A Three Phase Load Flow Algorithm for Shipboard ...

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A Three Phase Load Flow Algorithm for Shipboard Power Systems (SPS) M. M. Medina, Student Member, IEEE, L. Qi, Student Member, IEEE, and K. L. Butler-Purry, Senior Member, IEEE

Abstract— Shipboard Power Systems (SPS) are island electric power systems. Generation, transmission, and distribution systems in SPS are tightly coupled. The transmission system consists of the lines that interconnect the generator buses in a ring configuration. Hence, there is no transmission system. Therefore, SPS can be considered radial a distribution system downstream of the generators. In this paper, a three phase load flow algorithm developed for SPS is proposed. Due to the nature of SPS, the load flow algorithm must solve for systems with multiple sources and ring configured systems. This algorithm combined three methods that addressed the issues of multiple sources, ring configuration, and radial load flow. For systems with multiple sources in the same proximity, the sources are collapsed to one source bus. The breakpoint compensation method is used for ring configured systems, and the BackwardForward Sweep method is used to perform a radial load flow. This algorithm was tested on the IEEE 37 Bus Radial Distribution Test Feeder and a simplified Shipboard Power Test System. The results produced minimal percent error when compared to the actual output results. Index Terms—load flow analysis, shipboard power system, collapse, breakpoint compensation, Backward Forward Sweep.

I. INTRODUCTION hipboard power systems (SPS) in U.S. naval ships supply electric power to weapon, navigation, communication and operation systems. SPS are island (finite inertia) electric power systems. Generation, transmission, and distribution systems in SPS are tightly coupled. The transmission system consists of the lines that interconnect the generator buses in a ring configuration. The ring configuration allows any generator to provide power to any load. Hence, there is no transmission system. Therefore, SPS are considered a radial distribution system downstream of the generators. AC radial SPS differ in nature from conventional terrestrial based utility systems in many ways. Cable lengths in SPS are limited to the length of the ship. Therefore, transmission line dynamics are not significant [1]. They consist of ungrounded delta connected systems with closely coupled components. For single-phase to ground faults, the ungrounded delta configuration ensures continuous operation of the electrical

S

This work was supported by the Office of Naval Research under grant N00014-9901-0704. M. M. Medina (e-mail: [email protected]), L. Qi (e-mail: [email protected]), and K. L. Butler-Purry (e-mail: [email protected]). are with Department of Electrical Engineering, Texas A&M University, College Station, TX 77843, USA

0-7803-8110-6/03/$17.00(C)2003 IEEE

system. When compared to utility systems that have many generators contributing to the total capacity, ships have larger dynamic loads relative to generator size due to the limited capacity [1]. SPS prime movers are faster than utility systems relative to dynamic times of interest. The size and proximity of components require fast frequency and voltage controls. Load Flow is a power system analysis approach that determines the steady state system operating conditions. Traditional load flow methods, which incorporate GaussSeidel and/or Newton Raphson techniques, were primarily developed for transmission system analysis. Distribution load flow analysis must incorporate the unique characteristics of distribution systems such as unbalanced loads, distributed loads, radial network structure, grounded or ungrounded systems, and one, two, or three phase lines. Due to the high R/X ratios and unbalanced operation in distribution systems, the Newton Raphson and Fast Decoupled may provide inaccurate results and not converge. Therefore, traditional load flow methods cannot be directly applied to distribution systems since the assumptions made for transmission systems are not valid for the unique characteristics of distribution systems. Recently there has been a lot of interest in the area of three phase distribution load flow and several methods have been developed in [2]-[6]. One algorithm is the Backward-Forward Sweep method, which performs load flow analysis on radial distribution systems. However, most algorithms were developed for conventional grounded radial configured utility distribution systems. In our research work, a three phase load flow algorithm that considers the distribution system characteristics as well as the distinct features of SPS is necessary. This paper proposes a three phase load flow algorithm developed for SPS. The algorithm combines the three features of collapsed multiple sources, Breakpoint Compensation, and BackwardForward Sweep. SPS are ring configured systems. By introducing the Breakpoint Compensation method, ring configured SPS are converted into pseudo radial systems so that the load flow calculation of the Backward-Forward Sweep method can be applied. SPS have multiple sources in the same proximity (small impedance between them) at nearly the same voltage. These sources are collapsed into one source node so that there is only one source feeding the system. The methods of collapsed multiple sources, Breakpoint Compensation, and Backward-Forward Sweep are discussed in Section II. Implementation of the algorithm and component modeling are described in Section III. In Section IV, the algorithm was tested on the IEEE 37 Bus Radial Distribution Test Feeder and a simplified Shipboard Power System, and

