Improvement of Power Flow Calculation with Optimization Factor Based on Current Injection Method

Abstract

This paper presents an improvement in power flow calculation based on current injection method by introducing optimization factor. In the method proposed by this paper, the PQ buses are represented by current mismatches while the PV buses are represented by power mismatches. It is different from the representations in conventional current injection power flow equations. By using the combined power and current injection mismatches method, the number of the equations required can be decreased to only one for each PV bus. The optimization factor is used to improve the iteration process and to ensure the effectiveness of the improved method proposed when the system is ill-conditioned. To verify the effectiveness of the method, the IEEE test systems are tested by conventional current injection method and the improved method proposed separately. Then the results are compared. The comparisons show that the optimization factor improves the convergence character effectively, especially that when the system is at high loading level and R/X ratio, the iteration number is one or two times less than the conventional current injection method. When the overloading condition of the system is serious, the iteration number in this paper appears 4 times less than the conventional current injection method.

1. Introduction

Power flow studies are necessary for planning, operating, economic scheduling, and other analysis, such as transient stability, voltage stability, and contingency studies. The tasks of power flow calculation are to solve the steady-state operating conditions of power systems based on the operation modes and the wiring of the systems. Different power flow methods have emerged and continue to be developed with the changing needs.

The conventional power flow solution comprises power equations expressed in terms of rectangular or polar coordinates. Many important contributions have been reported in this area [1–3]. To improve the convergence characteristic of Gauss-Seidel method, the Newton-Raphson method was introduced. The NR method was once considered the state of the art power-flow technique and widely accepted in industry applications. However, the main disadvantage of the Newton-Raphson method is the necessity for factorizing and updating the Jacobian matrix during the iterative solution process [4]. To solve this problem, Fast Decoupled power flow method was proposed to speed up the iteration process of the NR method and decrease the required minimum memory storage. Nevertheless, the convergence rate of the decoupled NR is influenced by the ill-conditioned case, such as when a system has high ratio of line R/X [5–8]. Some other methods have been presented in other forms, such as the use of sequence component frame [9–11] and the method based on the loop frame of reference [12].

In [13], a new power flow method based on current injection was presented, in which the current injection equations are written in rectangular coordinates. The Jacobian matrix is composed of 6×6 block matrices and has the same structure as the nodal admittance matrix. By using this method, the Jacobian matrix can be updated faster than using conventional NR power flow method in the case of PQ buses. However, in the case of the existence of PV buses, the current injection mismatch power flow method was presented to improve the convergence character of current injection method [14, 15]. The improvement in [16] increased the number of required equations to three for each PV bus. The paper [17] presented a new representation of PV buses in revised current injection power flow method to reduce the number of required equations and to improve the convergence character of well- and ill-conditioned systems, while the elements related to PV buses have to be changed during the iteration process. In [18] a development of power flow calculation for ill-conditioned system was presented. The fact that “the Taylor series expansion of the load flow equations is expressed up to the third term completely and the final term has the same form but different variable as the first term” was used in [18]. Following the main ideas presented in [18], which is called Iwamoto’s method in [15], a new second order power flow method was proposed. This method is useful for heavily loaded and overloaded systems as well as ill-conditioned distribution systems.

This paper improved the current injection method presented in [17] following the main idea in [18]. The optimization presented in [18] is briefly described here firstly to make the paper clear. Then the improved method, results, and conclusions are presented.

