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1 Modeling: Line Impedance • Shunt Admittance • Line Segment Models • Step-Voltage Regulators • Transformer Bank Connections • Load Models • Shunt Capacitor Models
2 Analysis: Power-Flow Analysis
Radial distribution feeders are characterized by having only one path for power to flow from the source (distribution substation) to each customer. A typical distribution system will consist of one or more distribution substations consisting of one or more "feeders." Components of the feeder may consist of the following:
• Three-phase primary "main" feeder
• Three-phase, two-phase ("V" phase), and single-phase laterals
• Step-type voltage regulators or load tap changing transformer (LTC)
• In-line transformers
• Shunt capacitor banks
• Three-phase, two-phase, and single-phase loads
• Distribution transformers (step-down to customer's voltage)
The loading of a distribution feeder is inherently unbalanced because of the large number of unequal single-phase loads that must be served. An additional unbalance is introduced by the non-equilateral conductor spacings of the three-phase overhead and underground line segments.
Because of the nature of the distribution system, conventional power-flow and short-circuit programs used for transmission system studies are not adequate. Such programs display poor convergence characteristics for radial systems. The programs also assume a perfectly balanced system so that a single-phase equivalent system is used.
If a distribution engineer is to be able to perform accurate power-flow and short-circuit studies, it’s imperative that the distribution feeder be modeled as accurately as possible. This means that three phase models of the major components must be utilized. Three-phase models for the major components will be developed in the following sections. The models will be developed in the "phase frame" rather than applying the method of symmetrical components.
Fgr. 1 shows a simple one-line diagram of a three-phase feeder; it illustrates the major components of a distribution system. The connecting points of the components will be referred to as "nodes."
Note in the figure that the phasing of the line segments is shown. This is important if the most accurate models are to be developed.
The following sections will present generalized three-phase models for the "series" components of a feeder (line segments, voltage regulators, transformer banks). Additionally, models are presented for the "shunt" components (loads, capacitor banks). Finally, the "ladder iterative technique" for power-flow studies using the models is presented along with a method for computing short-circuit currents for all types of faults.
1.1 Line Impedance
The determination of the impedances for overhead and underground lines is a critical step before analysis of the distribution feeder can begin. Depending upon the degree of accuracy required, impedances can be calculated using Carson's equations where no assumptions are made, or the impedances can be determined from tables where a wide variety of assumptions are made. Between these two limits are other techniques, each with their own set of assumptions.
FGR. 1 Distribution feeder.: Substation transformer; Voltage regulator; Underground cables Capacitor bank; Voltage regulator; Three-phase lateral
1.1.1 Carson's Equations
Since a distribution feeder is inherently unbalanced, the most accurate analysis should not make any assumptions regarding the spacing between conductors, conductor sizes, or transposition. In a classic paper, John Carson developed a technique in 1926 whereby the self and mutual impedances for second overhead conductors can be determined. The equations can also be applied to underground cables. In 1926, this technique was not met with a lot of enthusiasm because of the tedious calculations that had to be done on the slide rule and by hand. With the advent of the digital computer, Carson's equations have now become widely used.
In his paper, Carson assumes the earth is an infinite, uniform solid, with a flat uniform upper surface and a constant resistivity. Any "end effects" introduced at the neutral grounding points are not large at power frequencies, and therefore are neglected. The original Carson equations are given in Equations 1 and 2.
1.1.6 Underground Lines
Fgr. 5 shows the general configuration of three underground cables (concentric neutral, or tape shielded) with an additional neutral conductor.
Carson's equations can be applied to underground cables in much the same manner as for overhead lines. The circuit of Fgr. 5 will result in a 7 × 7 primitive impedance matrix. For underground circuits that don’t have the additional neutral conductor, the primitive impedance matrix will be 6 × 6.
Two popular types of underground cables in use today are the "concentric neutral cable" and the "tape shield cable." To apply Carson's equations, the resistance and GMR of the phase conductor and the equivalent neutral must be known.
FGR. 6 Concentric neutral cable.
Phase conductor Insulation; Insulation screen; Concentric neutral strand
1.1.7 Concentric Neutral Cable
Fgr. 6 shows a simple detail of a concentric neutral cable. The cable consists of a central phase conductor covered by a thin layer of nonmetallic semiconducting screen to which is bonded the insulating material. The insulation is then covered by a semiconducting insulation screen. The solid strands of concentric neutral are spiraled around the semiconducting screen with a uniform spacing between strands. Some cables will also have an insulating "jacket" encircling the neutral strands.
