Читать книгу Power Flow Control Solutions for a Modern Grid Using SMART Power Flow Controllers - Kalyan K. Sen - Страница 20
1.2 Traditional Power Flow Control Concepts
ОглавлениеThe flow of AC power, irrespective of its source, has two components: active power and reactive power. A transmission line consists of electrical conductors that may be composed of many sections. Each section consists of a resistance (R′), inductive reactance (X′L), and line‐to‐ground (shunt) capacitive reactance (X′C) as shown in Figure 1-3. Since there is no significant storage of electric power at the utility scale, the active power, except for the loss in the resistance of the conductor, reaches from one end of the line to the other end. This active power can be used for lighting, heating, cooling, motion force in electric motors, and so on. The distributed series inductive and shunt capacitive reactances of the line absorb and generate reactive power, respectively. This reactive power flow causes an extra loss in the resistance of the line.
Figure 1-3 Representation of a transmission line between sending and receiving ends.
The symbols shown in the figure are
Vs is the sending‐end voltage with a magnitude (Vs) and a phase angle (δs),
Vr is the receiving‐end voltage with a magnitude (Vr) and a phase angle (δr),
R′ is the line resistance in each section,
X′L is the inductive reactance in each section of the line, and
X′C is the line‐to‐ground (shunt) capacitive reactance in each section.
Transmission lines with lengths less than 50 miles (80.5 km) are classified as being short lines; lines of lengths between 50 and 150 miles (80.5 and 241.4 km) are classified as medium‐length lines and lines above 150 miles (241.4 km) are considered long lines. Consider a line in the interconnected transmission system, connecting sources and loads as shown in Figure 1-1 as a relatively short line where the capacitive shunt reactance from the line to ground and among the lines can be ignored as shown in Figure 1-4. The resistances and inductive reactances from all the line sections are lumped together as shown in the figure. The natural power flow in an AC transmission line depends on (1) magnitudes of the sending and receiving‐end voltages, (2) phase angle between these voltages, and (3) line impedance.
The additional symbols shown in the figure are
VXn is the natural voltage across the line reactance with a magnitude (VXn) and a phase angle (θVXn),
VRn is the natural voltage across the line resistance with a magnitude (VRn) and a phase angle (θVRn),
In is the natural line current with a magnitude (In) and a phase angle (θIn),
Psn is the natural active power flow at the sending end,
Qsn is the natural reactive power flow at the sending end,Figure 1-4 Power flow along a transmission line between sending and receiving ends.
Prn is the natural active power flow at the receiving end,
Qrn is the natural reactive power flow at the receiving end,
R is the line resistance (R > 0 and represents a positive resistance), and
X is the line reactance (X > 0 and represents an inductive reactance).
The natural active and reactive power flows (Psn and Qsn) at the sending end are derived in Appendix B as
(B‐12)
and
(B‐14)
where
(B‐13)
and the power angle is given in Chapter 2 as
(2‐27)
The natural active and reactive power flows (Prn and Qrn) at the receiving end are
(B‐21)
and
(B‐22)
Ignoring the line resistance as shown in Figure 1-5a, the natural active and reactive power flows (Psn and Qsn) at the sending end and the natural active and reactive power flows (Prn and Qrn) at the receiving end for a relatively short lossless line are
(2‐40)
(2‐43)
(2‐46)
and
(2‐48)
where
(2‐41)
In addition to using these formulae to characterize a two‐generator/single‐line power system network, they may be used when designing an electrical generator where the Vs and Vr are the generator’s internal voltage and terminal voltage, respectively, and X is the internal reactance of the generator as shown in Figure 1-5b. When designing an inverter, Vs represents the inverter’s output voltage, which is typically created using a Pulse‐Width Modulation (PWM) technique and passed through a filter that consists of an inductor with a reactance (X) and a capacitor (Cf) to create a filtered voltage, Vr, as shown in Figure 1-5c.
Figure 1-5 (a) Electric grid: power flow along a lossless transmission line between sending and receiving ends; (b) equivalent representation of an electrical machine; (c) equivalent representation of an inverter.
