Load Flow and Power System Simulation [
5.6.1 Uses of Load Flow
Section 5.4 looked at the maths describing an individual overhead line or underground cable. Mature power systems are likely to have thousands of such lines, all interconnected. The same basic maths applies to each and every line, but now the equations must be solved simultaneously. Structured procedures for such calculations are known as load flow.
A basic load flow calculation provides information about the voltages and currents and complex power flows throughout a network, at a particular point in time, with a given set of load and generation conditions. Additional information, such as losses or line loadings, can then be easily calculated.
Load flow analysis is an essential tool that provides the following vital information for the design as well as the operation and control of power systems:
• checking whether equipment run within their rated capacity;
• checking that voltages throughout the network are kept within acceptable limits;
• ensuring that the power system is run as efficiently as possible;
• ensuring that the protection system will act appropriately under fault conditions and that under likely contingencies the system will remain secure and operational;
• assisting in the planning of the expansion of conventional and renewable generation and the necessary strengthening of the transmission and distribution system to meet future increases in power demand.
Load flow calculations are central to the numerous software packages available for power system simulation, including PSS/E and DIgSILENT. Most such packages also include many additional functions, such as the following:
• Fault analysis determines the currents that will flow in the event of a short circuit. Such information is critical in the design of switchgear and protection systems.
• Unbalanced analysis allows modelling of networks where the three phases are not perfectly balanced. This is particularly relevant in networks where singlephase loads are connected.
• Dynamic simulation allows transients and stability to be examined. This is particularly important in the design of transmission systems. The modelling has to include the dynamic characteristics of the generators and their associated control systems.
5.6.2 A Particular Case
The oneline diagram of the Icelandic national power system shown in Figure 5.12 is powered mostly by renewable energy, primarily hydro, but with an increasing contribution from geothermal. The generators are indicated by the © symbol, while the loads are shown as triangles. In practice, these loads are the bulk supply substations that feed power to the distribution networks, but for now they will be thought of as simple point loads. The thick horizontal lines are nodes (also known as buses) and, although they have a finite length in the diagram, electrically they are just connection points: they have zero length and zero impedance. The thinner lines in the diagram are the actual transmission lines and each one has a finite impedance, which may be estimated from the line characteristics and its length. The system uses a mix of 220, 132 and 66 kV and the transformers connecting these are indicated by the usual § symbol.
In the design and operation of a power system it is important to know the power flows, both active and reactive, the voltages (which will normally differ slightly from the nominal
values mentioned above) and the currents, throughout the system. The almost illegible numbers in Figure 5.12 are the values of these variables, which have been calculated by load flow analysis. How such calculations are performed will now be considered.
Consider the transmission line running across the middle of the diagram from node Brennimelur to node Hrauneyjarfoss. The power transfer over this transmission line will obey the mathematical analysis presented earlier in this unit. Indeed, the power transfer over every individual transmission line in the whole system must obey the equations presented earlier. The challenge is that, in general, the equations require to be solved simultaneously.
While the mathematical techniques and the software packages for performing load flow analysis are readily available, obtaining suitable input data can be much more difficult. The required input data can be divided into network data and load/generation data.
5.6.3 Network Data
The impedances of the individual lines are usually calculated from knowledge of the line type and the length of the line. In practice, particularly in lower voltage networks, these are not always known with absolute certainly. The line may have been installed fifty years ago and could have been modified since. Utilities do not always have perfect records. Also, particularly with underground cables, the details of the installation can significantly affect the characteristics; for example the dampness of the ground and the proximity of other cables can have a significant effect.
The ratios and impedances of the transformers must also be known and, again, records can lack details. Any tap changing mechanisms on transformers must also be included in the model.
5.6.4 Load/Generation Data
Time Dependence
One load flow analysis solution provides the results for one operational instant. If any of the loads or generator outputs are changed, then the analysis must be run again. With modern computers, rerunning the analysis is trivial; the challenge is in deciding what data should be used for loads and generators, given that they are continuously varying.