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the comparison results of the load flows and actual results are given. In Section V, the conclusions are presented. II. METHODS USED IN LOAD FLOW ALGORITHM FOR SPS The proposed load flow algorithm for SPS incorporates three methods. The Backward-Forward Sweep iterative method was adopted to solve the radial load flow analysis of AC SPS. Ring configured SPS must then be converted to a radial system for analysis purposes. The Breakpoint Compensation method was used to convert the system to a pseudo radial system to perform the radial load flow method. For systems with multiple sources, the conventional PV compensation method was studied. However, this method was not suitable for sources in the same proximity at nearly the same voltage as they are on SPS. Hence, a method was used that collapses the source buses to one node. The rest of this section discusses the three methods in detail. A. Backward-Forward Sweep Method The Backward-Forward Sweep method has been used by [2], [3], and [4] as an iterative means to solving the load flow equations of radial distribution systems. The BackwardForward Sweep method exploits the radial topology of a distribution network. There are two steps in this method, the Backward sweep, which updates currents using Kirchoff’s Current Law (KCL), and the Forward sweep, which updates voltage using voltage drop calculations. These two steps are repeated until convergence is achieved. The Backward Sweep calculates the current injected into each branch as a function of the end node voltages. It performs a current summation while updating voltages. Bus voltages at the end nodes are initialized for the first iteration. Starting at the end buses, each branch is traversed toward the source bus updating the voltage and calculating the current injected into each bus. These calculated currents are stored and used in the subsequent Forward Sweep calculations. The calculated source voltage is used for mismatch calculation as the termination criteria by comparing it to the specified source voltage. The Forward Sweep calculates node voltages as a function of the currents injected into each bus. The Forward Sweep is a voltage drop calculation with the constraint that the source voltage used is the specified nominal voltage at the beginning of each forward sweep. The voltage is calculated at each bus, beginning at the source bus and traversing out to the end buses using the currents calculated in previous the Backward Sweep. Convergence is achieved when the magnitude of the voltage mismatch, |[∆VLF]|, between the calculated source voltage in the Backward Sweep, [Vs,BK], and the specified source voltage, [VS], is less than or equal to a specified tolerance as shown in (1). If it is greater than a specified tolerance, then load flow on the system is run again. The Backward and Forward Sweep is repeated until this convergence is met. ∆

  

VLF  

=

 V  − V  S   S,BK 

≤ ε1

(1)

B. Breakpoint Compensation Method SPS as well as some terrestrial distribution systems are connected in a ring configuration. Since the distribution load flow iterative method is used for analysis of radial configured systems, some adjustments must be made to use this algorithm on ring configured systems. There are many proposed techniques to solve ring (loop) configured distribution systems that follow the same methodology. Cheng and Shirmohammadi in [4] developed a breakpoint compensation method for ring configured distribution systems. The new load flow algorithm for SPS utilizes this compensation method. To accommodate for the ring configuration, the loop is broken and the system is made into a pseudo radial system for analysis purposes. The ring is separated at the farthest bus from the source based on electrical distance to form a pseudo radial system. The bus that is separated, called the breakpoint, becomes two identical buses with a fraction of the original load and capacitor values on each bus. There is no way to approximately determine the amount of load and capacitance that will be served from each feeder. Hence, this algorithm arbitrarily chooses to place half of the load and capacitance at each bus. The current fed to each breakpoint will be regulated by the breakpoint current. The loop is simulated by injecting a breakpoint current, [ I BK ], at each breakpoint bus. For example, the ring configured system shown in Fig. 1 is shown again in Fig. 2 as a radial system by creating a breakpoint at bus 3 into two identical buses, 3a and 3b. Assuming that the distance (impedance) from bus 1 and bus 3 is the same as the distance from bus 1 and bus 2, then either bus can be chosen as the breakpoint. A radial load flow can be applied to the pseudo radial system. G