The direct way to modify the effective line reactance (jXeff) between its two ends is to connect a compensating reactance (–jXse) in series with the line as shown in Figure 1-6. The active and reactive power flows (Pr and Qr) at the receiving end of the line are given by the following equations:
(2‐207)
and
(2‐208)
where
(2‐209)
(2‐210a)
Note that Xse > 0 represents a capacitive compensating reactance and Xse < 0 represents an inductive compensating reactance, respectively. However, Xeff > 0 represents an effective inductive reactance and Xeff < 0 represents an effective capacitive reactance, respectively.
Figure 1-6 Power flow in a lossless line with a series‐compensating reactance (Xse).
Depending on whether the compensating reactance (–jXse) is capacitive or inductive, the voltage (Vq = jVq) across the compensating reactance lags or leads the prevailing line current (I) by 90°. This leads to the concept of an emulated reactance, which is defined as
(1‐3a)
or
(1‐3b)
and is shown in Figure 1-7. In this concept, a compensating voltage is created, maintaining the quadrature relationship with the prevailing line current, and is connected in series with the line. Note that is defined to be a capacitive compensation and Vs′s < 0 is defined to be an inductive compensation, respectively.
The concept of an emulated reactance can be further extended to represent an emulated impedance when the compensating voltage is not restricted to be in quadrature, but at any phase angle with respect to the prevailing line current (I). That means the compensating voltage can be made to look like a virtual, four‐quadrant, compensating impedance (Zse = Rse − jXse) that consists of a resistance (Rse = +R or − R) and a reactance (Xse = XC or − XL) in series with the line without any discrete circuit component as shown in Figure 1-8b. This two‐parameter, resistive (Rse) and reactive (Xse), control makes it possible to modify the magnitude and the phase angle of the line voltage simultaneously, which results in an independent control of active and reactive power flows in the line.
Figure 1-7 Power flow in a lossless line with a series‐compensating voltage (Vs′s).
Figure 1-8 (a) Power transmission system with a series‐compensating voltage (Vs′s); (b) four‐quadrant emulated impedance.
The series‐compensating voltage (Vs′s) is related to (Vdq), such that
(1‐4)
and
(1‐5)
where Vd = Vd and Vq = jVq are the respective active or direct and reactive or quadrature components of the compensating voltage with load convention, meaning the line current (I) enters at the higher potential terminal of the voltages (Vd and Vq) as shown in Figure 1-8a.
The natural or uncompensated power flow through a transmission line in a power system network is, in general, not optimal. Any of the power flow control parameters (line voltage magnitude, its phase angle, and line reactance) can be regulated with the use of the following equipment:
Voltage‐Regulating Transformer (VRT), shunt or parallel‐connected switched reactor/capacitor, also known as Shunt Reactor (SR)/Shunt Capacitor (SC), Static Var Compensator (SVC), or STATic synchronous COMpensator (STATCOM) for voltage regulation as shown in Figure 1-9
PAR or Phase‐Shifting Transformer (PST) for phase angle regulation as shown in Figure 1-10
Thyristor‐Controlled Series Capacitor (TCSC) or Static Synchronous Series Compensator (SSSC) for series reactance regulation as shown in Figure 1-11.
The dynamic performance of a VRT is limited by the speed of operation of the mechanical Load Tap Changers (LTCs), which respond in seconds; this level of response time is acceptable in most utility applications. However, if a faster response is desired, the mechanical LTCs can be upgraded with power electronics‐based LTCs, called Thyristor‐Controlled (TC) LTCs as discussed in Chapter 2. More on LTCs can be found in the book, titled “On‐Load Tap‐Changers For Power Transformers: Operation, Principles, Applications and Selection,” by A. Krämer, Maschinenfabrik Reinhausen, 2000. The power electronics‐based solutions can be divided into two categories, based on the type of semiconductor switches: naturally commutated switch, such as a thyristor and forced‐commutated switch, such as Insulated Gate Bipolar Transistor (IGBT). Each of these solutions is based on engineering trade‐offs.
Figure 1-9 Transmission line Voltage Regulators: (a) Two‐winding Transformer, (b) Autotransformer, (c) Switched Reactor, (d) Switched Capacitor, (e) TSC + TCR = SVC, and (f) STATCOM.
Figure 1-10 Transmission line voltage Phase Angle Regulators: (a) asymmetric and (b) symmetric.
