Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

θh and vh and the rows associated with the derivatives of ph, qh have to be set to zero except for diagonal elements that have to set to 1. A simple script that implements this operation is presented in Script C.1 of Appendix C. Further discussion about islanded buses as well as the more general concept of network connectivity is given in Subsection 11.3.3 of Chapter 11.

10.1.2Areas, Zones, Regions and Systems

Apart from buses, most power flow analysis tools allow defining other classes of topological devices such as areas, zones, regions and/or systems. For example, the IEEE common data format defines interchange area data [350]. These are generally used for evaluating power transfers between areas. For example:

hBA

where pex is the neat power exchange, BA is the set of buses that belong to the area and pG,h and pL,h the generated and consumed powers within the area. If the area is exporting power pex > 0, while pex < 0 otherwise. In power flow analysis, one cannot impose a fixed power exchange, say pex = pex0, unless some generator active power is left undetermined. Thus, in general, power exchange limits are used only in OPF analysis, where market rules or agreements among area operators impose limits to the import/export of active power between areas.

Typical data for such areas are depicted in Table 10.2. In some cases, an area slack bus can be defined. This slack bus is generally just a “suggestion” and does not a ect or redefine vθ generators. Only in case the area separates from the remaining system as a consequence of line outages, the slack bus is used as reference bus. Similar data can be defined for zones, regions and systems.

Another use of areas is the possibility of grouping variables to be plotted or visualized in the power flow report file. This feature is particularly useful when dealing with networks containing thousands of buses since the complete power flow report would result too large to be understandable. Depicting only a reduced number of buses, for example those pertaining to a given area can simplify the interpretation of the results.

Table 10.2 Area parameters

10.2Static Generators

In the classical power flow analysis, generators are only PV and slack ones. This section revises and generalizes the concepts and the models of static generators from the viewpoint of the implementation in a general power system analysis tool.

10.2.1PV Generator

PV generators impose the voltage magnitude and the power injected at the buses where they are connected, as follows:

There are three ways of implementing PV generators for power flow analysis.

1.The classical model considers the voltage vh as a constant and thus, only imposes one equations for the active power ph. This model allows reducing the number of power flow equations as well as the size of the Jacobian matrix. On the other hand, handling reactive power limits is complex because the number of variables and equations changes whenever a reactive power limit becomes binding. However, if generator reactive power limits are not considered, this is the most e cient model.

2.To avoid the issue above, one can use two equations, one for the active

power ph and one for the reactive power qh. If qGmax < qh < qGmin, the reactive power equation has to impose that the reactive power balance at node h is satisfied and that the row corresponding to qh and the column corresponding to vh in the Jacobian matrix gy are zero. Only the diagonal element has to be set to 1 to avoid singularity. In this way, the voltage

vh is not varied in the iterations of the Newton’s method. If qh ≥ qGmax or qh ≤ qGmin, then the reactive power is set to qh = qGmax or qh = qGmin and the Jacobian matrix gy is not modified so that the voltage vh can vary. This approach has the advantage of maintaining constant dimensions of the variables, the equations and the Jacobian matrix. However, the number of zeros of the Jacobian matrix changes whenever a reactive power limit becomes binding. This can be an issue if symbolic factorization is used.

3.Using three equations solves the issues of the previous model. Two equations are for the active and reactive power injections ph and qh, while the third one imposes the voltage value or the reactive power value as follows:

In this model, the reactive power qG is an internal variable of the PV generator. The drawback of this model is that each PV generator introduces an additional variable. Another drawback is that only one PV generator can be defined at each bus. In fact equation vh = vG0 would be duplicated in case of defining two PV generators at bus h. This is generally not a real issue, since it is not a good practice defining more than one PV generator at the same bus.

In case of using the distributed slack bus model, the active power equation becomes:

where kG is the distributed slack bus variable and γ is the loss participation factor. Table 10.3 depicts PV generator parameters, which include reactive power and voltage limits needed for optimal power flow and continuation load flow analysis. Refer to Chapters 5 and 6 for details.

Table 10.3 PV generator parameters

Example 10.1 Enforcing Generator Reactive Power Limits

This example focuses on the handling of PV generator reactive power limits. With this aim, consider the IEEE 14-bus system with the following modifications with respect to the base case:

1.The shunt capacity at bus 9 is removed.

