Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

Example 4.5 Distributed Slack Bus Power Flow for the IEEE 14-Bus System

Table 4.5 shows the results of the power flow with distributed slack bus model for the IEEE 14-bus test system. The loss participation factors of generators 1 and 2 are γ1 = γ2 = 1. Furthermore, pG0,1 = 2.3 pu for the slack bus. In this case, the final generated powers are very similar to the single slack bus model. This result is a consequence of the fact that pG0,1 pG0,2.

Table 4.5 Power flow results for the IEEE 14-bus system with distributed slack bus model

4.5A General Framework for Power Flow Solvers

Let us consider a set of autonomous ODE:

The simplest method of numerical integrating (4.68) is the explicit Euler’s method, as follows:

where Δt is a given step length.

The analogy between (4.40) and (4.69) is straightforward if one defines:

Equations (4.40) can thus be viewed as the ith step of the Euler’s forward method where Δt = 1 [127]. Furthermore, robust Newton’s methods (4.48) are nothing but the ith step of the Euler’s integration method where Δt = α. In other words, the computation of the optimal multiplier α corresponds to a variable step forward Euler’s method.

Equations (4.68) and (4.70) leads to:

Thus, assuming that gy is not singular, y0 is also the solution of (4.8). Assuming a non-singular Jacobian matrix for the power flow equations is an implicit hypothesis of any power flow analysis based on the factorization of gy (see also the discussion in Section 5.4 of the Chapter 5).

4.5.1Stability of the Continuous Newton’s Method

Di erentiating (4.70) with respect to y leads to:

=−[gy ]−1gy − ( Ty ([gy ]−1))g(y)

=−Iny − ( Ty ([gy ]−1))g(y)

where Iny is the identity matrix of order ny . Since the equilibrium point y0 is a solution for g(y0) = 0, one has:

To prove equation (4.74) let us define the following quantities, based on tensor notation:

the region of attraction. A case study that confirms this hypothesis is given in [196].

Example 4.6 Runge-Kutta’s Formula for Solving the Power Flow Problem

It is well-known that the forward Euler’s method, even with variable time step, can be numerically unstable. Reference [127] suggests that, given the analogy between the power flow equations (4.8) and the ODE (4.68), any well-assessed numerical method can be used for integrating (4.68). It is thus intriguing to use some e cient integration method for solving (4.8). Since

˜ − −1

the computation of f = [gy ] g implies the inversion of the power flow Jacobian matrix, only explicit integration methods are suitable and computationally e cient, since one does not need to compute the Jacobian matrix

˜

of f . Explicit numerical integration methods, including a variety of RungeKutta’s formulæ are described in Subsection 8.3.1 of Chapter 8.

It is worth noting that any order and any version of explicit integration method can be used, and any of these methods is expected to be numerically more stable than the Euler’s forward method. For example, Figure 4.10 compares the convergence behavior of continuous Newton’s method solved using the classical 4th order Runge-Kutta’s formula (8.35) (see Example 8.2

Fig. 4.10 Convergence behavior of Runge-Kutta’s 4th order formula and the Iwamoto’s method for the IEEE 14-bus system

of Chapter 8) and the Iwamoto’s method for the IEEE 14-bus system. The use of a robust power flow solver algorithm is not required in this case since this is a well-conditioned case. However, the results confirm that the RungeKutta’s formula converges faster than a variable step Euler’s method (e.g., Iwamoto’s method).

Script 4.6 Runge-Kutta’s Formula for Solving the Power Flow Problem

The Python implementation of the generic ith iteration of the Runge-Kutta’s 4th order formula (8.35) provided in Chapter 8 looks as follows:

yold = system.DAE.y

k1 = alfa*calcInc() system.DAE.y = yold + 0.5*k1

k2 = alfa*calcInc() system.DAE.y = yold + 0.5*k2

k3 = alfa*calcInc() system.DAE.y = yold + k3

4.6 Summary

This section summarizes most relevant concepts related to power flow analysis.

Power flow problem vs. classical circuit analysis: The power flow analysis di ers from classical circuit analysis in what concerns input data. Typical circuit input data are generator voltage phasors and load impedances. Thus, circuit analysis problem formulation is based on current injections that result in a linear problem if all parameters are constant. On the

other hand, typical power flow data are load power consumptions and generator active power supplies. Thus, the power flow problem formulation is based on power injections that result in a nonlinear set of algebraic equations. The intrinsic nonlinearity of the power flow problem makes its solution a challenge.

System model: The classical power flow problem formulation considers only constant PV generators and constant PQ loads. Furthermore, one generator, the slack, is defined as slack and is used as phase angle reference. These models are deduced based on practical assumptions and operation of transmission systems. However, the classical power flow problem formulation can be improved in a variety of ways. For example, the distributed slack model is a better description of the behavior of the system than the standard single slack model. Other improvements of the classical models concerns the inclusion of detailed models of regulating transformers and dynamic devices. The entire Part III is devoted to provide further insights on detailed power flow models.

Power flow solvers: The classical formulation of the power flow problem can be solved only numerically. With this aim, several methods are available, including but not limited to: Gauss-Seidel’s, Jacobi’s, Newton’s, FDPF, and dc power flow methods. The Newton’s method is available in several versions: inexact (e.g., GMRES), dishonest and robust.

Dc power flow: The dc power flow formulation avoids the issues originated by the nonlinearity of the classical power flow problem. The dc formulation allows simplifying the equations so that only active power mismatches and bus voltage phase angles are considered. The resulting problem is linear but completely neglects reactive power flows and bus voltage magnitude deviations. For this reasons, the error introduced by the dc power flow model are not negligible.

General framework for power flow analysis solvers: Power flow algorithms based on the Newton’s method and its variants can be viewed as particular implementations of a general class of solvers, namely the continuous Newton’s method. This approach shows that the map of an iterative method is formally equivalent to find the equilibrium point of a set of differential equations. The equilibrium point of the equivalent ODE system is the solution of the power flow problem.

p2 for which the reactive power generated by the slack bus is the maximum one, say q1max. Since the system has no other reactive power source, the load power cannot be increased any further. If the generator reactive power limit is reached before the transmission system limit, the former yields the point of collapse or, which is the same, the pmax2 value.

In order to determine the collapse point due to reactive power limit for the 2-bus system, assume that the slack is modelled as a constant QV generator, where q1 = q1max. The resulting power flow equations are:

In this case, the voltage magnitude v1 is a variable. Using the second and the third equations of (5.7), one has:

v1 = xLq1max + v22 + xLp2 tan φ2 (5.8)

Then, substituting (5.8) in (5.2), the expression of p2(v2) becomes:

If (5.6) and (5.9) intersect in the upper part of (5.6), then the generator reactive power limit occurs before the transmission system limit. This situation is illustrated in Figure 5.3. The interpretation of Figure 5.3 is as follows: as far as q1 < q1max, the system is described by (5.6); then, at q1 = q1max, the slack bus model changes from constant vθ to constant qv and the system behavior is described by the equation (5.9).

The determination of PV curves and, in particular, of the maximum loading condition, has great relevance in security analysis. In fact, the knowledge of the maximum loading condition allows defining the distance of the current working condition to the collapse. If this distance is too small, the system operator has to take corrective actions to provide a minimum security margin, i.e. a minimum distance of the current operating point to the collapse.

Unfortunately, analytical formulæ such as (5.6) or (5.9) cannot be found for a generic system. Even for the 2-bus system considered so far, including a resistance in the transmission line prevents from obtaining an explicit expression for p2(v2). Thus, a general numerical method for determining the maximum loading condition is desirable. The following sections describe systematic approaches to tackle this problem.