Nowadays power flow mathematical modeling underlines power system operation. Traditionally, software packages for power flow calculation use a mathematical model based on nodal admittance equation. The authors have developed a software package for calculating power flow using an alternative power-energy flow model. The model has several advantages compared with the traditional approach. In particular, it allows combining single-phase and three-phase network representation. The authors have analyzed the limits of applicability of this method, as well as the errors that it introduces.
1 Introduction
A classical problem of circuit theory is to find all branch currents and all node voltages of an assigned
circuit. Typical input data are generator voltages as well as the impedances of all branches. If all
impedances are constant, the resulting set of equations that describe the circuit is linear. The power flow
problem is conceptually the same problem as solving a steadystate ac circuit [
The main problem of this approach is connected with the fact that iteration procedure of Newton method may diverge. It depends on initial conditions and structure of grid. In addition, this method has sufficiently high computational complexity for large grid. This circumstance can be significant when using power flow calculation in control systems, which performs variant calculations.
The authors proposed the formulation of the problem based on the power-energy flow model [
High voltage transmission network are generally modeled using single-phase model. It is believed that power flow of
backbone networks are symmetric. In this way, exact consideration of all three phases is not required. In normal condition,
this assumption does not lead to a high error. For the analysis of asymmetric condition, such as short circuits, it is customary
to use the symmetric component method[
Traditionally management of distribution and transmission grids was divided administratively and technically. As a
consequence, there was no need for a joint consideration of two types networks. Modern trends in digital smart grid [
In general, approach for compilation equation is the same for single-phase and three-phase models. In proposed model active Ps and reactive Qs line start power, voltage magnitudeV in bus are accepted as unknown variables. Three types of equation describe power flow. In the beginning, they will be described in general form for a single-phase model, and then fundamental differences will be presented for a three-phase model. 2.1
First type of equation is relationship between voltages of the beginning Vi and the end Vj of line: Where ' V ' and ' V '' are expressed as:
Vj
Vi
Vi '
V ' 2 '
V ''
2 ' V ' ' V ''
X ij
Vi Vi Where: Vi – voltage of line start, Vj – voltage of line end, Pijs and Qisj – active and reactive line start power, Rij and X ij – active and reactive line resistance.
Second type of equations are active and reactive power balance in bus: (Q sji '
Pji )¦ Q ji )¦ (Piks ) (Qisk )
P i 0 Where: Pi and Qi – load of bus; Piks and Qisk – start power of lines, that start number coinciding with the node number; Pjsi and Q sji – start power of lines, that end number coinciding with the node number; ' Pji and ' Qji – power loses in lines , that end number coinciding with the node number. Power loses are expressed as follows: Third type of equations use condition that sum of the angle difference is equal to zero in closed loop: For variables under consideration, this equation can be transformed as follows: ' Pji ' Q ji
Ps 2 ji Ps 2 ji V j2 Vj 2
Q sji2 R ,
ji
Qsji2 X ji ¦ ' G
0 ¦
§ ' V '' · atg ©¨ Vi ' V ¹¸' 0 All voltages in single-phase model are phase-to-phase voltages, all power in single phase are the sum of each phase power.
In three-phase model, all this type of equations is used for each phase. The main difference is connected with component ' Pji , ' Qji , ' V ' and ' V '' . Below are the expressions for phase A, expressions for phases B and C can be obtained by analogy. Active power loses is expressed as: ' Pa Pa2 RUea[2Zaa ] Qa2 URea2[Zaa ] (PaQb PbQa )Im[e i(12q0 G ab)Zab ] (Pa Pb QaQb ) Re[e iq(120 G ab) Zab ]
UaUb (PaQc PcQa )Im[ei(120q G ac)Zac ] (Pa Pc QaQc ) Re[ei(12q0 G ac)Zac ]
U U
a c Reactive power loses is expressed as: ' Qa (PaQb (PaQc
Pa2 Im[Zaa ] Qa2 Im[Zaa ]
U a2 U a2 PbQa ) Re[e i(12q0 G ab) Zab ] (Pa Pb
UaUb PcQa ) Re[ei(120q G ac) Zac ] (Pa Pc
U U a c
QaQb ) Im[e iq(120 G ab)Zab ] QaQc ) Im[ei(12q0 G ac)Zac ] Voltage drop components are expressed as: ' V '
Pa Re[Zaa ]
U a
Qa Im[Zaa ]
To analyze the proposed approach, a software package was developed in the Wolfram Mathematica [
To analyze the algorithms underlying the software package, a test network was used, which is presented in Figure 1. The network combines both a single-phase representation for the backbone network and a three-phase representation of the distribution network. Firstly, power flow calculation in three phases with regard to mutual induction was analyzed. For this, a set of asymmetrical loads was generated, the maximum amplitude values of the loads are shown in Figure 1. Then, based on these loads, the model without mutual induction and the reference model were compared. An exact three-phase model was used as a reference calculation. The results of the calculation of the modes are shown in Figure 2. In this case, each point in Figure 1 corresponds to the steady-state power flow, the standard deviation of the voltage in the reference calculation is plotted along the X-axis, and the error resulting from the calculation is on the Y-axis. As you can see, the error does not exceed 5 percent, if the standard deviation of the voltage does not exceed 3%. These data can be considered the limit of applicability of the three-phase model.
The article described a software package for calculating power flow of electric power systems. It is based on a powerenergy flow model. The model allows, depending on the available information and the requirements for the problem being solved, applying the phase or linear form of the described model. The transition from one model to another is carried out without a significant change in the design model itself. Within this model, assumptions are used at the interface point between the single-phase and three-phase network representation. Error analysis was performed on a test system. Calculations showed if at the interface point the differences between the voltages do not exceed 10% percent, then the proposed approach does not lead to a significant error.
The reported study was funded by the Ministry of Education and Science of the Russian Federation under Federal Targeted Programme according to the agreement № 14.578.21.0226 (Project identifier: RFMEFI57817X0226).