G 1

2

Z3

M

M

Load

Load Load

Z1

Z2

M 3 G

Fig. 1. Two Source Ring Configured System G

G 1

2

Z3

M

M

Load

Load

Z1

Z2

G

G 3a Ibk

M/2

Half of Load Values on each breakpoint

3b Ibk M/2

Load/2

Load/2

Fig. 2 Ring Configured System with Breakpoints

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This formation introduces an additional convergence criterion for each breakpoint, which is that the voltage at the breakpoint buses must be equal within a specified tolerance. This is checked after the load flow has converged. The magnitude of the voltage mismatch, |[∆VBK]|, is calculated using (2) where [VBK,a] and [VBK,b] are the voltages at each breakpoint bus i, split into a and b. If the voltage mismatch is less than or equal to a specified tolerance then the voltage at the breakpoint has successfully converged. If the voltage mismatch at the breakpoint buses is greater than a specified tolerance, then the breakpoint current is drawn from breakpoint bus a and injected into breakpoint bus b simulating the ring configuration of the system as shown in Fig. 2.This process is repeated until the load flow has converged and the breakpoint convergence criterion has been met. The breakpoint current is calculated by using (3) where the voltage mismatch is |[∆VBK]|, and the breakpoint impedance matrix is [ZBK]. The breakpoint impedance matrix is formed as described in [4] where diagonal elements are the sum of the series line impedance of each branch on the ring and off diagonal elements are the sum of the series line impedance between shared paths of two different rings in the system. This matrix only needs to be formed once since it remains constant throughout the analysis

[∆VBK ] =

[VBK,a ] − [VBK,b ] ≤ ε 2

[I BK ] = [Z BK ]−1 * [∆VBK ]

(2) (3)

C. Collapsing Multiple Sources on SPS A characteristic of SPS is that there are multiple generators (sources) in the same proximity in the network, some generators operate online and some generators operate offline acting as emergency generators. Due to the tight coupling between the generators and loads (mostly motors), very sophisticated automatic voltage regulators (AVRs) are included with the generators. Further, since the generators are typically very close in electrical distance (no more than a few hundred feet and connected in a ring), the generator buses operate at nearly the same voltage. One approach that was studied was a PV compensation method. One source was specified as the slack bus and the other sources as PV buses. Since the voltages at both buses are similar, the current on the cable between the generator buses is small. However, convergence in the Load Flow program could not be achieved because the current on the cable between the generator buses was very large. Therefore, the PV Compensation method could not be used for analysis of SPS. Another approach was to make some reductions based on the topology of the generators in SPS. Since the generators are operating at nearly the same voltage then the currents between the generator buses are negligible. Therefore the two generator buses can be assumed to be the same bus and treated as one collapsed bus. For the algorithm presented in this paper, the online source buses are combined into one generator bus. For example, in the three generator system shown in Fig. 1 with the generator at bus 3 assumed to be

