The Physics of Energy, page 173
Figure 38.17 500 kV electric power transmission TSPs and LSTs. Both single circuit (three cables) and double circuit (six cables) structures are shown. The heights shown for the LSTs are typical. Cable locations are marked. In addition, each tower carries a ground wire (not shown) [302].
Figure 38.18 Aluminum conductor steel reinforced (ACSR) high-voltage transmission cables. Note that the cables are braided from many strands making them more flexible and easier to manipulate. (Credit: Lisa Cable)
A number of constraints must be considered in configuring transmission lines. As described in §3, transmission lines must be designed to minimize resistive losses. It is also important to minimize the reactance of the lines, which reduces the power factor and results in wasteful reactive power. Excess reactance in transmission lines connecting generators can also destabilize the system.
Estimating resistive losses is quite straightforward. According to eq. (38.12), the resistive heating per unit length in a transmission cable is approximately given by
(38.15)
where ρ and A are the resistivity and cross-sectional area of the conductor. Limiting resistive losses places stronger constraints on longer transmission lines, since total resistive loss grows linearly with length. There is also a constraint associated with eq. (38.15) that is independent of the length of the line. If gets too big, the cable may heat enough to stretch and sag below its allowed ground clearance or past its limit of elastic deformation. Whatever the length, the cross-sectional area of a cable and its response to resistive heating can be designed for the anticipated load (and power factor).
The constraints on transmission lines due to complex impedance effects are less transparent. The magnetic fields created by the currents in the conductors give rise to self-inductance in each conductor and mutual inductance between conductors. The mutual inductance dominates because it is proportional to , where D is the distance between the conductors and r is their radius. As mentioned above, is typically kept large to insulate the conductors from one another. This line inductance adds an imaginary component to the real impedance from the line resistance, effectively acting in series. Both terms grow linearly with the line length, but typically the line inductance dominates. For example, a typical arrangement of aluminum conductors used in a 345 kV transmission line may have a resistance per unit length of /km, and an inductive reactance per unit length of /km, more than ten times as large (Problem 38.15). In fact, for many purposes – in particular in the analysis of grid stability where phases are of central importance – it is a good approximation to ignore line resistance compared to inductive reactance and consider lossless lines.
Similarly, electric fields generated by charges on the wires give rise to line capacitance between the conductors. The capacitance is proportional to , so the large separation between conductors tends to diminish the importance of line capacitance. In high-voltage transmission, line capacitance can be ignored entirely for lines with length less than ~80 km. At greater lengths, capacitance must be included along with line resistance and inductance in order to obtain an accurate computation of line impedance.
Constraints on High-voltage Transmission Lines
Both resistive and reactive effects limit the power-carrying capacity of high-voltage AC transmission lines. Resistance causes power losses and also heating in the conductors. Reactance, primarily due to induction between conductors, reduces the power factor and can lead to instabilities. Line inductance, like resistive losses, grows linearly with the length of the transmission line; inductance is often the limiting factor for long-distance power transmission.
Line inductance becomes problematic when it makes a significant contribution to the total impedance. For example, for a purely resistive load with , line inductance gives a substantial contribution to the impedance – lowering the power factor – when . When the inductance is too large on a line connecting two generators, it not only affects reactive power but can actually destabilize the network. This is described further in §38.4.4. We have considered here limits on steady-state performance. In some cases the ability of the system to respond to transients places more severe limits, particularly on the longest transmission lines [298].
For very long lines, one cost-effective way to avoid inductance problems is to convert from AC to DC at the generator, transmit the power as direct current, for which inductance is meaningless, and then convert back to AC before the step-down transformer at the head of the distribution system.
Transformers Transformers are required in order to raise or lower AC voltages, and are generally used in at least three stages along an electric power network. First, step-up transformers convert electricity generated in the 10–20 kV range to voltages suitable for long distance transmission. These transformers are usually located at the power plant. Step-down transformers are needed to convert from transmission to distribution voltage levels. These transformers are typically found at substations (see Figure 38.19), along with switches, circuit breakers, capacitor banks (see below), and other electronic devices to monitor and control electric power. Finally, smaller transformers are located at the interface between primary and secondary distribution networks. Transmission at low voltage is wasteful, so these final transformers are located as close to the retail customer as possible. Different patterns of settlement have led to different secondary distribution systems in the US and Europe (see below) that utilize transformers differently.