2.The load at bus 4 is inductive instead of capacitive and is consuming 0.04 pu of reactive power.

252 10 Power Flow Devices

Table 10.4 Power flow results for the IEEE 14-bus system with generator reactive power limit violations

The power flow results, without enforcing generator reactive power limits are shown in Table 10.4. The PV generator at bus 8 is producing more reactive power than the maximum limit 0.24 pu (see Appendix D for the complete data of the IEEE 14-bus system). Thus, this solution is not acceptable.

Enforcing reactive power limits is a delicate task since it requires to switch the PV generator model to a constant PQ generator, fixing the generated reactive power to qG = qGmax or to qG = qGmin, depending on the limit that is binding. The principal di culty is that limits should be modelled as inequalities:

qGmin ≤ qG ≤ qGmax

Unfortunately, no power flow method discussed in Chapter 4 allows directly modelling and handling inequalities.

A possible approach is to solve a preliminary power flow without enforcing reactive power limit, to check the solution and, if there are reactive power limit violations, to re-run the power flow analysis changing binding PV generators to PQ ones. With this aim, it is not advisable to switch all critical PV generators at a time since doing so could lead to switch more generators than strictly necessary. In fact, switching a PV to a PQ generator leads to a redistribution of all power flows in the network and, as a consequence of this redistribution, some reactive power limit may not be binding anymore. The most secure strategy is to switch one PV generator per iteration, for example starting from the one that exceeds most its reactive power limit. Then, the power flow problem is solved again, and if some reactive power limit is violated, the procedure is repeated.

The method described above is not e cient, especially if the network contains thousands of buses and hundreds of PV generators. In order to save time, a common strategy is to check PV generator reactive powers on the fly, i.e., while executing the iterative method used for solving the power flow problem. The idea is to check the reactive power production of the PV generators at each iteration and if there is some limit violation, switch the PV generator to a PQ one. The main issue is to decide not only how many generators have to be switched per iteration, but also when it is convenient to apply the model switch. In fact, some limit violation can be due to a temporary power mismatch that disappear in the following iterations. On the other hand, if one waits too much, the e ciency can be low.

Table 10.5 shows the result of the power flow analysis for the IEEE 14bus system allowing switching PV generators since the first iteration of the Newton’s method. The reactive power limits of three generators are binding. However, this does not seem a reasonable result since, in Table 10.5, only one PV generator is exceeding its reactive power limit by a relatively small amount.

Table 10.5 Base case power flow results for the IEEE 14-bus system enforcing reactive power limits since the first iteration

Solving once again the power flow problem and enabling the check of PV generator reactive powers only after the second iteration provide the results that are shown in Table 10.6. As expected, only the generator at bus 8 is switched to a constant PQ model.

As a final remark, consider the following question: which is the better solution between the two depicted in Tables 10.5 and 10.6? From the mathematical point of view, both solve the power flow problem and provide a solution

Table 10.6 Base case power flow results for the IEEE 14-bus system enforcing generator reactive power limits

within technical limits. Thus, if one looks at each solution separately, both are acceptable. Actually, the solution shown in Table 10.5 is characterized by 0.1412 pu of active losses and by 0.3050 pu of reactive losses while the solution shown in Table 10.5 is characterized by 0.1360 pu of active losses and by 0.2743 pu of reactive losses. Thus, the latter solution is preferable if the goal is to minimize losses.

In conclusion, enforcing generator reactive power limit cannot be conveniently solved using a simple power flow problem. As shown in this example, using the Newton’s method or similar iterative techniques, one can obtain a feasible solution, but there is no guarantee that there not exist a better solution. Only formulating the power flow problem as a nonlinear programming optimization problem as described in Chapter 6 can provide, under certain hypotheses, the best solution with respect to a given objective function.

10.2.2Constant Voltage Phasor Generator

In principle, any number of constant vθ generators can be included in a network. In fact, consider the results shown in Table 10.6. The power flow

solution would not change if one assumes that the system has two vθ generators, say at bus 1 and bus 2, where:

v1 = 1.060 θ1 = 0

v2 = 1.045 θ2 = −0.0871 rad

However, since bus voltage phase angles are generally not known a priori, it is quite uncommon to define more than one vθ generator per interconnected ac network. The unique vθ generator is generally called slack bus. As discussed in Chapter 4, a slack bus is not fully justified unless it is an equivalent of a strong network with unlimited active and reactive power capacity. In general, a distributed slack bus model should be preferred.