offline, the generators at buses 1 and 2 are combined as shown in Fig. 3. G 1 M

M Load

Load Z1

Z2 Load M 3 G

Fig. 3. Two Source Ring Configured System

III. THREE PHASE LOAD FLOW ALGORITHM FOR SPS This section describes the implementation of the three phase load flow algorithm [7]. A flow chart is shown that depicts how the three methods from Section II are combined to create a load flow algorithm for SPS. An efficient load flow requires detailed component modeling. This section also describes the component models used in the algorithm for constant loads, capacitors, induction motor loads, voltage regulators, transformers, and lines. A. Implementation of Algorithm The load flow algorithm for SPS was implemented using MATLAB [8]. One assumption made when implementing this algorithm was that there could only be one ring in the system. Another assumption is that if there are multiple sources that they are nearly the same voltage on a SPS. Fig. 4 shows the flow chart of the algorithm and how the three methods discussed in Section II are combined. First if there are multiple sources in the system, then they are collapsed into one source bus. Next, if the system is ring configured, then the Breakpoint Compensation method is applied to convert it to a pseudo radial system. The Backward-Forward Sweep method is then applied to compute a radial load flow. If the load flow (Backward-Forward Sweep) does not converge, it is repeated until the load flow convergence criterion is met. Once the load flow has converged, then the breakpoint compensation criterion must be met. If the breakpoint compensation criterion is not met, then the breakpoint current injection is calculated and the load flow is rerun on the system. If the breakpoint compensation criterion is met, then the load flow is successfully calculated. B. Component Modeling Voltage and current were represented as line to line voltages and line currents, respectively. The calculated phase currents for delta connected components were converted to the equivalent line currents to be used in the radial load flow calculations. All values were converted to per unit.

4 Read Input File

yes

Collapse Multiple Sources

Are there muiltple sources no Is system ring configured? no Compute load flow of radial system

yes

Find breakpoint buses

([ a ], [ d ], [ A ]) were calculated as in [2]. Delta-delta transformers were modeled by their constant admittance ([Ypp], [Yps], [Yss], [Ysp]) as in [9]. Lines were modeled using the standard Π equivalent circuit. The update equations for these components are shown in Table I. These update equations will be used in the Backward-Forward Sweep method to solve for primary side voltage and current in the Backward Sweep and for secondary voltage in the Forward Sweep. Vi-1

Ii

Create breakpoints

Vi

I' i

Line, Transformer, or Voltage Regulator Induction Motor

IMi

IL i ICi Load

Sublaterals Ij Breakpoint Ibk i Capacitor

Fig. 5. Voltage and Current Flows on a Branch

no Calculate and update breakpoint current injection

no

Did load flow converge? yes Did breakpoint converge? yes

[I ] = [IL ]+ [IC ]+ [IM ]+ [Ibk ]+ ∑ I ' i

i

i

 j  i 

(4)

TABLE I COMPONENT MODELS AND UPDATE EQUATIONS Direction of Update Equation Calculation Constant Power Load 

 IL   i 

Backward Sweep

  S =  i  Vi  

Constant Current Load

Fig. 4. Flow Chart of SPS Load Flow Algorithm Backward Sweep

motors, respectively, [ I i j ] is the sum of the current of the sublaterals branching from bus i, and [ Ibki ] is the breakpoint current at bus i. An open delta connected type B voltage regulator was modeled. The constant effective turns ratios

i

j∈B

Print Output

Detailed modeling is needed for the various components in SPS. Component models were derived from [2], [3], and [9]. Load and capacitor models were built with the input complex power absorbed and nominal (rated) voltage. The constant parameters of power ([S]) and impedance ([Z]) were built with the input values. For constant current loads, the current magnitude (|[ILnom]|) and power factor ([ θ ]) are the constant parameters built with the input values. Capacitors were modeled by their constant admittance ([ Y ]). Induction motor loads were modeled by the positive and negative input sequence admittance [2]. The constant sequence admittance was converted to the equivalent constant phase admittance ([YLL]). These constant parameter models and bus voltage were used to calculate the injected current for each iteration at bus i for loads ([ ILi ]), capacitors ([ ICi ]), and motors ([ IM i ]) as shown in Table I. Branches are modeled by voltage regulators, transformers, and lines as shown in Fig. 5 where i-1and i are the primary and secondary buses of a branch, respectively [3]. To solve for the voltage and current on the primary side of a branch, the current injected into the secondary side bus ([ I i' ]) needs to be calculated as shown in (4) where [ ILi ], [ ICi ], and [ IM i ] are the injected currents from loads, capacitors, and

i

IL  =   i   

[IL ] *cos δ nom i

    i − θi    

 +  

*         

[ ]



j * ILnom *sin δi  − θi  i  





 

