfect operation for the branch-node model. Finally, the reliability indices calculated in the first and second stages are analytically combined to obtain the reliability indices for the subtransmission system (system reliability indices) and for the load nodes of the distribution system (load-node reliability indices). Test results show that the proposed methodology is suitable for both planning studies and 24 h-ahead security assessment.

Improved formulations of and solution techniques for the alternating current optimal power flow (ACOPF) problem are critical to improving current market practices in economic dispatch. We introduce the IV-ACOPF formulation that unlike canonical ACOPF formulations–which represent network balancing through nonlinear coupling–is based on a current injections approach that linearly couple the quadratic constraints at each bus; yet, the IV-ACOPF is mathematically equivalent to the canonical ACOPF formulation. We propose a successive linear programming (SLP) approach to solve the IV-ACOPF, which we refer to as the SLP IV-ACOPF algorithm. The SLP IV-ACOPF leverages commercial LP solvers and can be readily extended and integrated into more complex decision processes, e.g., unit commitment and transmission switching. We demonstrate with the standard MATPOWER test suite an acceptable quality of convergence to a best-known solution and linear scaling of computational time in proportion to network size.