In order to apply Carson's equations to this cable, the following data needs to be extracted from a table of underground cables:
dc is the phase conductor diameter (in.) dod is the nominal outside diameter of the cable (in.) ds is the diameter of a concentric neutral strand (in.) GMRc is the geometric mean radius of the phase conductor (ft) GMRs is the geometric mean radius of a neutral strand (ft) rc is the resistance of the phase conductor (O/mile) rs is the resistance of a solid neutral strand (O/mile) k is the number of concentric neutral strands
1.4 Step-Voltage Regulators
A step-voltage regulator consists of an autotransformer and a LTC mechanism. The voltage change is obtained by changing the taps of the series winding of the autotransformer. The position of the tap is determined by a control circuit (line drop compensator). Standard step regulators contain a reversing switch enabling a ±10% regulator range, usually in 32 steps. This amounts to a 5/8% change per step or 0.75 V change per step on a 120 V base.
A type B step-voltage regulator is shown in Fgr. 13. There is also a type A step-voltage regulator where the load and source sides of the regulator are reversed from that shown in Fgr. 13. Since the type B regulator is more common, the remainder of this section will address the type B step-voltage regulator.
FGR. 13 Type B step-voltage regulator. Preventive autotransformer
FGR. 14 Regulator control circuit. Potential transformer; Line current; Current transformer. Voltage relay Time delay Motor operating circuit Line drop compensator
The tap changing is controlled by a control circuit shown in the block diagram of Fgr. 14. The control circuit requires the following settings:
1. Voltage level: The desired voltage (on 120 V base) to be held at the "load center." The load center may be the output terminal of the regulator or a remote node on the feeder.
2. Bandwidth: The allowed variance of the load center voltage from the set voltage level. The voltage held at the load center will be ±1 2/ of the bandwidth. For example, if the voltage level is set to 122 V and the bandwidth set to 2 V, the regulator will change taps until the load center voltage lies between 121 and 123 V.
3. Time delay: Length of time that a raise or lower operation is called for before the actual execution of the command. This prevents taps changing during a transient or short time change in current.
4. Line drop compensator: Set to compensate for the voltage drop (line drop) between the regulator and the load center. The settings consist of R and X settings in volts corresponding to the equivalent impedance between the regulator and the load center. This setting may be zero if the regulator output terminals are the load center.
The rating of a regulator is based on the kVA transformed, not the kVA rating of the line. In general this will be 10% of the line rating since rated current flows through the series winding that represents the ±10% voltage change.
1.4.1 Voltage Regulator in the Raise Position
Fgr. 15 shows a detailed and abbreviated drawing of a type B regulator in the raise position. The defining voltage and current equations for the type B regulator in the raise position are as follows:
Voltage equations; Current equations
FGR. 15 Type B voltage regulator in the raise position.
Equations 82 and 83 are the necessary defining equations for modeling a regulator in the raise position.
1.4.2 Voltage Regulator in the Lower Position
Fgr. 16 shows the detailed and abbreviated drawings of a regulator in the lower position. Note in the figure that the only difference between the lower and the raise models is that the polarity of the series winding and how it’s connected to the shunt winding is reversed.
The defining voltage and current equations for a regulator in the lower position are as follows:
Voltage equations Current equations
FGR. 16 Type B regulator in the lower position.
Equations 83 and.90 give the value of the effective regulator ratio as a function of the ratio of the number of turns on the series winding (N2) to the number of turns on the shunt winding (N1). The actual turns ratio of the windings is not known. However, the particular position will be known. Equations 83 and 90 can be modified to give the effective regulator ratio as a function of the tap position.
Each tap changes the voltage by 5/8% or 0.00625 per unit. On a 120 V base, each step change results in a change of voltage of 0.75 V. The effective regulator ratio can be given by...
1.4.3 Line Drop Compensator
FGR. 17 Line drop compensator circuit.
The changing of taps on a regulator is controlled by the "line drop compensator." Fgr. 17 shows a simplified sketch of the compensator circuit and how it’s connected to the circuit through a potential transformer and a current transformer.
