Solution to Application Example 8.1

The reliability indices are

SAIDI == 0.003 h/customer, where interruptions with 0 V are taken into account,

SAIFI = 0.004 interruptions/customer, where interruptions with 0 V are taken into account,

${\mathrm{SARFI}}_{\%\mathrm{V}}=\Sigma N_i/N_T=67.5\%\text{,}$

In this case %V = ± 10%. For the SARFI index, the voltage threshold allows assessment of compatibility for voltage-sensitive devices. Typical annual value for SARFI is about 15% for a 70% voltage (%V) event.

8.2.4.2 Application Example 8.2: Lateral Profile of Electric Field at Ground Level below a Three-Phase Transmission Line

If the calculation of the electric field at ground is repeated at different points in a section perpendicular to the transmission line, the lateral profile of the transmission-line electric field is obtained. Examples of calculated lateral profiles are presented in Fig. E8.2.1. Particularly important for the line design is the maximum (worst case) field and the field at the edge of the transmission corridor. The maximum field occurs within the transmission corridor, though for flat configurations it might occur slightly outside the outer phases. Figure E8.2.2 shows universal curves using nondimensional quantities that may be used to calculate the maximum electric field at ground for a flat configuration. Results for other configurations are given in [47].

Find the maximum (worst case) electric field at ground for a rated line-to-line voltage of V_L–L = 525 kV, a 3 × 3.3 cm conductor bundle with 45 cm spacing, phase-to-phase distance S = 10 m, the equivalent bundle diameter of D = 0.3 m, and height of the center of the bundle to ground H = 10.6 m.

8.2.4.3 Application Example 8.3: Lateral Profile of Magnetic Field at Ground Level under a Three-Phase Transmission Line

In most practical cases, the magnetic field in proximity to balanced three-phase lines may be calculated, considering the currents in the conductors and in the ground wires and neglecting the earth currents. As a result one obtains a magnetic field ellipse with minor and major axes as shown in Fig. E8.3.1. For the worst case (maximum) ground level magnetic field, find the coorsponding values of the magnetic-field intensity H and flux density B.

8.2.4.4 Mechanism of Corona

Electrical discharges are usually triggered by an electric field |$\overrightarrow{E}$| accelerating free electrons through a gas. When these electrons acquire sufficient energy from an electric field, they may produce fresh ions by knocking electrons from atoms and molecules by collision. This process is called ionization by electron impact. The electrons multiply, as illustrated in Fig. 8.6, where secondary effects from the electrodes make the discharge self-sustaining. The initial electrons that start the ionizing process are often created by photoionization: a photon from some distant source imparts enough energy to an atom so that the atom breaks into an electron and a positively charged ion. During acceleration in the electric field, the electron collides with the atoms of nitrogen, oxygen, and other gases present. Most of these collisions are elastic collisions similar to the collision between two billiard balls. The electron loses only a small part of its kinetic energy with each collision. Occasionally, an electron may strike an atom sufficiently hard so that excitation occurs, and the atom shifts to a higher energy state. The orbital states of one or more electrons changes, and the impacting electron loses part of its kinetic energy. Later, these excited atoms may revert to normal state, resulting in a radiation of the excess energy in the form of light (visible corona) and electromagnetic waves. An electron may also collide (recombine) with a positive ion, converting the ion to a neutral atom. Figure 8.7 shows the development of corona along an energized wet conductor, and Fig. 8.8 illustrates the frequency of corona discharge.

8.2.4.6 Factors Influencing Generation of Corona

At a given voltage, corona is determined by conductor diameter, line configuration, type of conductor (e.g., stranded), condition of its surface, and weather. Rain is by far the most important aspect of weather in increasing corona. The effect of atmospheric pressure and temperature is generally considered to modify the critical disruptive voltage (where the corona effects set in) of a conductor directly, or as the 2/3 exponent of the air density factor, δ, which is given by [35]

$\delta =\frac{17.9\cdot \left(\mathrm{barometric}\mathrm{pressure}\mathrm{in}\mathrm{inches}\mathrm{H}\mathrm{g}\right)}{459+°\mathrm{Fahrenheit}}\text{.}$

The equation for the critical disruptive voltage is

$V_{L-N\_o}=g_o\cdot {\delta }^{2/3}\cdot r\cdot m\cdot \mathrm{In}\left(\frac{D}{r}\right)\text{,}$

where

V_{L–N_o} = critical disruptive voltage in kV from line to neutral,

g_o = critical gradient in kV per centimeter,

r = radius of conductor in centimeters,

D = distance in centimeters between two conductors, and

m = surface factor.

Corona in fair weather is negligible or moderate up to a voltage near the disruptive voltage for a particular conductor. Above this voltage corona effects increase very rapidly. The calculated disruptive voltage is an indicator of corona performance. A high value of critical disruptive voltage is not the only criterion of satisfactory corona performance. Consideration should also be given to the sensitivity of the conductor to foul weather.

