A New Method to Solve Reactive Power Optimization Problems of Power System by Introducing the Branch Current

The mathematical model used in this paper is not only the traditional node voltage equation but also the introduction of the branch current equation when research the power system reactive power optimization, and so establishes a mathematical model of hybrid power network node voltages and branch currents. The state variables in this model are the node voltages and branch currents, the network flow is explicitly expressed, and they play a key role for the simplification of the solving model. Solving the model will be broken down into two sub-problems, one is a network loss minimization objective augmented Lagrange function, forming the Kuhn-Tucker conditions, and the other is a linear equation. IEEE 30-bus system example shows that the complexity and high dimension of the model solution have been significantly improved, the solving process becomes easier, and the solution is close to the global optimal solution. Compared with traditional optimal power flow algorithm, this algorithm can improve the computational efficiency of reactive power optimization. DOI: http://dx.doi.org/10.5755/j01.eee.19.9.2523


I. INTRODUCTION
Reactive power optimization of the power network is a dynamic, multi-bound and nonlinear nixed planning which involves the choice of reactive power compensation location, reactive compensation capacity, transformer tap adjustments and the generator terminal voltage tie with other aspects.Reactive power optimization of power system is on the condition of ensuring the system reactive power balance, reduce loss of the entire network, save system operating costs, and improve the voltage quality by regulating the generator bus voltage, on-line tap changer tap stalls, and the capacity of reactive power compensation equipment.
The mathematical model includes the choice of objective function and constraints of the agreement, generally expressed as (1).
In (1), x is the state variable, refers to the node voltage, u Manuscript received September 29, 2012; accepted September 27, 2013.This research was funded by a grant (No. 51307108) from the Youth Science Fund Project of National Natural Science Foundation of China.This research was performed in cooperation with the Institution.is control variable, means that the node injection of reactive power, the functions ( , ) f x u , ( , ) g x u and ( , ) h x u respectively represent the network loss (objective function), the equality and inequality constraints min ( , ) ( , ) 0, ..
There are several ways to solve (1), including linear programming [1], [2], nonlinear programming (including quadratic programming method [3], [4], interior point methods [5]- [8], evolutionary programming [9]- [11], etc.) and intelligent algorithms [12]- [17].All these algorithms are based on the node voltages equation, in which the node voltage, nodal injected active or reactive power as variables.Although the node voltage analysis method is effective, but there are some problems and shortcomings: 1) As the most obvious feature of the electricity network, the trend of the amount is not directly reflected, during power network analysis, the success of many network flow theory and not being used; 2) Due to the large number of inequality constraints, there is the problem of the "curse of dimensionality", and therefore it is necessary to improve its computational efficiency.
Power state through the node voltage, node injection active power, node injection reactive power and branch currents reflect these physical quantities.References [18], [19] are based on branch-current model for distribution network, the flow convergence is better, but the model ignores the admittance, and the use of a constant load impedance model, it is not suitable for transmission network.
Therefore, it is necessary to introduce branch current as the state variables in order to overcome the above problems.This paper has established a grounding branch as a current source, and forms the expansion power network equations including node voltage variable and branch current variable.Thus, the objective function can be written as the product of the line current and impedance.The reactive power optimization problem can be decomposed into two sub problems, one is the minimum cost flow model and the other is a linear equation group.The former can be solved by quadratic programming method.The method in this paper has high calculation efficiency of the algorithm, and close to the global optimal solution.It has got better results by analysis IEEE-30 system.This paper has the following assumptions: 1) The reactive power optimization inject reactive power to all nodes in the optimal target; 2) The net loss reduce small when adjust transformer tap, so ignore the optimization of the transformer turns ratio; 3) The active power is regarded as a constant, reactive power of all nodes and node voltage are considered variables except slack bus; 4) The power load model is constant; 5) Assumed that the power grid is a three-phase balanced network.