A common approach is to identify the worst cases, and typical choices are the winter maximum peak demand and the summer minimum, except in the US where the maximum is due to summer airconditioning. At the winter peak, the system is stretched to its limit and it is necessary to ensure that no equipment is overloaded or any node voltage is below the minimum permitted. During the summer minimum, because of the low loading of transmission lines and cables, there may be an excessive reactive power available that would manifest itself as high voltage at nodes.
In a transmission system, the loads and generation normally vary smoothly and the relevant maximum and minimum figures can readily be identified. In lower voltage networks, the loads are much more variable and the worst case conditions can be very difficult to identify.
Types of Nodes (Buses) PQ Buses
The obvious way of describing the load at a node is by specifying it as an impedance. Simple uncontrolled loads, such as heaters and incandescent light bulbs, tend to behave as nearconstant impedances and thus the power they draw varies with the square of the voltage: a ±10% variation in voltage, which is not uncommon, gives powers ranging from 81 to 121%. Many loads, however, include some sort of control mechanism such that the power they draw does remain near constant irrespective of the voltage changes. Most electronic based devices behave in this way.
Another important example is a distribution network that is supplied via a tapchanging transformer. Such a transformer senses the voltage on the lower voltage side and changes the taps so that this is maintained at nearly nominal value. A fixed load impedance on the transformer secondary appears on the primary side as a fixed P and Q demand irrespective of the transformer ratio. As the taps change the voltage on the primary side of the transformer may change substantially, but the P and Q are invariant. For example the tap changing transformer at the primary substation can be adjusted to achieve a constant voltage on the 11 kV side irrespective of (modest) voltage changes on the 33 kV side. Thus, the power drawn by the 11 kV network from the 33 kV network is also held near constant. If a load flow calculation was being performed on the 33kV network, which would typically feed several 11kV networks, it would be reasonable to represent each primary substation as a constant P and Q. The same is true in load flow modelling of the higher voltage networks.
To summarize, assuming that the voltage at a load bus is kept constant, the load can be expressed as a fixed P and Q demand; such nodes are referred to as PQ buses. As will be explained later, this is a very convenient formulation for the purposes of carrying out the load flow.
Small renewable energy generators also fall in the PQ bus category. Distributed generators are infrequently called upon to control the network voltage. Instead, they are often configured to operate at nearunity power factor (Q = 0); in this case, it may be appropriate to label the node to which they are connected as a PQ node. In the case of fixed speed wind turbines, however, the reactive power consumed by the induction generator will be dependent on voltage, as will the reactive power generated by the power factor correction capacitors. Many load flow software packages include facilities to model induction machines and related equipment appropriately. The situation is similar with small hydro systems interfaced to the grid through induction generators. Energy from photovoltaic, wave and tidal schemes and MW sized wind turbines is fed to the grid through a power electronic converter. This provides the facility of reactive power injection/extraction at the point of connection.
To summarize, for relatively small embedded RE generators the P injection depends solely on the RE source (wind, sun, water) level at the time and the Q injection either on the bus voltage or on the setting of the power electronic converter. In the latter case the converter could be regulated to inject active power at a chosen power factor.
According to the generally accepted convention summarized in Figure A.20, at PQ nodes generators inject active power and so P is positive, whereas for loads, P is negative. The reactive power Q direction is defined similarly.
PV Buses
For large synchronous generators, the Q is often not specified, and instead it is the voltage V that is known. This is because such generators are fitted with AVRs that hold the V constant. To accommodate this, in load flow analysis nodes where such generators are connected are referred to as PV buses and are dealt with a little differently in the maths. Unfortunately, PV buses are sometimes described as generator buses. which makes sense so long as all the generators are large synchronous generators with AVRs.
As renewable energy generators increase in size, utilities have been developing regulations requiring that such generators behave in a traditional manner. Multimegawatt wind turbines connected to the network through PWM inverters may therefore be required to regulate the local bus voltage. In such cases the node has to be treated as a PV bus.