The purpose of the line drop compensator is to model the voltage drop of the distribution line from the regulator to the load center. Typically, the compensator circuit is modeled on a 120 V base. This requires the potential transformer to transform rated voltage (line-to-neutral or line-to-line) down to 120 V. The current transformer turns ratio (CTp : CTs ) where the primary rating (CTp) will typically be the rated current of the feeder. The setting that is most critical is that of R' and X' . These values must rep resent the equivalent impedance from the regulator to the load center. Knowing the equivalent impedance in Ohms from the regulator to the load center (Rline_ohms and Xline_ohms), the required value for the compensator settings are calibrated in volts and determined by...
1.4.4 Wye Connected Regulators
FGR. 18 Wye connected type B regulators.
Three single-phase regulators connected in wye are shown in Fgr. 18. In Fgr. 18 the polarities of the windings are shown in the raise position. When the regulator is in the lower position, a reversing switch will have reconnected the series winding so that the polarity on the series winding is now at the output terminal.
Regardless of whether the regulator is raising or lowering the voltage, the following equations apply.
1.4.5 Voltage Equations
1.5 Transformer Bank Connections
Unique models of three-phase transformer banks applicable to radial distribution feeders have been developed. Models for the following three-phase connections are included in this document:
• Delta-grounded wye
• Grounded wye-delta
• Ungrounded wye-delta
• Grounded wye-grounded wye
Fgr. 21 defines the various voltages and currents for the transformer bank models. The models can represent a step-down (source side to load side) or a step-up (source side to load side) transformer bank. The notation is such that the capital letters A, B, C, N will always refer to the source side of the bank and the lower case letters a, b, c, n will always refer to the load side of the bank. It’s assumed that ...
2.1 Power-Flow Analysis
The power-flow analysis of a distribution feeder is similar to that of an interconnected transmission sys tem. Typically what will be known prior to the analysis will be the three-phase voltages at the substation and the complex power of all the loads and the load model (constant complex power, constant impedance, constant current, or a combination). Sometimes, the input complex power supplied to the feeder from the substation is also known.
In Sections 1.3 through 1.5, phase frame models were presented for the series components of a distribution feeder. In Sections 1.6 and 1.7, models were presented for the shunt components (loads and capacitor banks). These models are used in the "power-flow" analysis of a distribution feeder.
A power-flow analysis of a feeder can determine the following by phase and total three-phase:
• Voltage magnitudes and angles at all nodes of the feeder
• Line flow in each line section specified in kW and kVAr, amps and degrees, or amps and power factor
• Power loss in each line section
• Total feeder input kW and kVAr
• Total feeder power losses
• Load kW and kVAr based upon the specified model for the load
Because the feeder is radial, iterative techniques commonly used in transmission network power-flow studies are not used because of poor convergence characteristics (Trevino, 1970). Instead, an iterative technique specifically designed for a radial system is used. The ladder iterative technique will be presented here.
2.1.1 The Ladder Iterative Technique
188.8.131.52 Linear Network
FGR. 31 Typical distribution feeder.
A modification of the ladder network theory of linear systems provides a robust iterative technique for power-flow analysis. A distribution feeder is nonlinear because most loads are assumed to be constant kW and kVAr. However, the approach taken for the linear system can be modified to take into account the nonlinear characteristics of the distribution feeder.
For the ladder network in Fgr. 29, it’s assumed that all of the line impedances and load impedances are known along with the voltage at the source (Vs ...)
The backward sweep will determine a computed source voltage V1. As in the linear case, this first iteration will produce a voltage that is not equal to the specified source voltage Vs. Because the network is nonlinear, multiplying currents and voltages by the ratio of the specified voltage to the computed voltage won’t give the solution. The most direct modification to the ladder network theory is to perform a for ward sweep. The forward d sweep commences by using the specified source voltage and the line currents from the backward sweep. KVL is used to compute the voltage at node 2 by…
This procedure is repeated for each line segment until a "new" voltage is determined at node 5. Using the new voltage at node 5, a second backward sweep is started that will lead to a new computed voltage at the source. The backward and forward sweep process is continued until the difference between the computed and specified voltage at the source is within a given tolerance.
184.108.40.206 General Feeder
A typical distribution feeder will consist of the "primary main" with laterals tapped off the primary main, and sub-laterals tapped off the laterals, etc., Fgr. 30 shows an example of a typical feeder.