II. POWER SYSTEM NETWORK MODELLING
The electric power system network is typically used by an impedance branch and two grounded branch consisting of a π-type equivalent circuit approximation (Fig. 1).Therefore, electric power network equations make up of the impedance equation and the earthed branch circuit equation.Which, each node load can have a variety of equivalent forms, and the load is equivalent to the voltage source in this paper.In Fig. 1
The mixed power network branch current -node voltage equations can be obtained by deducing (2): cos cos 0, sin sin 0.
For node i , the node injection power equals the product of node voltage and branch current conjugate when the load is treated as voltage source.Node injection current is divided into two parts, one is earth branch and the other is load branch, and then the power equation is where , which means the sum of all branch currents associated with node i .( j ) is the sum of ground branch currents associated with node i , and ( j ) is the sum of ground branch admittances associated with node i .Then the current of load branch equals node injection current minus ground branch current.
In order to simplify the equation derivation and calculation, ignore the node ground branch conductance l G , and suppose 2 0 cos sin .
There are the same form equations for the node j .We can obtain (8) by deducing (3) and ( 4): ( cos cos ) ( cos cos ), where, , GB is the admittance of branch l .Brings it into (6) and ( 7) can get the traditional forms of node voltage equation: The difference in ( 9) is that there do not contain ground branch susceptance in the node self-susceptance ii B .This also verified that the introduction of the branch current does not change the nature of the equations.And with the introduction of the branch current the directly observed values ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 19, NO. 9, 2013 increase.It can be directly expressed by branch current like the question of network loss.
Mixed equations of the electricity network are composed by ( 3), ( 4), ( 6), (7), and node voltage and branch current are the state variables.The equations are linear function of the branch current and nonlinear function of node voltage.Figure 2 is a π-type equivalent circuit of current source simulated ground branch, where, Gi I  is the ground branch current of node i .
Equation ( 13) can be obtained by deducing (6) cos sin and then min max cos sin and from (7): The reactive power optimization mathematical model is (10), (3), ( 4), ( 6), ( 7), ( 14) and (15), it has the following characteristics: 1) The target function is a quadratic function of branch current variables; 2) The constraint is the linear function of branch current variables. The Due to the partial derivatives /0 q i Lq      of the control variable i q , and i q is only in (7), so (7) can be omitted, and the optimal solution has nothing to do with the i q .So, the reactive power optimization model can be simplified by the (10), (3), ( 4), ( 6), ( 14) and (15), and optimize only for state variables , . Eventually optimal node inject reactive power i q can be calculated by (7).

IV. MODEL SOLUTION
We assume that cos cos 3) and ( 4), the (17) can be obtained .
Equations ( 17) are linear, and each node voltage value can be calculated if known the real part value and the imaginary part value of branch current.
Problems A can be described as (1), where u is expressed as a real part and an imaginary part of the branch current vector, x is expressed in amplitude and phase angle of nodal voltage vector.( , ) 0 h x u  is the inequality constraints, in particular are ( 14) and ( 15). ( , ) 0 g x u  is the equality constraints, in particular ( 6) and (17).So the Kuhn-Tucker condition of problems A is: 0, 0, ( , ) 0, 0, ( , ) 0, ( , ) 0.
Obviously, the solutions calculated by solving the sub-problem S and linear equations ( 6) and ( 17) are not equivalent with that of problem A because of ' u  .In order to obtain the exact solution of the reactive power optimization, the variable u must be modified according to (22) after solving the sub-problem S.
Solution of the whole model iteration steps are as follows: 1) Assuming 0 as state variables for solving sub-problem S, get branch current ( 1)   k I  ; 3) If ( 1)   ( ) kk II    (  is a small positive number) finish iteration and go to step 5), otherwise continue; 4) The node voltages can be calculated by using ( 6) and ( 17), 1 kk  and return to step 2); 5) The node reactive power injection will be calculated by using (7).
The solution of sub-problem S is global optimal because it is about convex quadratic programming problem of line resistance, and ( 6), (17) are linear equations which have the only solution.Therefore, the optimal solution obtained finally is closely enough to the global optimal solution.