8.2.4.7 Application Example 8.4: Onset of Corona in a Transmission Line

A V_rated = 345 kV_L–L transmission line with g_o = 21.1 kV/cm (e.g., see Fig. E8.4.1b), r = 1.6 cm, d = 760 cm, m = 0.84 where at an altitude of 6000 feet the barometric pressure is 23.98 inches Hg (mercury) results at a temperature of −30°F in an air density factor of δ = 1.00. Calculate the critical disruptive voltage V_{L–N_o}.

8.2.4.9 Solutions for the Minimization of EMFs, Corona, and Other Environmental Concerns in Newly Designed Transmission Lines

Designing a transmission line with minimum environmental impact requires a study of five key factors:

1. impact of electric $\overrightarrow{E}$ and magnetic fields $\overrightarrow{H}$,

2. visual impact of the design,

3. generation of corona and its influence on radio and television interference,

4. air pollution via the generation of ozone O₃ and NO_x, and

5. impact of physical location.

A properly designed transmission line will have almost no perceptible negative environmental effects during fair weather, thus reducing disturbances, if they occur at all, to a small fraction of time.

Audible Noise, Radio Interference (RI), and Television Interference (TVI) Caused by Corona

In the United States, there exist no regulations on a local or federal level that expressly limit the level of radio noise that a transmission line may produce. The Federal Communications Commission (FCC) places power transmission lines in the category of “incidental radiation device”. As such, the FCC requires that the “device shall be operated so that the radio frequency energy that is emitted does not cause any emissions, radiation or induction which endangers the function of a radio navigation service or other safety services, or seriously degrades, obstructs or repeatedly interrupts a radio communication service” operating in accordance with FCC regulations. Figure 8.10 shows typical frequency spectra of corona-induced radio noise, and Figure 8.11 illustrates a typical lateral profile of radio noise. Figure 8.12 illustrates radio noise tolerability criteria caused by corona, one for fair weather and one for foul weather; in addition there exists a 120 Hz hum (at 60Hz) due to the changing energy content of the lines. Video signals are more affected than sound signals because they are amplitude modulated whereas the latter are frequency modulated. HDTV is less affected by corona. A reduction of the radio noise entails a reduction of corona effects (e.g., bundle conductors). Note that the number of decibels (dB) is defined by dB = 20log₁₀(v_out/v_in).

Ozone (O₃) and NO_x Generation

NO_x is a generic term for the mono-nitrogen oxides NO and NO₂ (nitric oxide and nitrogen dioxide). Figure 8.13 shows an example of ozone recordings for a 1300 kV line with large corona loss of 650 kW/km. Although ozone is the primary gaseous corona product, nitrogen oxides (NO_x) may also be generated. The percentage of nitrogen oxide production is about one-sixth of that of ozone. The generation of ozone due to disassociation of oxygen in the corona region around the conductor is not of sufficient magnitude to be harmful. At ground level it has been found to be less than the daily natural variation. Conductor bundling is one method of reducing corona and its associated ozone generation.

8.2.4.10 Economic Considerations

Corona loss of a well-designed transmission line should be below 1 kW per three-phase mile. This loss can be controlled by selecting proper configurations of bundle conductors (see Fig. E8.4.3 and Table E8.4.1). A bundle conductor is a conductor made up of two or more subconductors, and is used as one-phase conductor. Bundle conductors are not economical at or below 200 kV, but for rated voltages of 400 kV or more they are the best solution for overhead transmission lines. The increase in transmitting capacity justifies economically the use of two-conductor bundles on 220 kV lines.

The advantages of bundle conductors are higher disruptive voltage with conductors of reasonable dimensions, reduced surge impedance and consequent higher power capabilities, and less rapid increase of corona loss and RI with increased voltage. These advantages must be weighed against increased circuit cost and increased charging apparent power (kVA). Theoretically there is an optimum subconductor separation for bundle conductors that will give the minimum crest gradient on the surface of a subconductor and hence highest disruptive voltage.

8.2.4.11 No-Cost/Low-Cost EMF Mitigation Hearings of PUC of California [52]

Electric fields $\overrightarrow{E}$ measured in V/m are created whenever power lines are energized with a voltage, whereas magnetic fields $\overrightarrow{H}$ measured in A/m (or $\overrightarrow{B}$ measured in milligauss) are created when current flows through the lines. Both electric and magnetic fields attenuate rapidly with distance from the source. Electric fields are effectively shielded by materials such as trees or buildings, whereas magnetic fields are not easily shielded by objects or materials. Therefore, concerns regarding potential power line EMF health effects arise primarily due to exposure to magnetic fields $\overrightarrow{H}\text{.}$

On January 15, 1991, the California Public Utilities Commission (CPUC) began an investigation to consider the Commission’s potential role in mitigating health effects, if any, of EMFs created by

1. electric utility power lines, and

2. cellular radiotelephone facilities.

Due to the lack of scientific or medical conclusions about potential health effects from utility electric facilities and power lines, the CPUC adopted seven interim measures that help to address public concern on this subject.

• No-cost and low-cost steps to reduce EMF levels: When regulated utilities design new projects or upgrade existing facilities, approximately 4% of the project’s budget may be used for reducing EMFs. The CPUC did not set specific reduction levels for EMFs. It was considered to be inappropriate to set a specific numerical standard until a scientific basis for doing so exists. No-cost measures are those steps taken in the design stage, including changes in standard practice, that will not increase the project cost but will reduce the magnetic flux density. Maximum (worst case) magnetic flux densities that could be permitted are ranging from 1 to 10 mG. Low-cost measures are those steps that will cost about 4% or less of the total project cost and will reduce the magnetic field flux density in an area (e.g., by a school, near residence) by approximately 15% or more at the edge of the right-of-way.