2. Current Injection Power Flow

The basic current injection method proposed in [17] is used in this paper. The combined power and current injection mismatches power flow formulation can be calculated from   the following:Δ𝐼_𝑚1Δ𝐼_𝑟1Δ𝐼_𝑚2Δ𝐼_𝑟2⋮Δ𝑃_𝑘⋮Δ𝐼_𝑚𝑛Δ𝐼_𝑟𝑛=𝜕𝐼_𝑚1𝜕𝑉_𝑟1𝜕𝐼_𝑚1𝜕𝑉_𝑚1𝜕𝐼_𝑚1𝜕𝑉_𝑟2𝜕𝐼_𝑚1𝜕𝑉_𝑚2⋯𝜕𝐼_𝑚1𝜕𝛿_𝑘⋯𝜕𝐼_𝑚1𝜕𝑉_𝑟𝑛𝜕𝐼_𝑚1𝜕𝑉_𝑚𝑛𝜕𝐼_𝑟1𝜕𝑉_𝑟1𝜕𝐼_𝑟1𝜕𝑉_𝑚1𝜕𝐼_𝑟1𝜕𝑉_𝑟2𝜕𝐼_𝑟1𝜕𝑉_𝑚2⋯𝜕𝐼_𝑟1𝜕𝛿_𝑘⋯𝜕𝐼_𝑟1𝜕𝑉_𝑟𝑛𝜕𝐼_𝑟1𝜕𝑉_𝑚𝑛𝜕𝐼_𝑚2𝜕𝑉_𝑟1𝜕𝐼_𝑚2𝜕𝑉_𝑚1𝜕𝐼_𝑚2𝜕𝑉_𝑟2𝜕𝐼_𝑚2𝜕𝑉_𝑚2⋯𝜕𝐼_𝑚2𝜕𝛿_𝑘⋯𝜕𝐼_𝑚2𝜕𝑉_𝑟𝑛𝜕𝐼_𝑚2𝜕𝑉_𝑚𝑛𝜕𝐼_𝑟2𝜕𝑉_𝑟1𝜕𝐼_𝑟2𝜕𝑉_𝑚1𝜕𝐼_𝑟2𝜕𝑉_𝑟2𝜕𝐼_𝑟2𝜕𝑉_𝑚2⋯𝜕𝐼_𝑟2𝜕𝛿_𝑘⋯𝜕𝐼_𝑟2𝜕𝑉_𝑟𝑛𝜕𝐼_𝑟2𝜕𝑉_𝑚𝑛⋮⋮⋮⋮⋮⋮⋮⋮⋮𝜕𝑃_𝑘𝜕𝑉_𝑟1𝜕𝑃_𝑘𝜕𝑉_𝑚1𝜕𝑃_𝑘𝜕𝑉_𝑟2𝜕𝑃_𝑘𝜕𝑉_𝑚2⋯𝜕𝑃_𝑘𝜕𝛿_𝑘⋯𝜕𝑃_𝑘𝜕𝑉_𝑟𝑛𝜕𝑃_𝑘𝜕𝑉_𝑚𝑛⋮⋮⋮⋮⋮⋮⋮⋮⋮𝜕𝐼_𝑚𝑛𝜕𝑉_𝑟1𝜕𝐼_𝑚𝑛𝜕𝑉_𝑚1𝜕𝐼_𝑚𝑛𝜕𝑉_𝑟2𝜕𝐼_𝑚𝑛𝜕𝑉_𝑚2⋯𝜕𝐼_𝑚𝑛𝜕𝛿_𝑘⋯𝜕𝐼_𝑚𝑛𝜕𝑉_𝑟𝑛𝜕𝐼_𝑚𝑛𝜕𝑉_𝑚𝑛𝜕𝐼_𝑟𝑛𝜕𝑉_𝑟1𝜕𝐼_𝑟𝑛𝜕𝑉_𝑚1𝜕𝐼_𝑟𝑛𝜕𝑉_𝑟2𝜕𝐼_𝑟𝑛𝜕𝑉_𝑚2⋯𝜕𝐼_𝑟𝑛𝜕𝛿_𝑘⋯𝜕𝐼_𝑟𝑛𝜕𝑉_𝑟𝑛𝜕𝐼_𝑟𝑛𝜕𝑉_𝑚𝑛Δ𝑉_𝑟1Δ𝑉_𝑚1Δ𝑉_𝑟2Δ𝑉_𝑚2⋮Δ𝛿_𝑘⋮Δ𝑉_𝑟𝑛Δ𝑉_𝑚𝑛, (1)

where   Δ𝐼_𝑟𝑘: real parts of current mismatch at bus 𝑘, Δ𝐼_𝑚𝑘: imaginary parts of current mismatch at bus 𝑘, Δ𝑃_𝑘: active power mismatch at bus 𝑘, Δ𝑉_𝑟𝑘: real voltage component correction at bus 𝑘, Δ𝑉_𝑚𝑘: imaginary voltage component correction at bus 𝑘, and Δ𝛿_𝑘: voltage angle correction at bus 𝑘.