Constant Impedance Load IL   i 

Backward Sweep

Backward Sweep

V = i Z  i

   

Induction Motor Load [IM i ] = [YLLi ][Vi ]

Backward Sweep

Capacitor [ICi ] = [YiVi ]

Backward Sweep

Voltage Regulators [Vi −1] = [a][Vi ]

[Ii ] = [d ]* [Ii' ] [Vi ] = [A][Vi −1]

Forward Sweep

Transformers

Backward Sweep

Forward Sweep

[Vi −1 ] = ( [Ysp ] )−1( [I 'i ] − [Yss ][Vi ] )

[I i ] = [Ysp ][Vi −1] + [Yss ][Vi ] [Vi ] = ( [Y ps ] )−1 ( [I i ] − [Y pp ] [Vi −1 ] ) Lines

Backward Sweep

Forward Sweep

[Vi −1] = [Vi ] + [Ii' ][Zi ]

[Ii ] = [I i' ] [Vi ] = [Vi −1] − [Ii ][Zi ] IV. RESULTS AND COMPARISONS

A. IEEE 37 Bus Radial Distribution Test Feeder The standard IEEE 37 Bus Radial Distribution Test Feeder was developed by the IEEE Distribution System Analysis

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Subcommittee to have a common set of valid data to verify solutions of developed programs [10]. In this paper, it was used to validate the proposed load flow algorithm for its use with delta connected radial networks and component modeling. The system diagram is shown in Fig. 6. The system data can be found in [10]. It is a three wire ungrounded delta connected system operating at a nominal voltage of 4.8 kV. It includes a voltage regulator that consists of two single-phase units connected in open delta on phases AB and BC. There is unbalanced loading from spot loads of constant power, current, and impedance. 799

707

701 713 704 702 714

742 705 729 744 727

703

percent error =

724

722

712

720 706

718

725

728 730 732 708 736 710 735

709

733 775 734 737

738

731

740 711

angles for nine of the buses are shown in Table II. Dashes (-) indicate that a phase is not present at that bus. The percent error was calculated using (5) where the calculated values are the results of the proposed load flow algorithm and the actual values are from [10]. As seen in Table II, the percent error is less than one except for some of the voltage magnitudes for phase A. This is because the actual and simulated voltage angles are small values and very close to zero. Yet, due to the way that the percent error is calculated, division by such small numbers makes it seem that there is a large error, when in fact the absolute values are almost the same.

741

Fig. 6. IEEE 37 Bus Distribution Test Feeder

The per unit voltage and current magnitudes and phase

actual value − calculated value × 100 actual value

B. Shipboard Power Test System A Shipboard Power Test System was developed based on the configuration and description of a typical SPS described previously. It incorporates the necessary characteristics that are included in SPS such as multiple sources in close proximity, ring configuration, delta-delta transformer, load centers, and a delta configured system. Components and cables outlined in dashed lines are designated as offline (out of service) alternate cable routes and emergency components. For this test system, there are three generators, two online (G1 and G2) and the generator at bus 3 (G3) is an offline emergency generator. The generators are connected in a ring configuration between switchboards (SB1, SB2, SB3). It is a delta connected system made up of three phase cables (C12, C13, C23, C24).