V. EXAMPLE ANALYSIS
The case study is made at IEEE-30 system with active set arithmetic solved the sub-problem S. The upper limit of voltage at node 10 is set as 1.0421 and the upper limit of voltage at node 24 is set as 1.0261 while upper limit of voltage is set as 1.1 and lower limit of voltage is set as 0.95 at all nodes.
The final calculating results are listed in Table and the reactive powers at node 10 and are 0.144558 and 0.0914809 calculated by the nodal injective current.The thirteen iterations are needed with step 3.24 listed in Table II.The upper limit of voltage at node 10 is violated in iteration 4, then the search direction is changed and the violation is eliminated with Lagrange factor 0.0178555.Up to iteration 12, the upper limit of voltage at 24 is violated and it is eliminated with Lagrange factor 0.0079243.The final network losses are reduced from 0.0879016 to 0.0873921.The comparing results of proposed approach with Newton method and reduced gradient algorithm are listed in Table III while the inequality constraints are ignored.It can be seen that optimization effect of proposed approach is better than Newton method and reduced gradient algorithm.The comparing results of voltage magnitude with Newton method and reduced gradient algorithm are listed in Table IV.The results show that the node voltage magnitude is more closed to the standard data calculated by the proposed algorithm in this paper, so the error of voltage magnitude is smaller while optimizing the reactive power.The hybrid electric power network equations composed of node voltage and branch current can also provide useful ideas to solve practical problems of power system in several other areas in addition to the excellent performance in the reactive power optimization.

A. Explicit expression of node voltage high and low solutions
Deducing (5) ) The meaning of ( 24) is: 1) When the square of the amplitude of the injection current of nodes is out of the circle of which the center is 0 2 ii Bq and the radius is , that is only '>' condition has been met in (24), the high and low solutions of (23) exist, and the system is stable; 2) When the square of the amplitude of the injection current of nodes is in the circle, that is only '<' condition has been met in (24), no solutions of (23) exist, so the system is unstable; 3) Equation ( 23) has a unique solution and the solution is on the circle when '=' condition has been met in (24), so the unique solution is the stable margin of system.The voltage collapse point can be found if calculate the power flow equations under this conditions.

C. Voltage stability critical condition
When the equality of (24) meets the two solution curves of node voltage intersect, and reach the voltage stability critical point.So, the voltage stability critical condition is When the branch current and node voltage are as variables the expressive information is more abundant, and can solve the problem difficult to resolve in the past.So the hybrid electric network equations can be applied to research in different fields of power system.

VII. CONCLUSIONS
The dimension of network equations increases while introduce node voltage and branch current as a state variable during power system reactive power optimization.At the same time the computational complexity increases because need to consider the constraint of node voltage and node injection power.So it brings some problems and difficulties compared with the conventional node voltage equations.But the above issues are handled through the improvement of the algorithm.Ultimately, conclude as follows: 1) The power network equations can be expressed as a mixed form based on node voltages and branch currents, and the mixed equations contain more abundant information to improve the observability of the power system, and the calculation efficiency enhances also because the branch power flow can be calculated in the same computing cycle with node voltage; 2) In order to simplify the solution of reactive power optimization the problem has been decomposed into two easy solving sub-problems, and then reduce the complexity of the problem and the dimension of the equations.So the computational efficiency has been improved.
3) After comparing with the traditional method the proposed algorithm is closer to the global optimal solution because of the further correction to the state variable, and the node voltage error is smaller, too.
4) Because the convergence of the proposed algorithm is good the mixed equation mathematical model based on node voltages and branch currents proposed in this paper can be applied to other optimization problems in the power system.
injection active power is a constant value for all nodes in reactive power optimization model, and variables are divided into branch current, node voltage and reactive power injection.Branch currents and node voltages represent the network status, they are considered as state variables, and node injected reactive power is the control variable.The augmented Lagrange function of reactive power optimization model is

TABLE I .
IEEE-30 SYSTEM NODE CALCULATION RESULTS.

TABLE II .
CALCULATING PROCESS INFORMATION.

TABLE III .
COMPARING RESULTS WITH OTHERS.

TABLE IV .
COMPARING NODE VOLTAGE MAGNITUDE WITH OTHERS.