• New designs to reduce EMF levels: The CPUC initiated workshops to establish guidelines for new and rebuilt facilities. These guidelines incorporate alternative sites, increase the size of rights-of-way, place facilities underground, and use other suggested methods for reducing EMF levels at transmission, distribution, and substation facilities.

• Uniform residential and workplace EMF measurement methods are to be established.

• Promotion of education and research: The Department of Health Services (DHS) should manage such programs supported by public and utility funds.

CPUC-regulated utilities and municipal utilities may use ratepayer funds to pay their share of development costs for the following programs:

• EMF education,

• EMF research, and

• other research.

It has been many years since the CPUC devised the above-mentioned seven interim measures. During this time a significant body of scientific research has developed. In 2002 the DHS completed its comprehensive evaluation of existing scientific studies. Although there is still no consensus in the scientific community regarding health risks of EMF exposure, DHS reported troubling indications that EMF exposure may increase risk of certain diseases (e.g., childhood and to some extent adult leukemia, adult brain cancer, Lou Gehrig’s disease) and other health problems.

8.2.4.12 Summary and Conclusions

The results of the many papers and books published in the area of generation of EMFs and corona by transmission lines can be summarized as follows:

• Transmission lines with line-to-line voltages of up to 200 kV do not generate any significant amount of corona; however, such lines can generate significant amounts of EMFs. For transmission lines with line-to-line voltages above 200 kV the following conclusions can be drawn: Maximum (worst case) AC electric field strength $\overrightarrow{E}$ at ground level below transmission line is up to |${\overrightarrow{E}}_{\mathrm{worst}}$| = 15 V/mm at a line-to-line voltage of 1050 kV. The maximum occurs at the edge of transmission corridor. An electric field strength of |${\overrightarrow{E}}_{\mathrm{acceptable}}$| = 5 V/mm is acceptable. Maximum (worst case) AC magnetic field strength $\overrightarrow{H}$ at ground level below transmission line is |${\overrightarrow{H}}_{\mathrm{worst}}$| = 14 A/m corresponding to a maximum flux density |${\overrightarrow{B}}_{\mathrm{worst}}$| = 0.176 G = 176 mG at a phase current of 2000 A. Note the DC earth field is about B_earth = 0.5 G. A flux density of |${\overrightarrow{B}}_{\mathrm{acceptable}}$| = 3 mG is acceptable.

• Corona depends on construction of transmission towers including type of conductors, e.g., hollow copper cable (e.g., HH) or aluminum cable steel reinforced (ACSR), surface of cable, type of bundle, fair or foul weather conditions, temperature, barometric pressure, and air density.

• Corona loss of 1 kW/three-phase mile is acceptable.

• Radio and television interference occurs within the 0.1 MHz to 1 GHz range.

• Audible noise ranges from 50 Hz to 120 Hz (power-frequency hum) to 10 kHz (hissing noise).

• Additional ozone (O₃) production is in the range of 15 ppb (parts per billion). The naturally occurring level is 25 ppb. That is, O₃ production by power lines is insignificant.

• Nitrogen oxide (NO_x) production is about one-sixth of the ozone generation. That is, NO_x production by power lines is insignificant.

• Active compensation (e.g., through compensation of EMFs by arranging several transmission systems on one tower, by reducing, compensating, or canceling EMFs in houses through compensation circuits) is possible.

• Passive (e.g., horizontal or vertical mesh structures, conductive suits) and natural shielding (e.g., trees, houses) is possible.

• Overall control of EMFs and corona through transmission line (e.g., bundle conductors, reverse-phased double circuit) and tower construction (e.g., height) is feasible; it is, however, more expensive.

• For newly designed transmission lines all environmental concerns (e.g., limits for EMFs, RI, and TVI) and technical constraints (corona loss of 1 kW/three-phase mile) can be satisfied.

• The mitigation of detrimental environmental, technical, or economic effects of existing transmission lines will be difficult to achieve at no or low cost as proposed by the Public Utilities Commission of California.

• The issue has surfaced that corona on transmission lines may create X-rays. A practical lower bound of X-rays is at least 2 keV. However, the electrical discharges of coronal activity on transmission lines can produce electron energies on the order of about 1–10 eV. This is sufficient for production of visible light but is orders of magnitude below electron energies necessary for the production of X-rays by transmission line corona [53]. That is, transmission lines do not generate X-rays.

8.2.5.1 Application Example 8.5: Frequency Control of an Interconnected Power System Broken into Two Areas: The First One with a 300 MW Coal-Fired Plant and the Other One with a 5 MW Wind-Power Plant

Figure P4.12 of Chapter 4 shows the block diagram of two generating plants interconnected by a tie line (transmission line).