TABLE II RESULTS AND COMPARISON FOR IEEE 37 BUS DISTRIBUTION TEST FEEDER Voltage Actual Values Bus

Mag.

a RG7 b c a 705 b c a 703 b c a 704 b c a 720 b c a 775 b c a 722 b c a 734 b c a 711 b c

1.0437 1.025 1.0345 1.0241 1.0075 1.0088 1.0178 1.0051 1.0034 1.0217 1.0044 1.0065 1.0205 1.0011 1.0041 1.0111 1.0012 0.9967 1.0185 0.9954 1.0023 1.0029 0.9978 0.9893 0.9982 0.9963 0.9852

Phase Angle 0 -120 120.9 -0.13 -120.59 120.46 -0.17 -120.7 120.2 -0.17 -120.61 120.46 -0.21 -120.66 120.53 -0.11 -120.73 120.07 -0.3 -120.62 120.68 -0.01 -120.74 119.88 0.06 -120.74 119.76

Current

Calculated Values

Percent Error

Phase Angle 0.00 -120.00 120.96 -0.13 -120.59 120.52 -0.17 -120.70 120.26 -0.17 -120.61 120.53 -0.21 -120.65 120.59 -0.10 -120.73 120.14 -0.30 -120.62 120.75 -0.01 -120.73 119.95 0.06 -120.73 119.83

Phase Angle 0.00 0.00 0.05 0.92 0.00 0.05 0.00 0.00 0.05 0.00 0.00 0.06 1.90 0.01 0.05 5.36 0.00 0.06 0.53 0.00 0.06 0.00 0.00 0.06 0.00 0.01 0.06

Mag.

1.0458 1.0256 1.0358 1.0261 1.0081 1.0101 1.0198 1.0057 1.0047 1.0236 1.005 1.0078 1.0224 1.0017 1.0054 1.0131 1.0018 0.998 1.0204 0.996 1.0037 1.0049 0.9984 0.9906 1.0001 0.9969 0.9865

Mag.

0.201 0.059 0.126 0.195 0.060 0.129 0.197 0.060 0.130 0.186 0.060 0.129 0.186 0.060 0.129 0.198 0.060 0.130 0.187 0.060 0.140 0.199 0.060 0.131 0.190 0.060 0.132

(5)

Actual Values Mag.

382.22 283.97 356.27 20.47 20.70 33.69 189.70 134.29 131.76 41.84 72.40 71.48 24.26 52.13 66.88 4.85 32.61 35.12 105.26 67.33 64.58 29.64 29.66

Phase Angle -62.28 -179.65 72.66 -80.18 -150.36 64.51 -55.62 168.37 79.32 -55.17 -163.27 50.53 -84.57 -147.09 51.68 -84.78 -147.19 39.84 -55.34 160.83 86.69 -85.87 94.15

Calculated Values Mag.

383.23 284.67 356.43 20.47 20.71 33.67 189.81 134.46 131.77 41.92 72.46 71.53 24.31 52.12 66.90 4.85 32.61 35.11 105.29 67.38 64.55 29.67 29.67

Phase Angle -62.34 -179.88 72.57 -80.13 -150.41 64.49 -55.69 168.26 79.21 -55.26 -163.40 50.45 -84.63 -147.19 51.63 -84.71 -147.18 39.85 -55.38 160.76 86.63 -85.81 94.19

Percent Error Mag.

0.26 0.25 0.04 0.02 0.05 0.05 0.06 0.13 0.01 0.19 0.08 0.07 0.22 0.01 0.03 0.09 0.00 0.02 0.03 0.08 0.04 0.11 0.05

Phase Angle 0.00 0.13 0.13 0.06 0.03 0.03 0.13 0.06 0.14 0.17 0.08 0.16 0.07 0.07 0.10 0.08 0.01 0.02 0.07 0.04 0.07 0.07 0.04