Figure 38.19 Diagram showing components at an electric power distribution substation. The step-down transformer converts power from the primary to the secondary voltage. Some of the other components shown are disconnect switches, circuit breakers, and lightning arrestors. (Credit: Shigeni23 reproduced under CC-BY-SA 3.0 via license Wikimedia Commons)
For electricity transmitted as three-phase power, each phase requires a separate transformer. Transformers therefore generally occur in banks of three, except at the final stage of distribution, where a single phase may be brought to a residential customer (see Figure 38.20). Transformers used in the electric transmission system are typically more than 99% efficient. Nevertheless, because they handle hundreds of megawatts of power, transformers must be designed to dissipate heat efficiently. This requirement raises its own set of environmental problems (Question 38.4).
Figure 38.20 Left: A large-capacity three-phase generator-step-up (GSU) transformer for nuclear power plants manufactured by Mitsubishi Electric. It is rated at 760 MVA (see §38.2.3) and transforms electric power generated at 25 kV to 345 kV transmission voltage. Note the three phases entering and the forced air cooling fans. Right: A pole-mounted single-phase distribution step-down transformer for supplying power at 120 V to a few residences in the US. Note the single-phase input on top. This canister transformer is oil cooled. (Credit: (Left) Mitsubishi Electric Power Products, Inc; (Right) Steve Mann)
Voltage Support As electric power flows through transmission and distribution networks, its voltage degrades. Line resistance decreases the magnitude of the voltage. Reactive effects reduce the power factor, which increases transmission current and leads to further line voltage reduction. Electrical devices are designed to accept a certain range of supplied voltage. In the US, for example, the standard is , so a device rated for 120 V is designed to function with voltages between 114 and 126 V. To meet these standards, voltage support devices are added to the grid, usually located at the same substations where transformers are needed. Variable transformers have an output voltage that can be adjusted by tapping the output coil at different locations, effectively changing the number of windings. These are the most straightforward and inexpensive voltage support devices, and are used to adjust the magnitude (but not the phase) of the voltage as needed. Traditional transformer taps must be physically moved and typically have some limited number of settings, making this a cumbersome way to support voltage; settings are typically either set to fixed positions or pre-programmed to change in response to anticipated daily fluctuations in load. Capacitor banks allow operators to correct the phase of the line voltage. Adding a capacitance in parallel with inductive loads corrects the power factor, and provides a local source of reactive power that reduces the transmission current needed, consequently decreasing the resistive voltage drop in the network (Problem 38.8). Advances in technology in recent years continue to improve the level of detail and automation that is achievable in grid control in areas such as voltage support; in particular, more modern devices known as static VAR compensators or SVCs use semiconductor electronics to provide voltage support over short response times. These devices can rapidly compensate for changes in reactive load without operator intervention, though they are more expensive than mechanically switched capacitors.
Other Components: Safety and Control A variety of other components, including switches, circuit breakers, lightning arresters, and many types of measurement and monitoring equipment are employed throughout an electric grid. Circuit breakers are placed so as to isolate other components such as generators, transformers, or loads when tripped due to overloads in the system. Switches allow operators some control over how power flows through a network, although as already mentioned it is not possible to dictate the flow of power in complete detail.
38.4.2 Distribution of Three-phase Power
Three-phase electricity can be connected to power loads in a number of different ways. This versatility, along with reduced resistive losses, provide additional benefits of three-phase power beyond the virtues in power generation mentioned earlier.
Direct three-phase use The simplest way to connect and use three-phase power is to directly connect all three phases to compatible devices. Three-phase power is perfectly matched to motors with the same phase structure, allowing them to provide constant torque.
Wye connection – phase to ground With three phases running along distribution lines, three customers or devices can each be supplied with the output of a single phase, with voltage (120 V in the US or 230 V in Europe) relative to ground. This is the wye configuration shown in Figure 38.12(a). Naively, it might seem that the Wye configuration would require six wires: three feeds and three returns. If the loads on each phase are equal, however, then the phases cancel and the sum of the three currents is zero as in eq. (38.14). In this situation, no additional return wires are needed. Even if the loads are not perfectly balanced, the sum of the return currents is generally small enough to be carried through the ground (literally). Thus, when three-phase power is distributed to customers, utilities strive to balance the loads on the three phases. A glance at overhead wires carried by many utility poles in residential neighborhoods will confirm the presence of three conductors. In some cases, a fourth common neutral wire is used to return excess current when loads are out of balance; this can be necessary particularly in local distribution networks where a small number of varying loads have larger fluctuations as a fraction of total power.