Data for generation set #1 (steam turbine and generator) are as follows:

Angular frequency change (Δω₁) per change in generator output power (ΔΡ₁) having the droop characteristic of $R_1=\mathrm{\Delta }{\omega }_1/\mathrm{\Delta }{P}_1$ = 0.01 pu, load change (ΔP_L1) per frequency change (Δω₁) resulting in $D_1=\mathrm{\Delta }{P}_{L1}/\mathrm{\Delta }{\omega }_1$ = 0.8 pu, step load change $\mathrm{\Delta }{P}_{L1}\left(s\right)=\mathrm{\Delta }{P}_{L1}/s\mathrm{p}\mathrm{u}=0.2/s\mathrm{p}\mathrm{u}\text{,}$ angular momentum of steam turbine and generator set M₁ = 4.5, base apparent power S_base = 500 MVA, governor time constant T_G1 = 0.01 s, valve changing (charging) time constant T_CH1 = 0.5 s, and (load reference set point)₁ = 0.8 pu.

Data for generation set #2 (wind turbine and generator) are as follows:

Frequency change (Δω₂) per change in generator output power (ΔP₂) having the droop characteristic $R_2=\mathrm{\Delta }{\omega }_2/\mathrm{\Delta }{P}_2\mathrm{p}\mathrm{u}$ = 0.02 pu, load change (ΔP_L2) per frequency change (Δω₂) resulting in $D_2=\mathrm{\Delta }{P}_{L2}/\mathrm{\Delta }{\omega }_2$ = 1.0 pu, step load change $\mathrm{\Delta }{P}_{L2}\left(s\right)=\mathrm{\Delta }{P}_{L2}/s$ pu = $-0.2/s$ pu, angular momentum of wind turbine and generator set M₂ = 6, base apparent power S_base = 500 MVA, governor time constant T_G2 = 0.02 s, valve (blade control) changing (charging) time constant T_CH2 = 0.75 s, and (load reference set point)₂ = 0.8 pu.

Data for tie line: $T=377/X_{\mathit{tie}}$ with X_tie = 0.2 pu.

a) List the ordinary differential equations and the algebraic equations of the block diagram of Fig. P4.12 of Chapter 4.

b) Use either Mathematica or MATLAB to establish transient and steady-state conditions by imposing a step function for load reference set point (s)₁ = $0.8/s$ pu, load reference set point (s)₂ = $0.8/s$ pu, and run the program with zero step load changes ΔΡ_L1 = 0, ΔΡ_L2 = 0 for 5 s. After 5 s impose the positive step-load change ΔΡ_L1(s) = $\mathrm{\Delta }{P}_{L1}/s$ pu =$0.2/s$ pu, and after 7 s impose the negative step-load change ΔΡ_L2(s) = $\mathrm{\Delta }{P}_{L2}/s$ pu =$-0.2/s$ pu, for the droop characteristics R₁ = 0.01 pu and R₂ = 0.02 pu to find the transient and steady-state responses Δω₁(t) and Δω₂(t).

c) Use either Mathematica or MATLAB to establish transient and steady-state conditions by imposing a step function for load reference set point ${\left(s\right)}_1=0.8/s\mathrm{p}\mathrm{u}\text{,}$ load reference set point ${\left(s\right)}_2=0.8/s\mathrm{p}\mathrm{u}\text{,}$ and run the program with zero step load changes ΔΡ_L1 = 0, ΔΡ_L2 = 0 for 5 s. After 5 s impose positive step-load change ΔΡ_L1(s) = $\mathrm{\Delta }{P}_{L1}/s$ pu =$0.1/s$ pu, and after 7 s impose the negative step-load change ΔΡ_L2(s) = $\mathrm{\Delta }{P}_{L2}/s$ pu =$-0.1/s$ pu, for the droop characteristics R₁ = 0.01 pu (coal-fired plant) and R₂ = 0.5 pu (wind-power plant) to find the transient and steady-state responses Δω₁(t) and Δω₂(t).

8.2.5.2 Application Example 8.6: Frequency Control of an Interconnected Power System Broken into Two Areas: The First One with a 5 MW Wind-Power Plant and the Other One with a 5 MW Photovoltaic Plant

Figure P4.12 of Chapter 4 shows the block diagram of two generators interconnected by a tie line (transmission line).

Data for generation set #1 (wind turbine and generator) as follows:

Frequency change (Δω₁) per change in generator output power (ΔΡ₁) having the droop characteristic $R_1=\mathrm{\Delta }{\omega }_1/\mathrm{\Delta }{P}_1\mathrm{p}\mathrm{u}=0.5\mathrm{p}\mathrm{u}\text{,}$ load change (ΔP_L1) per frequency change (Δω₁) yields $D_1=\mathrm{\Delta }{P}_{L1}/\mathrm{\Delta }{\omega }_1$ = 0.8pu, step-load change ΔΡ_L1(s) = $\mathrm{\Delta }{P}_{L1}/s$ pu = $0.1/s$ pu, angular momentum of wind turbine and generator set M₁ = 4.5, base apparent power S_base = 50 MVA, governor time constant T_G1 = 0.01 s, valve (blade control) changing (charging) time constant T_CH1 = 0.1 s, and (load reference set point)₁ = 0.8 pu.

