Toward a Unified Algorithm for Calculating Network Capacities
Abstract Generated abstract
This paper proposes a unified algorithm for calculating steady-state operating conditions in electric power networks by using generalized expressions for branch active and reactive powers within nodal power balance equations. It extends the treatment from symmetric four-terminal networks to nonsymmetric equivalents for two-winding transformers, deriving how transformer connection side, step-up or step-down operation, transformation ratio, leakage impedance, and magnetizing branch losses affect the coefficients used in the power formulas. The authors also outline how to form these coefficients during iterative solution and how to represent three-winding transformers and autotransformers as star equivalents. The approach is intended to permit calculation of complex system operating modes directly from transformer and network parameters, without preliminary reduction to common base conditions.
Full Text
Toward a Unified Algorithm for Calculating Network Capacities
R. I. Borisov, S. G. Slyusarenko
(Presented by the scientific seminar of the Department of Electrical Systems and Networks)
This paper proposes an algorithm for using a generalized expression for network powers in calculating steady-state operating conditions of electric systems on the basis of solving equations of nodal power balances.
For symmetric four-terminal networks, the generalized form for calculating network powers is
\[ P_{ij}=\frac{U_i^2 M_{ij}}{B_{ij}}\cos(\Psi_{B ij}-\Psi_{M ij}) -\frac{U_iU_j}{B_{ij}}\cos(\delta_i-\delta_j+\Psi_{B ij}); \tag{1} \]
\[ Q_{ij}=\frac{U_i^2 M_{ij}}{B_{ij}}\cos(\Psi'_{B ij}-\Psi_{M ij}) -\frac{U_iU_j}{B_{ij}}\cos(\delta_i-\delta_j+\Psi'_{B ij}), \tag{2} \]
where
\[ \dot M=\dot A-\dot D; \qquad \Psi'_{B ij}=\Psi_{B ij}-\frac{\pi}{2}. \]
Step-up and step-down two-winding transformers are replaced by T-equivalent circuits according to Fig. 1, i.e., they are nonsymmetric four-terminal networks. The calculation formulas for the coefficients of four-terminal networks are presented in Table 1. Below one of the
Table 1
| Step-down transformer | Step-down transformer | Step-up transformer | Step-up transformer | |
|---|---|---|---|---|
| $\dot Z_\mu=\infty$ | $\dot Z_\mu\ne\infty$ | $\dot Z_\mu=\infty$ | $\dot Z_\mu\ne\infty$ | |
| $\dot A$ | $K_{\mathrm T}$ | $K_{\mathrm T}$ | $\dfrac{1}{K_{\mathrm T}}$ | $\dfrac{1}{K_{\mathrm T}}$ |
| $\dot B$ | $\dfrac{\dot Z_{\mathrm T}}{K_{\mathrm T}}$ | $\dfrac{\dot Z_{\mathrm T}}{K_{\mathrm T}}$ | $\dfrac{\dot Z_{\mathrm T}}{K_{\mathrm T}}$ | $\dfrac{\dot Z_{\mathrm T}}{K_{\mathrm T}}$ |
| $\dot C$ | $0$ | $\dfrac{K_{\mathrm T}}{\dot Z_\mu}$ | $0$ | $\dfrac{K_{\mathrm T}}{\dot Z_\mu}$ |
| $\dot D$ | $\dfrac{1}{K_{\mathrm T}}$ | $\dfrac{1}{K_{\mathrm T}}\left(1+\dfrac{\dot Z_{\mathrm T}}{\dot Z_\mu}\right)$ | $K_{\mathrm T}$ | $K_{\mathrm T}\left(1+\dfrac{\dot Z_{\mathrm T}}{\dot Z_\mu}\right)$ |
variants for taking the indicated feature into account for the use of the generalized expressions for network powers (1, 2), using the example of a two-winding power transformer.
Figure 1 shows the equivalent circuits of a step-down (a) and a step-up (b) transformer. We assume that the leakage impedance \(\dot Z_T\) and the impedance of the magnetizing branch \(\dot Z_\mu\) have been referred to the high-voltage side, while the transformation ratio of the transformer is determined by the relation
\[ K_T=\frac{|\dot U_{TB}|}{|\dot U_{TH}|}. \]
Let us consider the expressions for the active component of the network power with allowance for no-load losses for step-up (3, 4) and step-down (5, 6) transformers. Since in all cases
Fig. 1.
(Table 1) the coefficient \(\dot B\) of the equivalent four-terminal network is determined in the same way, we shall describe it by the modulus \(B\) and the angle \(\Psi_B\), while the coefficients \(\dot A\) and \(\dot D\) will be written out in detail.
\[ P_B=\frac{1}{K_T}\frac{U_B^2}{B}\cos(\Psi_B-\Psi_A)-\frac{U_B U_H}{B}\cos(\delta_B-\delta_H+\Psi_B); \tag{3} \]
\[ P_H=K_T\left|1+\frac{\dot Z_T}{\dot Z_\mu}\right|\frac{U_H^2}{B}\cos(\Psi_B-\Psi_D)-\frac{U_B U_H}{B}\cos(\delta_H-\delta_B+\Psi_B); \tag{4} \]
\[ P_B=\frac{1}{K_T}\left|1+\frac{\dot Z_T}{\dot Z_\mu}\right|\frac{U_B^2}{B}\cos(\Psi_B-\Psi_D)-\frac{U_B U_H}{B}\cos(\delta_B-\delta_H+\Psi_B); \tag{5} \]
\[ P_H=K_T\frac{U_H^2}{B}\cos(\Psi_B-\Psi_A)-\frac{U_H U_B}{B}\cos(\delta_H-\delta_B+\Psi_B). \tag{6} \]
It follows from expressions (3, 4, 5, 6) that before calculating the network powers in the generalized form (1, 2) it is necessary to determine which winding, high- or low-voltage, of the transformer under consideration is connected to the calculation node, and also to establish whether this winding is primary or secondary.
Thus, for \(Z_\mu=\infty\), for a transformer connected to the calculation node by its high-voltage winding, \(M=\frac{1}{K_T}\); whereas in the case of connection by the low-voltage winding \(M=K\) (Fig. 2, a).
The formation of the coefficient \(\dot M\) with allowance for no-load losses should be carried out according to the circuit of Fig. 2, b.
The parameters \(U_i;\ U_j;\ \delta_i;\ \delta_j\) in the block for forming the coefficient \(M\) must be taken from the vectors of initial approximations. To avoid incorrect operation of the formation block, the dependent parameters of the operating condition must, at each step of the iterative calculation of the normal operating condition, be determined by summing the corresponding components of the solution vector and the correction vector.
Fig. 2.
The equivalent circuits of three-winding transformers and autotransformers should be represented in the form of a three-rayed star; the magnetizing branch should be taken into account from the side of the primary winding. The basic requirement in preparing the initial data remains the calculation of the leakage resistances of the windings and of the resistance of the magnetizing branch, reduced to the high-voltage side. For example, for an autotransformer with a low-voltage primary winding, the quadripole replacing the high-voltage winding will contain only the longitudinal reactance. The other two windings must be represented as two-winding transformers (Fig. 1). In this case the low-voltage winding will correspond to circuit (b), and the medium-voltage winding to circuit (a) with \(Z_\mu=\infty\).
Thus, the use of generalized expressions for network powers can ensure the calculation of operating conditions of complex power systems with transformers directly from the initial parameters, i.e., without first reducing them to base conditions.
REFERENCES
V. A. Venikov, I. P. Siuda. Calculations of Operating Conditions of Long-Distance AC Power Transmission Lines. Moscow, “Vysshaya shkola,” 1966.