2.1. Equations for PQ Buses

The current mismatch for a given bus 𝑘 is Δ𝐼_𝑘=𝑃_𝑘^𝑠𝑝−𝑗𝑄_𝑘^𝑠𝑝𝐸^∗_𝑘−^𝑛_𝑖=1𝑌_𝑘𝑖𝐸_𝑖, (2) where 𝐸^∗_𝑘: complex conjugated voltage phasor at bus 𝑘, 𝑃_𝑘^𝑠𝑝,𝑄_𝑘^𝑠𝑝: specified active and reactive net powers at bus 𝑘, 𝑌_𝑘𝑖=𝐺_𝑘𝑖+𝑗𝐵_𝑘𝑖: bus admittance matrix element, and 𝐸_𝑖: voltage phasor at bus 𝑖. Consider 𝑃_𝑘^𝑠𝑝=𝑃_𝑔𝑘−𝑃_𝑙𝑘,𝑄_𝑘^𝑠𝑝=𝑄_𝑔𝑘−𝑄_𝑙𝑘, (3) where 𝑃_𝑔𝑘, 𝑄_𝑔𝑘: active and reactive powers of generators for bus 𝑘 and 𝑃_𝑙𝑘, 𝑄_𝑙𝑘: active and reactive powers of loads for bus 𝑘.

Equation (2) can be expanded into its real and imaginary components: Δ𝐼_𝑟𝑘=𝑃_𝑘^𝑠𝑝𝑉_𝑟𝑘+𝑄_𝑘^𝑠𝑝𝑉_𝑚𝑘𝑉²_𝑟𝑘+𝑉²_𝑚𝑘−^𝑛_𝑖=1𝐺_𝑘𝑖𝑉_𝑟𝑖−𝐵_𝑘𝑖𝑉_𝑚𝑖,Δ𝐼_𝑚𝑘=𝑃_𝑘^𝑠𝑝𝑉_𝑚𝑘−𝑄_𝑘^𝑠𝑝𝑉_𝑟𝑘𝑉²_𝑟𝑘+𝑉²_𝑚𝑘−^𝑛_𝑖=1𝐺_𝑘𝑖𝑉_𝑚𝑖+𝐵_𝑘𝑖𝑉_𝑟𝑖, (4)(5) where 𝑉_𝑟𝑘: real voltage component at bus 𝑘 and 𝑉_𝑚𝑘: imaginary voltage component at bus 𝑘.

Equations (4) and (5) are written in compact forms: Δ𝐼_𝑟𝑘=𝐼^𝑠𝑝_𝑟𝑘−𝐼^calc_𝑟𝑘,Δ𝐼_𝑚𝑘=𝐼^𝑠𝑝_𝑚𝑘−𝐼^calc_𝑚𝑘, (6) where 𝐼^𝑠𝑝_𝑟𝑘, 𝐼^𝑠𝑝_𝑚𝑘: specified real and imaginary parts of current at bus 𝑘, 𝐼^calc_𝑟𝑘, 𝐼^calc_𝑚𝑘: calculated real and imaginary parts of current at bus 𝑘.

Using the Newton-Raphson solution algorithm, the elements in (1) for all buses as being of the PQ type are given by 𝜕𝐼_𝑚𝑘𝜕𝑉_𝑟𝑖=𝐵_𝑘𝑖,𝜕𝐼_𝑚𝑘𝜕𝑉_𝑚𝑖=𝐺_𝑘𝑖𝜕𝐼_𝑟𝑘𝜕𝑉_𝑟𝑖=𝐺_𝑘𝑖,𝜕𝐼_𝑟𝑘𝜕𝑉_𝑚𝑖=−𝐵_𝑘𝑖. (7) The diagonal elements are given by 𝜕𝐼_𝑚𝑘𝜕𝑉_𝑟𝑘=𝐵_𝑘𝑘−𝑎_𝑘,𝜕𝐼_𝑚𝑘𝜕𝑉_𝑚𝑘=𝐺_𝑘𝑘−𝑏_𝑘,𝜕𝐼_𝑟𝑘𝜕𝑉_𝑟𝑘=𝐺_𝑘𝑘−𝑐_𝑘,𝜕𝐼_𝑟𝑘𝜕𝑉_𝑚𝑘=−𝐵_𝑘𝑘−𝑑_𝑘. (8) The elements 𝑎_𝑘, 𝑏_𝑘, 𝑐_𝑘, and 𝑑_𝑘 are presented in Appendix A of [13].