Data for generation set #2 (photovoltaic array and inverter):

Frequency change (Δω₂) per change in inverter output power (ΔΡ₂) having the droop characteristic $R_2=\mathrm{\Delta }{\omega }_2/\mathrm{\Delta }{P}_2\mathrm{p}\mathrm{u}=0.5\mathrm{p}\mathrm{u}\text{,}$ load change (ΔP_L2) per frequency change (Δω₂) yields D₂ = $\mathrm{\Delta }{P}_{L2}/\mathrm{\Delta }{\omega }_2$ = 1.0 pu, step-load change ΔΡ_L2(s) = $\mathrm{\Delta }{P}_{L2}/s$ pu =$-0.1/s$ pu, equivalent angular momentum M₂ = 6, base apparent power S_base = 50 MVA, governor time constant T_G2 = 0.02 s, equivalent valve changing (charging) time constant T_CH2 = 0.1 s, and (load reference set point)₂ = 0.8 pu.

Data for tie line: $T=377/X_{\mathit{tie}}$ with X_tie = 0.2pu.

Use either Mathematica or MATLAB to establish transient and steady-state conditions by imposing a step function for load reference set point(s)₁ = $0.8/s$ pu, load reference set point(s)₂ = $0.8/s$ pu and run the program with a zero step load changes ΔP_L1 = 0, ΔP_L2 = 0 for 5 s. After 5 s impose step-load change ΔΡ_L1(s) = $\mathrm{\Delta }{P}_{L1}/s$ pu =$0.1/s$ pu, and after 7 s impose the step-load change ΔΡ_L2(s) = $\mathrm{\Delta }{P}_{L2}/s$ pu =$-0.1/s$ pu, to find the transient and steady-state responses Δω₁(t) and Δω₂(t).

8.3.3.1 Review of Existing Methods

The separate online monitoring of the iron-core and copper losses of single- and three-phase transformers is desirable because additional losses due to power quality problems (e.g., harmonics, DC excitation) can be readily detected before any significant damage due to additional temperature rises occurs. Those losses can accurately and economically be measured for single-phase transformers via computer-aided testing (CAT) [71,88].

The known approaches for on-line loss measurement of high-efficiency transformers (e.g., efficiency greater than 97%) are inaccurate because they measure input and output powers and derive the loss from the difference of these two large values. The usually employed indirect method consisting of no-load (iron-core loss) and short-circuit (copper loss) tests [35] cannot be performed online while the transformer is partially or fully loaded. The same is valid for the loading back method [89].

Arri et al. presented an analog measurement circuit [90]. It is well known that there may occur zero-sequence currents on the Y-grounded side of a three-phase transformer, and these currents will produce corresponding losses in the transformer. The measuring circuit presented by Arri et al. cannot measure these zero-sequence components, and therefore the method is not suitable for transformers with grounding on any side (either primary or secondary). Moreover, Arri et al. rely on many instrument transformers (nine CTs and nine PTs) to convert the Δ connection to Y connection for a Y/Δ connected transformer: as a result the measuring accuracy is decreased. In [91] an analog measuring circuit with wattmeters is presented within a single-phase circuit.

This section presents a more accurate real-time method for digitally and separately measuring the steady-state iron-core and copper losses of three-phase transformers with various grounded connections while the transformers are operating at any load condition. This digital measuring circuit is based on voltage and current sensors (voltage dividers, shunts, PTs, CTs, or Hall devices), A/D converter, and personal computer. Using a computer-aided testing program (CATEA) [70] losses, efficiency, harmonics, derating, and wave shapes of all voltages and currents can be monitored within a fraction of a second.

The maximum measurement errors of the losses are acceptably small and vary in the range from 0.5 to 15%, mainly contingent on the accuracy of the voltage and current sensors used. Depending on the application, voltage dividers with optocouplers [73], current shunts with optocouplers, potential and current transformers (error < 0.1%), as well as Hall sensors [92] (error < 0.5%) can be employed.

When both voltage and current signals are obtained from Hall devices [92], the copper losses will include DC losses, if they exist. Provided voltage and current signals are generated by PT and CT sensors, the DC losses are not included in the copper losses, and must be measured by additional DC voltmeters and ammeters or sensors. In this case, the total loss consists of the iron-core, AC, and DC copper losses.

8.3.3.2 Approach

It is well known that for any Y (ungrounded) or Δ connected three-phase windings, the two-wattmeter method is often used to measure the power based on the voltages and currents of any two phases. For a Y grounded three-phase winding, an additional wattmeter is necessary for the voltage and current of the third phase, or those of the zero sequence. Therefore, for a transformer without grounding on any side (primary or secondary), power loss is measured with eight signals (voltages and currents of two phases for each side). If the primary or/and the secondary are grounded, the power loss can be measured with ten or twelve signals. This section deals with ten- and twelve-channel measuring circuits. The eight-channel measuring circuits for Y/Y ungrounded or Δ/Δ and Δ–Y ungrounded three-phase transformers are discussed in [73].