Figure 1-11 Transmission line Reactance Regulators: (a) Thyristor‐Controlled Series Capacitor (TCSC) and (b) Static Synchronous Series Compensator (SSSC).
When discussing dynamic and transient events, mechanical LTCs react in ≈ 3–5 s (3,000–5,000 ms); TC LTCs react in < 1 s (1,000 ms) and IGBT‐based converters react in < 0.010 s (10 ms). These different technologies are referred to as slow, medium speed, and fast, respectively. Note that as the response time of a particular solution increases from slow using mechanical LTCs to medium speed using TC LTCs to fast using IGBTs, there is a corresponding increase in the solution’s life‐cycle costs (installation, operation, and maintenance), complexity, and impracticability of relocation. Other important features to consider are reliability, efficiency, component non‐obsolescence, and interoperability.
For more than a century, the transmission line voltage magnitude has been regulated with transformers and tap changers. They are referred to, in this book, as the VRT in the form of a two‐winding transformer with galvanically isolated primary and secondary windings, called a Shunt–Shunt configuration, and an autotransformer with an electrical connection between the primary and secondary windings, called a Shunt–Series configuration. In both types of transformers, the line voltage is applied to the primary windings. In the two‐winding transformer, the full line voltage is induced in the secondary windings, whereas, in the autotransformer, only a fraction of the line voltage is induced in the secondary windings that are connected to the primary windings to produce the full line voltage. In both cases, the magnitude of the line voltage is regulated. The secondary voltage is varied with the use of LTCs. An LTC can step up/down the voltage without interruption of the load current. Both primary and secondary windings in the two‐winding transformer carry the full transmitted power. Both primary and secondary windings in the autotransformer carry only a fraction of the full transmitted power. Therefore, if the galvanic isolation is not needed, the rating of the transformer can be significantly reduced with a Shunt–Series configuration as compared to a Shunt–Shunt configuration. Regardless of which configuration is used, the voltages at the input (primary) and output (secondary) terminals of both a two‐winding transformer and an autotransformer are identical as discussed in Chapter 4, Section 4.1.
The primary reason for voltage regulation is due to the exchange of reactive power at the Point of Connection (POC) to the utility. However, a transformer neither generates nor absorbs reactive power. If a transformer delivers reactive power at one side (primary or secondary), it absorbs the same amount of reactive power on the other side (secondary or primary). Therefore, in the process of increasing voltage on the secondary side, it reduces voltage on the primary side. The opposite is true as well when, in the process of decreasing voltage on the secondary side, it increases voltage on the primary side. Figure 1-12 shows that a compensating voltage of ±15% of the natural primary voltage (Vsn) of 0.988 pu results in a secondary voltage (Vs′) in the range of 0.872 to 1.095 pu. In the process, the primary voltage (Vs) varies in the range of 1.022 to 0.945 pu. Therefore, a desired 15% change in voltage at the secondary terminal may result in a net 10.7% increase due to the reduction of voltage at the primary terminal and a net 11.6% decrease due to the increase of voltage at the primary terminal as discussed in Chapter 4, Section 4.1.
The indirect way to regulate the magnitude of the line voltage is to connect a reactor or a capacitor in shunt with the line. A shunt‐connected reactor absorbs reactive power from the line and lowers the line voltage, whereas a shunt‐connected capacitor raises the line voltage with its generated reactive power as discussed in Chapter 2. With a series‐connected switch, such as back‐to‐back thyristors (triac), whose duty cycle can be varied, the shunt reactor can be made to operate as a variable reactor, which is called a Thyristor‐Controlled Reactor (TCR). A Thyristor‐Switched Capacitor (TSC) connects fixed capacitors in a step‐like manner in shunt with the line through triacs. Therefore, a combination of the variable reactor and a parallel capacitor acts as a variable compensating reactor or capacitor, which is called SVC.
Figure 1-12 Ranges of voltages (Vs and Vs′) at the primary and secondary sides of a Voltage‐Regulating Transformer.