The ladder iterative technique for the feeder of Fgr. 31 would proceed as follows:
1. Assume voltages (1.0 per unit) at the "end" nodes (6, 8, 9, 11, and 13).
2. Starting at node 13, compute the node current (load current plus capacitor current if present).
3. With this current, apply KVL to calculate the node voltages at 12 and 10.
4. Node 10 is referred to as a "junction" node since laterals branch in two directions from the node.
This feeder goes to node 11 and computes the node current. Use that current to compute the volt age at node 10. This will be referred to as "the most recent voltage at node 10."
5. Using the most recent value of the voltage at node 10, the node current at node 10 (if any) is computed.
6. Apply KCL to determine the current flowing from node 4 toward node 10.
7. Compute the voltage at node 4.
8. Node 4 is a junction node. An end-node downstream from node 4 is selected to start the forward sweep toward node 4.
9. Select node 6, compute the node current, and then compute the voltage at junction-node 5.
10. Go to downstream end-node 8. Compute the node current and then the voltage at junction-node 7.
11. Go to downstream end-node 9. Compute the node current and then the voltage at junction-node 7.
12. Compute the node current at node 7 using the most recent value of node 7 voltage.
13. Apply KCL at node 7 to compute the current flowing on the line segment from node 5 to node 7.
14. Compute the voltage at node 5.
15. Compute the node current at node 5.
16. Apply KCL at node 5 to determine the current flowing from node 4 toward node 5.
17. Compute the voltage at node 4.
18. Compute the node current at node 4.
19. Apply KCL at node 4 to compute the current flowing from node 3 to node 4.
20. Calculate the voltage at node 3.
21. Compute the node current at node 3.
22. Apply KCL at node 3 to compute the current flowing from node 2 to node 3.
23. Calculate the voltage at node 2.
24. Compute the node current at node 2.
Apply KCL at node 2.
26. Calculate the voltage at node 1.
27. Compare the calculated voltage at node 1 to the specified source voltage.
28. If not within tolerance, use the specified source voltage and the backward sweep current flowing from node 1 to node 2 and compute the new voltage at node 2.
29. The forward sweep continues using the new upstream voltage and line segment current from the forward sweep to compute the new downstream voltage.
30. The forward sweep is completed when new voltages at all end nodes have been completed.
31. This completes the first iteration.
32. Now repeat the backward sweep using the new end voltages rather than the assumed voltages as was done in the first iteration.
33. Continue the backward and forward sweeps until the calculated voltage at the source is within a specified tolerance of the source voltage.
34. At this point, the voltages are known at all nodes and the currents flowing in all line segments are known. An output report can be produced giving all desired results.
2.1.2 The Unbalanced Three-Phase Distribution Feeder
The previous section outlined the general procedure for performing the ladder iterative technique. This section will address how that procedure can be used for an unbalanced three-phase feeder.
Fgr. 32 is the one-line diagram of an unbalanced three-phase feeder. The topology of the feeder in Fgr. 32 is the same as the feeder in Fgr. 31. Fgr. 32 shows more detail of the feeder however.
The feeder in Fgr. 32 can be broken into the series components and the shunt components.
220.127.116.11 Series Components
The series components of a distribution feeder are
• Line segments
• Voltage regulators Models for each of the series components have been developed in prior areas of this section. In all cases, models (three-phase, two-phase, and single-phase) were developed in such a manner that they can be generalized. Fgr. 33 shows the "general model" for each of the series components.
18.104.22.168 Shunt Components
The shunt components of a distribution feeder are
• Spot loads
• Distributed loads
• Capacitor banks
Spot loads are located at a node and can be three-phase, two-phase, or single-phase and connected in either a wye or a delta connection. The loads can be modeled as constant complex power, constant cur rent, constant impedance, or a combination of the three.
Distributed loads are located at the midsection of a line segment. A distributed load is modeled when the loads on a line segment are uniformly distributed along the length of the segment. As in the spot load, the distributed load can be three-phase, two-phase, or single-phase and connected in either a wye or a delta connection. The loads can be modeled as constant complex power, constant current, constant impedance, or a combination of the three. To model the distributed load, a "dummy" node is created in the center of a line segment with the distributed load of the line section modeled at this dummy node.