A ten-channel CAT circuit applied to Δ–Y₀ (Y₀ means Y grounded) connected three-phase transformers is shown in Fig. 8.16. The flowchart for the ten-channel signals is shown in Fig. 8.17. The output power (at the Y₀ side) is

$\begin{array}{l}{P}_{\mathrm{out}}=\frac{1}{T}{\displaystyle {\int }_0^T\left(v_{\mathit{an}}^{\prime }{i}_a^{\prime }+v_{bn}^{\prime }{i}_b^{\prime }+v_{cn}^{\prime }{i}_c^{\prime }\right)dt}=\frac{1}{T}{\displaystyle {\int }_0^T\left[v_{\mathit{an}}^{\prime }{i}_a^{\prime }+\left(v_0^{\prime }-v_{\mathit{an}}^{\prime }-v_{cn}^{\prime }\right)i_b^{\prime }+v_{cn}^{\prime }{i}_c^{\prime }\right]dt}\\ =\frac{1}{T}{\displaystyle {\int }_0^T\left[v_{\mathit{an}}^{\prime }{i}_{\mathit{al}}^{\prime }-v_{cn}^{\prime }{i}_{bl}^{\prime }+v_0^{\prime }{i}_b^{\prime }\right]dt}\end{array}$

where $i_{\mathit{al}}^{\prime }=\left(i_a^{\prime }-i_b^{\prime }\right),i_{bl}^{\prime }=\left(i_b^{\prime }-i_c^{\prime }\right)\text{,}$ and $v_0^{\prime }=\left(v_{\mathit{an}}^{\prime }+v_{bn}^{\prime }+v_{cn}^{\prime }\right)\text{.}$ The transformer loss is

$\begin{array}{l}{P}_{\mathrm{in}}-P_{\mathrm{out}}=\frac{1}{T}{\displaystyle {\int }_0^T\left[\left(v_{\mathit{AC}}{i}_{Al}+v_{BC}{i}_{Bl}\right)-(v_{\mathit{an}}^{\prime }{i}_{\mathit{al}}^{\prime }-v_{cn}^{\prime }{i}_{bl}^{\prime }+v_0^{\prime }{i}_b^{\prime }\right)}]dt\\ =\frac{1}{T}{\displaystyle {\int }_0^T\left[\right(v_{\mathit{AC}}\left(i_{Al}-i_{\mathit{al}}^{\prime }\right)+v_{BC}\left(i_{Bl}-i_{bl}^{\prime }\right)+\left(v_{\mathit{AC}}-v_{\mathit{an}}^{\prime }\right)i_{\mathit{al}}^{\prime }+\left(v_{BC}+v_{cn}^{\prime }\right)i_{bl}^{\prime }-v_0^{\prime }{i}_b^{\prime }}]dt\text{.}\end{array}$

The loss component $\left(-v_0^{\prime }{i}_b^{\prime }\right)$ in Eq. 8-6 is included in the copper loss, which can be proved below:

$\begin{array}{l}{p}_{cu}=\left(v_{\mathit{AC}}-v_{\mathit{an}}^{\prime }\right)i_a^{\prime }+\left(v_{CB}-v_{cn}^{\prime }\right)i_c^{\prime }+\left(v_{\mathit{BA}}-v_{bn}^{\prime }\right)i_b^{\prime }\\ =\left(v_{\mathit{AC}}-v_{\mathit{an}}^{\prime }\right)i_a^{\prime }+\left(-v_{BC}-v_{cn}^{\prime }\right)i_c^{\prime }+\left(v_{BC}-v_{\mathit{AC}}-v_0^{\prime }+v_{\mathit{an}}^{\prime }+v_{cn}^{\prime }\right)i_b^{\prime }\\ =\left(v_{\mathit{AC}}-v_{\mathit{an}}^{\prime }\right)\left(i_a^{\prime }-i_b^{\prime }\right)+\left(v_{BC}+v_{cn}^{\prime }\right)\left(i_b^{\prime }-i_c^{\prime }\right)-v_0^{\prime }{i}_b^{\prime }\\ =\left(v_{\mathit{AC}}-v_{\mathit{an}}^{\prime }\right)i_{\mathit{al}}^{\prime }+\left(v_{BC}+v_{cn}^{\prime }\right)i_{bl}^{\prime }-v_0^{\prime }{i}_b^{\prime }\text{.}\end{array}$

Note that the analog measuring circuit of [90] does not take into account zero-sequence components and, therefore, does not apply to general asymmetric operating conditions.

For Y–Y₀ connected three-phase transformers (see Fig. 8.18), the output power can be written as

$\begin{array}{l}{P}_{\mathrm{out}}=\frac{1}{T}{\displaystyle {\int }_0^T\left(v_{\mathit{an}}^{\prime }{i}_a^{\prime }+v_{bn}^{\prime }{i}_b^{\prime }+v_{cn}^{\prime }{i}_c^{\prime }\right)dt}=\frac{1}{T}{\displaystyle {\int }_0^T\left[v_{\mathit{an}}^{\prime }{i}_a^{\prime }+v_{bn}^{\prime }{i}_b^{\prime }+v_{cn}^{\prime }\left(i_0^{\prime }-i_a^{\prime }-i_b^{\prime }\right)\right]dt}\\ =\frac{1}{T}{\displaystyle {\int }_0^T\left[v_{\mathit{an}}^{\prime }{i}_a^{\prime }+v_{bc}^{\prime }{i}_b^{\prime }+v_{cn}^{\prime }{i}_0^{\prime }\right]dt}\end{array}$

where $i_0^{\prime }=\left(i_a^{\prime }+i_b^{\prime }+i_c^{\prime }\right)$. The transformer loss can be formulated as