Delta configuration – phase to phase An alternative way to connect to three-phase power is to tap one phase with respect to a second. As illustrated in Figure 38.21(a), if all three combinations are used, the result is a Δ connection analogous to the Δ generator configuration of Figure 38.12. The interphase voltage, plotted in Figure 38.21(b), has (208 V in the US or 398 V in Europe). Higher voltage is often useful for customers with heavier machinery. The customer provided with two phases also has the option of connecting either of the supplied phases to ground to obtain single-phase power at .
Figure 38.21 (a) Delta connection for three-phase power. (b) Voltage in phases A (red) and C (green) and the difference (purple), which would be accessed if one were tapped relative to the other.
In addition to voltage versatility, three-phase power reduces resistive losses in transmission lines. Suppose, for example, that a single-phase generator supplies power to a purely resistive load over two conductors, one carrying current in each direction. Total resistive losses are , where R is the resistance of a single conductor. In contrast, if a three-phase generator provides to each of three resistive loads through three conductors, the current in each conductor is one third as large as in the single-phase case, and the resistive losses in the three conductors are . Thus using three-phase power it is possible to transmit the same power with only 1/6 of the resistive losses (Problem 38.17). This advantage of three-phase power could alternatively be exploited by using smaller-diameter conductors, while tolerating the same resistive losses as would occur for two-phase power. Although this analysis suggests that even more phases would lower resistive losses even further, fixed costs and additional complexity associated with the additional phases have limited practical implementation to three phases.
38.4.3 The Topology of Transmission and Distribution Systems
Transmission and distribution systems take quite different forms. Transmission systems are typically highly interconnected mesh networks in which power can flow along many alternative routes. The transmission network for a section of a regional power system is shown in Figure 38.22. Since many generating plants, scattered over a large region, must be able to serve a complex and distributed set of loads, interconnectivity is essential. It must be possible to take power lines and generators out of service without disconnecting any of the distribution systems or other generators from the transmission network, even if in some cases the direction of current flow in transmission lines reverses when sources and loads change.
Figure 38.22 Diagram of a section of a transmission system. Voltages at busbars are labeled. After [296].
The interconnected structure of transmission networks leads to a number of complications in managing the flow of energy through the network. Energy from a given generator can flow along a variety of paths, often passing through one or more other generators, before reaching the load(s). The details of how power flows in a given network with constantly varying sources and loads can be analyzed by solving a system of linear equations analogous to eq. (38.1) for each segment, with the additional constraints that at each junction the incoming current is equal to the outgoing current.7 The resulting energy flow can have unexpected features.
Decreasing the output of a generator may actually increase the current being carried on a line connecting that generator to another. There can also be loop flow, where there is a net flow of energy around a closed loop within the circuit. Loop flows contribute to resistive losses. In addition, loop flows can result in power inadvertently crossing jurisdictions, confounding regulatory and business practices.
Electricity distribution networks are somewhat simpler in principle than transmission networks. They are typically radial or star networks where primary feeders emerging from a distribution substation are in turn stepped-down to secondary distribution voltages that feed retail customers. Distribution networks may also contain loops in order to provide redundancy, particularly in densely populated areas. Only one of the loop paths is likely to be active at any given time, however, with the other inactivated by an open switch. The reason for limiting service to one line is to avoid a phenomenon known as islanding in which current continues to flow around a distribution loop after the upstream connection has been broken either intentionally or in a power outage. Islanding poses a safety hazard to utility workers who expect to find no activity in lines that have been isolated from the grid.
Distribution networks have distinctly different characters in North America compared to Europe and many other countries, as illustrated in Figure 38.23. North American distribution networks were established when population densities were low, and feeder lines had to be long. To minimize resistive losses, higher voltage primary lines were extended far down the distribution network, with transformers – typically of low capacity and mounted on poles – providing the final retail voltage at locations very near to the customer. Thus, secondary distribution lines in the US tend to be short. When new services are needed, an additional transformer mounted on a nearby pole initiates a new secondary distribution network. Electrification in Europe occurred when population densities were higher, so primary lines are shorter, and transformers, that are typically in vaults, have higher capacity and serve more customers. Secondary lines are longer, but secondary line losses are less because the voltage is higher than in the US. In the European model, adding more service requires running more secondary lines from the transformer and is easy until the capacity of the transformer is saturated, at which point a new primary line must be run. The European system has the advantage that the distribution lines are more easily put underground, because low-voltage secondary lines can more easily be run underground than the high-voltage primary lines. Thus the overhead electric power lines that are ubiquitous in North America are rare in Europe. As noted, however, the North American model accommodates growth more easily than the European model.
Figure 38.23 Comparison of North American and European distribution systems. The North American system propagates primary distribution further down the line and also provides power at the primary line voltage to some (commercial) customers. After [296].