The following equation is expression of current mismatches in (1): Δ𝐼_𝑚𝑘=𝑉_𝑚𝑘Δ𝑃_𝑘−𝑉_𝑟𝑘Δ𝑄_𝑘𝑉²_𝑟𝑘+𝑉²_𝑚𝑘,Δ𝐼_𝑟𝑘=𝑉_𝑟𝑘Δ𝑃_𝑘+𝑉_𝑚𝑘Δ𝑄_𝑘𝑉²_𝑟𝑘+𝑉²_𝑚𝑘. (9)

2.2. Representation of PV Buses

The calculated active power at assumed bus 𝑘 can be calculated by 𝑃^calc_𝑘=^𝑛_𝑖=1𝑉_𝑘𝑉_𝑖𝐺_𝑘𝑖cos𝛿_𝑘𝑖+𝐵_𝑘𝑖sin𝛿_𝑘𝑖, (10) where 𝛿_𝑘𝑖=𝛿_𝑘−𝛿_𝑖, 𝛿_𝑘: voltage phase angle at bus 𝑘.

In the Jacobian matrix of (1), the elements related to buses of PV type are given as follows.

Diagonal elements 𝜕𝑃_𝑘𝜕𝛿_𝑘=−𝑉_𝑘^𝑛_{𝑖=1𝑖≠𝑘}𝑉_𝑖𝐺_𝑘𝑖sin𝛿_𝑘𝑖−𝐵_𝑘𝑖cos𝛿_𝑘𝑖. (11)

Off diagonal elements 𝜕𝐼_𝑚𝑡𝜕𝛿_𝑘=𝑉_𝑘𝐺_𝑡𝑘cos𝛿_𝑘−𝐵_𝑡𝑘sin𝛿_𝑘,𝜕𝐼_𝑟𝑡𝜕𝛿_𝑘=−𝑉_𝑘𝐺_𝑡𝑘sin𝛿_𝑘+𝐵_𝑡𝑘cos𝛿_𝑘,𝜕𝑃_𝑘𝜕𝑉_𝑚𝑡=𝑉_𝑘𝐺_𝑘𝑡sin𝛿_𝑘−𝐵_𝑘𝑡cos𝛿_𝑘,𝜕𝑃_𝑘𝜕𝑉_𝑟𝑡=𝑉_𝑘𝐺_𝑘𝑡cos𝛿_𝑘+𝐵_𝑘𝑡sin𝛿_𝑘. (12)

3. Description of Iwamoto’s Method

The method presented in [18] is described briefly here to make the improved method proposed in next part clearly. The conventional power flow equations in the rectangular coordinates are 𝑦_𝑠=𝑦(𝑥). (13) The Taylor series expansion of (13) turns out to be 𝑦_𝑠=𝑦𝑥_𝑒+𝐽Δ𝑥+𝑦(Δ𝑥), (14) where 𝑥_𝑒: estimate of 𝑥, 𝐽: Jacobian matrix, and Δ𝑥: error (correction vector).

Moving all the right-hand side of (14) to the left-hand side 𝑦_𝑠−𝑦𝑥_𝑒−𝐽Δ𝑥−𝑦(Δ𝑥)=0. (15)

To obtain the solution of this equation, it denotes by 𝜇, the step size optimization factor. Applying this factor to this equation, then it follows that 𝑦_𝑠−𝑦𝑥_𝑒−𝐽𝜇Δ𝑥−𝑦𝜇Δ𝑥=0. (16) Or 𝑦_𝑠−𝑦𝑥_𝑒−𝜇𝐽Δ𝑥−𝜇²𝑦(Δ𝑥)=0. (17) It can be rewritten as 𝑎+𝜇𝑏+𝜇²𝑐=0, (18) where 𝑎=𝑦_𝑠−𝑦𝑥_𝑒,().𝑏=−𝐽Δ𝑥=−𝑎,𝑐=−𝑦Δ𝑥 (19) The following cost function is considered to determine the value of the 𝜇 in a least squared sense: 1𝐹=2^𝑛_𝑖=1𝑎_𝑖+𝜇𝑏_𝑖+𝜇²𝑐_𝑖²⟶Minimize. (20) The optimization solution is given by 𝜕𝐹𝜕𝜇=0. (21) Namely, 𝑔₀+𝑔₁𝜇+𝑔₂𝜇²+𝑔₃𝜇³=0, (22) where 𝑔₀=^𝑛_𝑖=1𝑎_𝑖𝑏_𝑖,𝑔₁=^𝑛_𝑖=1𝑏²_𝑖+2𝑎_𝑖𝑐_𝑖,𝑔₂=3^𝑛_𝑖=1𝑏_𝑖𝑐_𝑖,𝑔₃=2^𝑛_𝑖=1𝑐²_𝑖. (23) The state vector for the next iteration is given by 𝑥^𝑘+1=𝑥^𝑘+𝜇^𝑘Δ𝑥^𝑘. (24)