$\begin{array}{l}{P}_{\mathrm{in}}-P_{\mathrm{out}}=\frac{1}{T}{\displaystyle {\int }_0^T\left[\left(v_{\mathit{AC}}{i}_A+v_{BC}{i}_B\right)-\left(v_{ac}^{\prime }{i}_a^{\prime }+v_{bc}^{\prime }{i}_b^{\prime }+v_{cn}^{\prime }{i}_0^{\prime }\right)\right]dt}\\ =\frac{1}{T}{\displaystyle {\int }_0^T\left[v_{\mathit{AC}}\left(i_A-i_a^{\prime }\right)+v_{BC}\left(i_B-i_b^{\prime }\right)-v_{cn}^{\prime }{i}_0^{\prime }+\left(v_{\mathit{AC}}-v_{\mathit{ac}}^{\prime }\right)i_a^{\prime }+\left(v_{BC}-v_{bc}^{\prime }\right)i_b^{\prime }\right]dt}\end{array}$

where ($-v_{cn}^{\prime }{i}_0^{\prime }$) is included in the iron-core loss. This can be shown as follows:

$\begin{array}{l}{p}_{fe}=v_{\mathit{An}}\left(i_A-i_a^{\prime }\right)+v_{Bn}\left(i_B-i_b^{\prime }\right)+v_{Cn}\left(i_C-i_c^{\prime }\right)\\ =v_{\mathit{An}}\left(i_A-i_a^{\prime }\right)+v_{Bn}\left(i_B-i_b^{\prime }\right)-v_{Cn}\left(i_A+i_B+i_0^{\prime }-i_a^{\prime }-i_b^{\prime }\right)\\ =\left(v_{\mathit{An}}-v_{Cn}\right)\left(i_A-i_a^{\prime }\right)+\left(v_{Bn}-v_{Cn}\right)\left(i_B-i_b^{\prime }\right)-v_{cn}^{\prime }{i}_0^{\prime }\\ =v_{\mathit{AC}}\left(i_A-i_a^{\prime }\right)+v_{BC}\left(i_B-i_b^{\prime }\right)-v_{cn}^{\prime }{i}_0^{\prime }\text{.}\end{array}$

For Y₀–Y₀ connected transformers, a twelve-channel circuit is used as shown in Fig. 8.19. The loss of the transformer can be derived directly from

$\begin{array}{l}{P}_{\mathrm{in}}-P_{\mathrm{out}}=\frac{1}{T}{\displaystyle {\int }_0^T\left[\left(v_{\mathit{An}}{i}_A+v_{Bn}{i}_B+v_{Cn}{i}_C\right)-\left(v_{\mathit{an}}^{\prime }{i}_a^{\prime }+v_{bn}^{\prime }{i}_b^{\prime }+v_{cn}^{\prime }{i}_c^{\prime }\right)\right]}dt\\ =\frac{1}{T}{\displaystyle {\int }_0^T[v_{\mathit{An}}\left(i_A-i_a^{\prime }\right)+v_{Bn}\left(i_B-i_b^{\prime }\right)+v_{Cn}\left(i_C-i_c^{\prime }\right)+}\\ \left(v_{\mathit{An}}-v_{\mathit{an}}^{\prime }\right)i_a^{\prime }+\left(v_{Bn}-v_{bn}^{\prime }\right)i_b^{\prime }+\left(v_{Cn}-v_{cn}^{\prime }\right)i_c^{\prime }]dt\text{.}\end{array}$

The instantaneous expression p_fe(t) in Eq. 8-10 includes two power components: the instantaneous iron-core loss and the instantaneous exciting reactive power. Similarly, p_cu(t) in Eq. 8-7 includes the instantaneous copper loss and the instantaneous reactive power due to the leakage flux. At steady state, the average value of reactive power during one electric cycle is zero; therefore, the average values of p_fe(t) and p_cu(t) during one electric cycle reflect the steady-state iron-core and copper losses.

The instantaneous expressions of p_fe(t) and p_cu(t) of various three-phase transformer connections are listed in Table 8.2, where all current and voltage differences can be calibrated directly under either sinusoidal or nonsinusoidal conditions.