From the implementation viewpoint, a vθ generator can inherit from the PV generators the voltage/reactive power model. In other words, the vθ generator can be a subclass of the PV one. Then, similarly to the PV generator, the angle/active power model can be defined in three ways (see also the previous section).

1.To assume that θh is a constant parameter. Thus no equations for the active power injection is needed. This model allows reducing the number of equations and the size of the Jacobian matrix. If an active power limit becomes binding, both equations and Jacobian matrix have to be re-sized. If no active power limits are considered, which is the standard case, this is the most e cient model.

2.To impose that the active power balance at bus h is always satisfied. In this case, one has to impose that the active power mismatch at bus h is zero and that the row corresponding to ph and the column corresponding to θh in the Jacobian matrix gy are zero. Only diagonal elements have to be set to 1 to avoid singularity. In this way, the voltage θh is not varied during the Newton’s method. This approach has the advantage of maintaining constant dimensions of the variables, the equations and the Jacobian matrix.

3.To add auxiliary variables and equations for the active and reactive powers produced by the generator. This model includes two additional variables, kG and qG, for the active and reactive powers, respectively:

As remarked above, it is unusual to consider active power limits for the slack bus. In fact, if the slack cannot provide the required active power, the power flow problem has no solution. However, this model has the advantage of providing a unique formulation for the single and the distributed slack bus model. In case of distributed slack bus model, the active power injection becomes:

where pG0 is the scheduled active power production for the vθ generator and all other equation are unchanged. Introducing the variable kG allows writing an unique code for the single and the slack bus model. In fact the number of variables and equations is always the same.

In case of the single slack bus model, kG is the active power production of the single slack bus.

Table 10.7 depicts the constant vθ generator parameters, which also contains data used in optimal power flow and continuation power flow analysis. In case of distributed slack bus model, the parameters pG0 and γ are required.

Table 10.7 Slack generator parameters

10.2.3 PQ Generator

PQ generators are modeled as constant active and reactive powers:

as long as voltages are within the specified limits. If a voltage limit is violated, PQ generators are converted into constant impedances, as follows:

qh = qG0v2/(vGlim)2

where vGlim is vGmax or vGmin depending on the case. For example, maximum and minimum voltage limits can be assumed 1.1 and 0.9 pu, respectively.

Table 10.8 depicts PQ generator parameters. The maximum and minimum reactive powers qGmax and qGmin can be defined in analogy with PV and vθ generators and can be used in CPF and OPF analyses.

From the implementation viewpoint, PQ generators have the same model as a PQ loads, which are described in the next section. Thus, PQ generators can be implemented as a subclass of the PQ load class. The only di erence is in the sign of active and reactive powers:

Alternatively, one can define a PQ generator using a PQ load and declaring negative power consumptions. However, a specific class for PQ generators is useful for separating power productions and power consumptions in the power flow report.

Table 10.8

10.3Static Loads

10.3.1PQ Load

(10.11)

as long as voltages are within the specified limits. If a voltage limit is violated, PQ loads are converted into constant impedances,1 as follows:

where vLlim is vLmax or vLmin depending on the case. For example, maximum and minimum voltage limits can be assumed 1.1 and 0.9 pu, respectively. Table 10.9 depicts PQ load parameters.

Table 10.9 PQ load parameters

In the standard transient stability analysis, PQ loads are converted to constant impedances after the power flow solution (see Section 8.2 of Chapter 8). In this case, PQ loads are forced to switch to constant admittances, as follows:

where v0 is the voltage value obtained through the power flow analysis. However, the adequacy of constant admittance or other load models depends on the simulation time frame as it is discussed in Chapter 14.

Example 10.2 Constant Power vs. Constant Impedance Load Models in Transient Stability Analysis for the IEEE 14-Bus System

Using a constant impedance or a constant power load model can drastically modify simulation results. Figure 10.1 shows the results for the IEEE 14-bus

1Some software package such as Eurostag allows defining the exponent of the voltage, as follows:

ph = −pL0v2/(vLlim)αp qh = −qL0v2/(vLlim)αq