4. Improvement in Current Injection Method

4.1. Parameters for PQ Buses

The improved method is based on the current injection method by defining a step size optimization factor. The equation (5) is the expression of imaginary current injection related to bus 𝑘. Neglecting all terms of order higher than two, we then get 𝐼^𝑠𝑝_𝑚𝑘=𝐼_𝑚𝑘+𝜕𝐼_𝑚𝑘𝜕𝑉_𝑟𝑘Δ𝑉_𝑟𝑘+𝜕𝐼_𝑚𝑘𝜕𝑉_𝑚𝑘Δ𝑉_𝑚𝑘+^𝑛_𝑖=1𝐵_𝑘𝑖Δ𝑉_𝑟𝑖+𝐺_𝑘𝑖Δ𝑉_𝑚𝑖+12^𝑛_𝑖=1𝜕²𝐼_𝑚𝑘𝜕𝑉_𝑟𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑟𝑘Δ𝑉_𝑟𝑖+𝜕²𝐼_𝑚𝑘𝜕𝑉_𝑟𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑟𝑘Δ𝑉_𝑚𝑖+𝜕²𝐼_𝑚𝑘𝜕𝑉_𝑚𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑚𝑘Δ𝑉_𝑟𝑖+𝜕²𝐼_𝑚𝑘𝜕𝑉_𝑚𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑚𝑘Δ𝑉_𝑚𝑖. (25) Hence, the following equations can be got based on the theory in part 3: 𝑎_𝐼𝑚𝑘=Δ𝐼_𝑚𝑘,𝑏_𝐼𝑚𝑘=−𝑎_𝐼𝑚𝑘,𝑐_𝐼𝑚𝑘1=−2^𝑛_𝑖=1𝜕²𝐼_𝑚𝑘𝜕𝑉_𝑟𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑟𝑘Δ𝑉_𝑟𝑖+𝜕²𝐼_𝑚𝑘𝜕𝑉_𝑟𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑟𝑘Δ𝑉_𝑚𝑖+𝜕²𝐼_𝑚𝑘𝜕𝑉_𝑚𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑚𝑘Δ𝑉_𝑟𝑖+𝜕²𝐼_𝑚𝑘𝜕𝑉_𝑚𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑚𝑘Δ𝑉_𝑚𝑖. (26) In a similar way, the following equations can be obtained 𝑎_𝐼𝑟𝑘=Δ𝐼_𝑟𝑘,𝑏_𝐼𝑟𝑘=−𝑎_𝐼𝑟𝑘,𝑐_𝐼𝑟𝑘1=−2^𝑛_𝑖=1𝜕²𝐼_𝑟𝑘𝜕𝑉_𝑟𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑟𝑘Δ𝑉_𝑟𝑖+𝜕²𝐼_𝑟𝑘𝜕𝑉_𝑟𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑟𝑘Δ𝑉_𝑚𝑖+𝜕²𝐼_𝑟𝑘𝜕𝑉_𝑚𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑚𝑘Δ𝑉_𝑟𝑖+𝜕²𝐼_𝑟𝑘𝜕𝑉_𝑚𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑚𝑘Δ𝑉_𝑚𝑖. (27) Thus, the parameters can be obtained as follows 𝑎_𝑘=Δ𝐼_𝑚𝑘Δ𝐼_𝑟𝑘^𝑇,𝑏_𝑘=−𝑎_𝑘,𝑐_𝑘=𝑐_𝐼𝑚𝑘𝑐_𝐼𝑟𝑘^𝑇. (28)