Capacitor banks are located at a node and can be three-phase, two-phase, or single-phase and can be connected in a wye or delta. Capacitor banks are modeled as constant admittances.
In Fgr. 32 the solid line segments represent overhead lines while the dashed lines represent underground lines. Note that the phasing is shown for all of the line segments. In the area of the Section 1.1, the application of Carson's equations for computing the line impedances for overhead and under ground lines was presented. There it was pointed out that two-phase and single-phase lines are represented by a 3 × 3 matrix with zeros set in the rows and columns of the missing phases.
In the area of the Section 1.2, the method for the computation of the shunt capacitive susceptance for overhead and underground lines was presented. Most of the time the shunt capacitance of the line segment can be ignored; however, for long underground segments, the shunt capacitance should be included.
The "node" currents may be three-phase, two-phase, or single-phase and consist of the sum of the load current at the node plus the capacitor current (if any) at the node.
2.1.3 Applying the Ladder Iterative Technique
The previous section outlined the steps required for the application of the ladder iterative technique. For the general feeder of Fgr. 32 the same outline applies. The only difference is that Equations 197 and 198 are used for computing the node voltages on the backward sweep and Equation 199 is used for computing the downstream voltages on the forward sweep. The [a], [b], [c], [d], [A], and [B] matrices for the various series components are defined in the following areas of this section:
• Line segments: Line segment models
• Voltage regulators: Step-voltage regulators
• Transformer banks: Transformer bank connections The node currents are defined in the following area:
• Loads: Load models
• Capacitors: Shunt capacitor models
2.1.4 Final Notes
22.214.171.124 Line Segment Impedances
It’s extremely important that the impedances and admittances of the line segments be computed using the exact spacings and phasing. Because of the unbalanced loading and resulting unbalanced line cur rents, the voltage drops due to the mutual coupling of the lines become very important. It’s not unusual to observe a voltage rise on a lightly loaded phase of a line segment that has an extreme current unbalance.
126.96.36.199 Power Loss
The real power losses of a line segment must be computed as the difference (by phase) of the input power to a line segment minus the output power of the line segment. It’s possible to observe a negative power loss on a phase that is lightly loaded compared to the other two phases. Computing power loss as the phase current squared times the phase resistance does not give the actual real power loss in the phases.
188.8.131.52 Load Allocation
Many times the input complex power (kW and kVAr) to a feeder is known because of the metering at the substation. This information can be either total three-phase or for each individual phase. In some cases the metered data may be the current and power factor in each phase.
It’s desirable to have the computed input to the feeder match the metered input. This can be accomplished (following a converged iterative solution) by computing the ratio of the metered input to the computed input. The phase loads can now be modified by multiplying the loads by this ratio. Because the losses of the feeder will change when the loads are changed, it’s necessary to go through the ladder iterative process to determine a new computed input to the feeder. This new computed input will be closer to the metered input, but most likely not within a specified tolerance. Again, a ratio can be determined and the loads modified. This process is repeated until the computed input is within a specified tolerance of the metered input.
2.1.5 Short-Circuit Analysis
The computation of short-circuit currents for unbalanced faults in a normally balanced three-phase sys tem has traditionally been accomplished by the application of symmetrical components. However, this method is not well-suited to a distribution feeder that is inherently unbalanced. The unequal mutual coupling between phases leads to mutual coupling between sequence networks. When this happens, there is no advantage to using symmetrical components. Another reason for not using symmetrical components is that the phases between which faults occur is limited. For example, using symmetrical components, line-to-ground faults are limited to phase a to ground. What happens if a single-phase lateral is connected to phase b or c? This section will present a method for short-circuit analysis of an unbalanced three-phase distribution feeder using the phase frame.
184.108.40.206 General Theory
Fgr. 34 shows the unbalanced feeder as modeled for short-circuit calculations. In Fgr. 34, the voltage sources Ea, Eb, and Ec represent the Thevenin equivalent line-to-ground voltages at the faulted bus. The matrix [ZTOT] represents the Thevenin equivalent impedance matrix at the faulted bus. The fault impedance is represented by Zf in Fgr. 34.
Kirchhoff's voltage law in matrix form can be applied to the circuit of Fgr. 33.
FGR. 34 Unbalanced feeder short-circuit analysis model.