Table 8.2

Iron-Core and Copper Losses of Three-Phase Transformers

	pfe(t)	pcu(t)
Y–Y (or Δ–Δ)	$v_{\mathit{AC}}\left(i_A-i_a^{\prime }\right)+v_{BC}\left(i_B-i_b^{\prime }\right)$	$\left(v_{\mathit{AC}}-v_{ac}^{\prime }\right)i_a^{\prime }+\left(v_{BC}-v_{bc}^{\prime }\right)i_b^{\prime }$
Δ–Y	$v_{\mathit{AC}}\left(i_{Al}-i_{\mathit{al}}^{\prime }\right)+v_{BC}\left(i_{Bl}-i_{bl}^{\prime }\right)$	$\left(v_{\mathit{AC}}-v_{\mathit{an}}^{\prime }\right)i_{\mathit{al}}^{\prime }+\left(v_{BC}+v_{cn}^{\prime }\right)i_{bl}^{\prime }$
Δ–Y0	$v_{\mathit{AC}}\left(i_{Al}-i_{\mathit{al}}^{\prime }\right)+v_{BC}\left(i_{Bl}-i_{bl}^{\prime }\right)$	$\left(v_{\mathit{AC}}-v_{\mathit{an}}^{\prime }\right)i_{\mathit{al}}^{\prime }+\left(v_{BC}+v_{cn}^{\prime }\right)i_{bl}^{\prime }-v_{bl}^{\prime }{i}_b^{\prime }$
Y–Y0	$v_{\mathit{AC}}\left(i_A-i_a^{\prime }\right)+v_{BC}\left(i_B-i_b^{\prime }\right)-v_{cn}^{\prime }{i}_o^{\prime }$	$\left(v_{\mathit{AC}}-v_{ac}^{\prime }\right)i_a^{\prime }+\left(v_{BC}-v_{bc}^{\prime }\right)i_b^{\prime }$
Y0–Y0	$v_{\mathit{An}}\left(i_A-i_a^{\prime }\right)+v_{Bn}\left(i_B-i_b^{\prime }\right)+v_{Cn}\left(i_C-i_c^{\prime }\right)$	$\left(v_{\mathit{An}}-v_{ac}^{\prime }\right)i_a^{\prime }+\left(v_{Bn}-v_{bn}^{\prime }\right)i_b^{\prime }+\left(v_{Cn}-v_{cn}^{\prime }\right)i_c^{\prime }$

If the neutral line is disconnected, and v₀′ and i_b′ are not measured, Fig. 8.16 becomes the eight-channel loss-measuring circuit of Δ–Y connected three-phase transformers (Fig. 8.20), and the transformer loss is obtained from Eq. 8-6 with v₀′= 0 [73]. Similarly, if the neutral line is disconnected, and i₀′ and $v_{cn}^{\prime }$ are not measured, Fig. 8.18 becomes the eight-channel loss-measuring circuit of Y–Y connected three-phase transformers, and the transformer loss is obtained from Eq. 8-9 with i₀′= 0 [73]. The circuits valid for Δ–Y, Δ–Y₀, and Y–Y₀ connected transformers also apply for Y–Δ, Y₀–Δ, and Y₀–Y connected three-phase transformers, respectively. In these cases, the capital letters A, B, and C represent the secondary, and the lower case letters a, b, and c denote the primary. The corresponding iron-core and copper losses, listed in Table 8.2, are then negative. Table 8.3 shows valid methods for three-phase transformers of various connections.

Note that in Figs. 8.16, 8.18, 8.19, and 8.20, CT and PT ratios on the primary side are 1 : 1, whereas on the secondary side they are N₂ : N₁. In this way, secondary signals are referred to the primary. In Fig. 8.20, only four CTs and five PTs are required for the loss measurement, comparing to the measurement circuit of [90] with nine CTs and nine PTs. Using fewer sensors means having higher measurement accuracy. In summary, the new approach adds the small copper and iron-core losses, while the conventional approach relies on the difference of two large values P_in-P_out.

8.3.3.3 Accuracy Requirements for Instruments

The efficiencies of high-power electrical apparatus such as single- and three-phase transformers in the kVA and MVA ranges are about 98% or higher. For a 25 kVA single-phase transformer [88] with an efficiency of 98.46% at rated operation, the total losses are P_{Loss_accurate} = 390 W, where the copper and iron-core losses are P_{cu_accurate} = 325 W and P_{fe_accurate} = 65 W, respectively [70]. The measurement errors of losses are derived for three different configurations: the conventional, new, and back-to-back approaches as discussed in Application Examples 8.8 to 8.10.

8.3.3.4 Application Example 8.8: Conventional Approach P_Loss = P_in – P_out

Even if current and voltage sensors (e.g., CTs, PTs) as well as current and voltmeters with maximum error limits of 0.1% are employed, the conventional approach—where the losses are the difference between the input and output powers—always leads to relatively large maximum errors (> 50%) in the loss measurements of highly efficient transformers.

Determine the maximum error in the measured losses for the 25 kVA, 240 V/7200 V single-phase transformer of Fig. E8.8.1, where the low-voltage winding is connected as primary, and rated currents are therefore I_2rat = 3.472 A, I_1rat = 104.167 A [72]. For a 25 kVA single-phase transformer [88] with an efficiency of 98.46% at rated operation, the total losses are P_{Loss_accurate} = 390 W, where the copper and iron-core losses are P_{cu_accurate} = 325 W and P_{fe_accurate} = 65 W, respectively [70].

The instruments and their maximum error limits are listed in Table E8.8.1, where errors of all current and potential transformers (e.g., CTs, PTs) are referred to the meter sides. In Table E8.8.1, error limits for all ammeters and voltmeters are given based on the full-scale errors, and those for all current and voltage transformers are derived based on the measured values of voltages and currents (see Appendix A4.1).