4.2. Parameters for PV Buses

From (1) for a PV bus one has 𝑃_𝑘^𝑠𝑝=𝑃_𝑘+𝜕𝑃_𝑘𝜕𝛿_𝑘+^𝑛_{𝑖=1𝑖≠𝑘}𝜕𝑃_𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑟𝑖+𝜕𝑃_𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑚𝑖+12𝜕²𝑃_𝑘𝜕𝛿²_𝑘Δ²𝛿_𝑘+^𝑛_{𝑖=1𝑖≠𝑘}𝜕²𝑃_𝑘𝜕𝑉_𝑟𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑟𝑘Δ𝑉_𝑟𝑖+𝜕²𝑃_𝑘𝜕𝑉_𝑟𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑟𝑘Δ𝑉_𝑚𝑖+𝜕²𝑃_𝑘𝜕𝑉_𝑚𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑚𝑘Δ𝑉_𝑟𝑖+𝜕²𝑃_𝑘𝜕𝑉_𝑚𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑚𝑘Δ𝑉_𝑚𝑖. (29) Using the same method as parameters for PQ buses, the following equations can be obtained 𝑎_𝑘=Δ𝑃_𝑘,𝑏_𝑘=−𝑎_𝑘,𝑐_𝑘1=−2𝜕²𝑃_𝑘𝜕𝛿²_𝑘Δ²𝛿_𝑘+^𝑛_{𝑖=1𝑖≠𝑘}𝜕²𝑃_𝑘𝜕𝑉_𝑟𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑟𝑘Δ𝑉_𝑟𝑖+𝜕²𝑃_𝑘𝜕𝑉_𝑟𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑟𝑘Δ𝑉_𝑚𝑖+𝜕²𝑃_𝑘𝜕𝑉_𝑚𝑘𝜕𝑉_𝑟𝑖Δ𝑉_𝑚𝑘Δ𝑉_𝑟𝑖+𝜕²𝑃_𝑘𝜕𝑉_𝑚𝑘𝜕𝑉_𝑚𝑖Δ𝑉_𝑚𝑘Δ𝑉_𝑚𝑖. (30)

5. Results

This paper used MATLAB to implement the program of power flow method proposed. IEEE 30-bus and 118-bus test system are tested here. Firstly the validity of the improved method proposed is proven. Here the systems are tested as symmetrical systems. Tables 1 and 2 show a part of the results. Then the results obtained by conventional current injection and the improved method proposed, respectively, are compared. Figures 1 and 2 show the iteration characteristics of IEEE 30-bus test system using the two different methods and Figures 3 and 4 show the results of IEEE 118-bus system in the same way.

Table 1 

Parts of the iteration results of 30-bus system.

Table 2 

Parts of the iteration results of 118-bus system.

Figure 1 

Iteration characteristics of IEEE 30-bus system using conventional current injection method.

Figure 2 

Iteration characteristics of IEEE 30-bus system using the improved method proposed.

Figure 3 

Iteration characteristics of IEEE 118-bus system using conventional current injection method.

Figure 4 

Iteration characteristics of IEEE 118-bus system using the improved method proposed.

Tables 3 and 4 compare the conventional current injection method and the improved method proposed here when the system has high R/X ratios and compare the two methods when the system is operating in overloaded condition.

Table 3 

Number of iterations for different 𝑅/𝑋 ratios-30-bus system.

Table 4 

Number of iterations for different loading levels-30-bus system.

(a) Validity of the Method. Tables 1 and 2 show the results of the iteration and present the validity of the improved method. The voltage amplitudes and angles are reasonable and acceptable. At the same time the feasibility of the method is proved.

(b) Convergence Characteristic. The iteration convergence criterion for the program of the improved method proposed is that the maximum active power mismatch and the maximum reactive power mismatch are both less than the tolerance. The results presented in Figures 1–4 show that the convergence characteristic of the improved method proposed is better than the conventional current injection method.

(c) High R/X Ratio. Many power flow methods always have bad convergence characteristic if the system has high R/X ratio. In Table 3 the R/X ratios are divided into five levels. When the system has high X/R ratio, both the two methods have good convergence characteristic, but with the increase of R/X ratio the advantage of the improved method proposed becomes obvious; the iteration number is one or two times less than the conventional current injection method.

(d) Overloading. The comparison of the iteration number for different loading cases is shown in Table 4. In this case the system is in overloading condition. In Table 4, five loading cases are presented. There has been little difference between the two methods when the overloading of the system is not serious. However, when the condition becomes more serious the iteration number of the improved method proposed appears 4 times less than the conventional current injection method. Even more, when the convergence characteristic of the improved method proposed is still acceptable when overloading is serious, the conventional current injection method shows no convergence.

6. Conclusions

This paper has presented improvement in power flow calculation based on current injection method by introducing an optimization factor. Unlike other current injection methods, the PV buses are represented by power mismatches here. The results have demonstrated the good performance of the improved method proposed in this paper. By using optimization factor the iteration character got improved. Even when the system has high R/X ratio or operates in overloaded conditions, good results could be got by using this method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (no. 51205046) and the Fundamental Research Funds for the Central Universities (nos. CDJZR170008, 106112014CDJZR175501). The constructive comments provided by the anonymous reviewers and the editors are also greatly appreciated.