Voltage regulation can also be achieved by the field control of a synchronous motor (Synchronous Condenser or SynCon) that generates or absorbs var as in the cases of a shunt‐connected capacitor or a shunt‐connected reactor. Voltage regulation can also be achieved when the back emf of the SynCon is replaced with a power electronics‐based Voltage‐Sourced Converter (VSC), which is called STATCOM as discussed in Chapter 2, Section 2.3.1.2. More discussion on this topic is given in “Introduction to FACTS Controllers: Theory, Modeling, and Applications,” by Sen and Sen, IEEE Press and John Wiley & Sons, 2009, Chapter 8, Section 8.1.
The power flow in a transmission line has traditionally been regulated with the use of a PAR. The line voltage is applied to the primary windings and the induced secondary voltage, called a compensating voltage that is varied with the use of LTCs is connected in series with the line. This compensating voltage is in quadrature with the phase‐to‐neutral voltage and as a result, the phase angle of the line voltage is regulated as discussed in Chapters 2 and 4. The PAR is configured in two forms – PAR asymmetric (asym) and symmetric (sym). In the process of varying the phase angle of the line voltage, a PAR (asym) also increases the magnitude of the line voltage. In a PAR (sym), while the phase angle is varied, the magnitude of the line voltage stays unchanged. When a high power flow enhancement is desired, the application of a PAR (sym) becomes limited, because of the need for a large amount of reactive power flow through the line. This large amount of reactive power flow creates significant additional losses, because of a large line current. Also, a larger‐than‐necessary rating of the PAR results when a large increase in active power flow is desired as discussed in Chapter 2, Section 2.5.2. Also as discussed in Chapter 2, Section 2.2.2.6, a PAR emulates an impedance in series with the line; however, this emulated impedance is not an independently controlled resistance and reactance; therefore, a PAR cannot control the active and reactive power flows in the line independently, whereas an IR offers an independent control of active and reactive power flows in the line as desired.
If the variable capacitor/reactor is connected in series with the line, the effective line reactance between the two ends of the line is regulated by the additional variable capacitor/reactor, which is called TCSC. The functionality of a TCSC can be realized with a series‐compensating voltage as in the case of a SSSC. The SSSC maintains the compensating voltage almost in quadrature with the prevailing line current. A leading voltage emulates a reactor; a lagging voltage emulates a capacitor. A TCSC or SSSC modifies the magnitude and phase angle of the line voltage, which are the combined functionalities of a VR and a PAR as shown in Figure 2‐28. Since, the series reactance compensation technique does not change the effective line resistance, it cannot control the active and reactive power flows in the line independently as shown in Figure 1‐30.
In a lightly loaded line, the reactive power absorbed by the series reactance of the line may be much less in comparison to the reactive power generated by the line‐to‐ground, shunt capacitance of the line. The resulting voltage increase in the line may reach or exceed the allowable limits for other loads that are connected to the grid. In a heavily loaded transmission line, the reactive power needed by the series reactance of the line may be much more in comparison to the reactive power generated by the shunt capacitance of the line. The resulting voltage along the line may decrease to a point that is below an acceptable limit when the full performance of the load is not possible. If the voltage along the transmission line is increased to be regulated at its nominal value by using a VR, the active power flow increases over the natural flow as discussed in Chapter 2, Section 2.6.1 (Shunt‐Compensating Reactance). If the phase angle between the voltages at the two ends of the transmission line is increased by using a PAR, the active power flow also increases. The unintended consequence of increasing active power flow by voltage regulation or phase angle regulation is that the reactive power flow in the line is also affected. When the line reactance is regulated, both the active and reactive power flows in the transmission line are varied simultaneously.
If the reactive power along the line is reduced, the freed‐up capacity of the line can be used to increase the revenue‐generating active power flow. As a result, the connected‐generators will be required to supply less reactive power. Furthermore, the efficiencies of the generators and step‐up transformers under a reduced reactive power condition will also increase. Since the loss in the line decreases due to less reactive power flow, the grid becomes more efficient. Therefore, it is desirable to compensate the lines to operate under independent, not simultaneous, control of the active and reactive power flows so that the line can facilitate the delivery of active power with the greatest value. The active and reactive power flows in a line can be regulated independently with a UPFC or an ST that controls the effective impedance of the line between its two ends, which is functionally equivalent to regulating both the magnitude and phase angle of the line voltage simultaneously.