Slack, Swing or Reference Bus
The two types of buses described earlier require that the Ps are specified at all the network nodes before the load flow calculations are initiated. This does not make sense in terms of the conservation of power principle because the transmission losses in the system are not known before the solution is arrived at! This conundrum is resolved by allowing one bus that has a generator connected to it to be specified in terms of the magnitude V and angle 5 of its voltage. Such a bus is known as a slack, swing or reference bus. The voltage at this bus acts as the reference with respect to which all other bus voltages are expressed. At the end of the load flow the calculated P and Q at this bus take up all the slack associated with the losses in the transmission.
5.6.5 The Load Flow Calculations
In small networks, it is often possible to obtain valid and useful results by direct application of the mathematical analysis presented earlier. Also, larger networks can often be reduced to equivalent circuits that can be solved in the same way. However, load flow analysis of any system beyond a few nodes is carried out by a computer.
Each bus of a power network is characterized by a number of variables. For a network with predominantly reactive transmission line impedances all these variables are linked by the complex Equation (4.7a), replicated below:
This equation describes the performance of a synchronous generator but it is equally applicable to a transmission line linking two buses. Note that there are four variables associated with each bus: the active and reactive power injected or extracted at the bus and the magnitude and angle of the bus voltage. The complex Equation (4.7a) can be split into two separate equations, one for the real part and another for the imaginary part. For a network with n nodes, there are 2n simultaneous equations to be solved; hence at each node two of the four variables have to be specified. All this fits neatly with the earlier arrangement of specifying buses in terms of P and Q, P and V or V and 5.
The solution of the 2n equations describing the network is not a trivial task! The basic Equation (4.7a) linking the node variables is nonlinear because it contains products of the
variables and trigonometric functions. The load flow solution to such nonlinear simultaneous equations requires iterative techniques and is beyond the terms of reference of this book. In general, therefore, the solution of most practical networks requires a very structured approach to the mathematics and is carried out by computers. Indeed, it was one of the first useful applications of computers during the 1950s. There are many mathematical techniques for performing load flow analysis. They are all iterative, and most begin by representing the network as a matrix. They often use NewtonRaphson or GaussSeidel iterative methods or embellished versions of these. They should all converge to very similar answers and the choice between the methods is mostly a matter of speed and reliability of convergence. The different methods can perform better or worse with different types of networks. For more information consult power system simulation packages PSS/E and DIgSILENT. A recent trend is that load flow software is being integrated into geographical information systems (GIS) and this integrates with the asset management side of the electricity utility business.
The assumption of constant voltage at the PQ buses inherent in the formulation of the load flow described above is not valid at the distribution level. The distribution transformers (11 kV to 400 V) shown in Figure 5.11 are not controlled and representing them as constant P and Q will lead to some inaccuracy. Load flow software that is intended for modelling the lower voltage levels of distribution networks often includes the facility to define loads that do vary with respect to voltage. However, obtaining load data that includes this characteristic may be less straightforward.
5.6.6 Results
The basic results provided by a load flow analysis are the power flows (both active and reactive) and the voltages and currents (both magnitude and phase) for all the lines and nodes throughout the system. (Strictly speaking, some of this information is redundant.) From these basic results, an engineer can readily identify or calculate:
• any overloaded lines or transformers;
• any voltages that are under or over acceptable limits;
• power losses, both active and reactive, for individual lines or for the whole system.
This information influences the design and operation of the network. Load flow studies are routinely performed to check the state of the network as the demand varies through the day or whenever changes to the network or to generation are being considered.
5.6.7 Unbalanced Load Flow
Ideally, the voltages and currents in threephase power systems are perfectly balanced, which leads to efficient operation and greatly simplifies any analysis. However, in low voltage networks particularly, the phases are not always balanced in practice and full analysis requires unbalanced load flow techniques. Unbalanced load flow usually employs the method of symmetrical components, which involves the description of the unbalanced network as the sum of three simpler networks. Many software packages can perform unbalanced load flow
analysis (perhaps sacrificing other features), but in practice the data they require is often unavailable.
Renewable Energy 101
Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of nonrenewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a nonrenewable supply, the nonrenewable energy sources release emissions into the air, which has an adverse effect on the environment.
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