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There is one load center (LC4) branching from switchboard 2 (SB2). The system supplies three induction motor loads (M1, M2, M3) and two balanced three phase static loads (SL1, SL3). There is a step down delta-delta connected transformer (450 to 120 V) located at the load center (LC4) supplying three, single phase, static loads (SL2). The system data is shown in Table III and the system is shown in Fig. 7. For this test system, the G1 and G2 were combined into one source. The breakpoint is at bus 3, therefore half of the load values are placed on each breakpoint. TABLE III SHIPBOARD POWER TEST SYSTEM DATA Component Data Sources Power Rating: 3125 kVA (G1, G2, G3) Voltage Rating: 0.45kV Cables Length: 53.85 ft. (C12, C13, Self Series Impedance: 0.3872+j3.2246 C23, C24) Mutual Series Impedance: 0.1042+j3.0654 Power Rating: 25kVa Transformer Voltage Rating: 0.45 – 0.12 kV (XFM –1) Impedance: 0.3477+j0.002478 Power Rating: 192.6 kW Voltage Rating 0.44 kV Induction Motor Slip: 4.84% Loads Stator Impedance: 0.0538953+j0.0004317 (M1, M2, M3) Rotor Impedance: 0.143625+j0.000382 Magnetizing Inductance: j0.0201333 Load Balanced Constant Impedance (SL1) Phase AB, BC, and CA: 312.5 kW Load Balanced Constant Impedance (SL3) Phase AB, BC, and CA: 250+j187.5 kVA Unbalanced Constant Impedance Single Phase Phase AB: 312.5 kW Loads Phase BC: 156.25 kW (SL2) Phase CA: 312.5 kW G1 SB1

Bus 1 SL1

M1

C12

C13

SB2

SB3 LC4 M3

M2 G2

C24

G3 T1 SL2

Bus 4 Bus 2

SL3

Bus 5 C23

Bus 3

Fig. 7. A Simplified Shipboard Power Test System

To compare the results of the proposed load flow algorithm, a set of simulated output data was needed. A time domain transient simulation program developed using MATLAB/SIMULINK, was run on the shipboard power test system. Since this was not a steady state analysis, some manipulations were made to extract the necessary values. The magnitude of each steady state output value was extracted for each phase at its peak value. The angle of each steady state output was extracted by calculating the difference of each

phase at its zero crossing. Additionally, the transformer model used in the time domain simulation includes the magnetizing impedance, whereas the model used in the load flow algorithm does not. Extraction manipulations and model differences should be taken into consideration when comparing the output values of the two simulations. The time domain simulations analysis run on the test system was a transient simulation. Hence, some parameters of dynamic loads, such as equivalent rotor resistance of the induction motors, take time to reach steady state values. Therefore, in order to compare the steady state results from the transient simulation to the load flow results, the constant impedance for each motor was calculated based on the steady state voltage and current results of the transient simulation. The calculated constant impedance values were then used for the induction motor model in the proposed load flow algorithm for proper comparison of the two sets of results. The magnitudes (Mag.) and phase angles of the representative bus voltages and branch currents are shown in Table IV. The percent error is calculated using (5). In Table IV, the voltage at bus 1 is represented by V1. The current injected into the breakpoint bus from bus 1 and bus 2 is I3a and I3b, respectively. Some of the percent errors for the phase angles of phase A are very large. The reason is the same as discussed in section B for the 37 Bus Test Feeder. TABLE IV RESULTS AND COMPARISON FOR SHIPBOARD POWER TEST SYSTEM Actual Value Calculated Value Percent Error Phase Phase Phase Mag. Mag. Mag. Angle Angle (°) Angle (°) a 0.9944 0.000 0.9944 0.000 0.0000 0.000 V1 b 0.9964 -118.800 0.9964 -118.800 0.0000 0.000 0.000 c 0.9961 120.240 0.9961 120.240 0.0000 a 0.9899 0.000 0.9901 0.009 0.0202 100.000 V3 b 0.9923 -118.800 0.9921 -118.787 0.0202 0.011 0.008 c 0.9915 120.240 0.9918 120.249 0.0303 a 0.986 0.000 0.9865 -0.114 0.0507 100.000 V4 b 0.9895 -118.800 0.9899 -118.904 0.0404 0.087 0.152 c 0.9869 120.240 0.9889 120.058 0.2027 a 0.1469 -61.152 0.1461 -61.307 0.5446 0.253 I3a b 0.1468 178.880 0.1460 178.722 0.5450 0.088 58.927 0.1461 58.757 0.5446 0.291 c 0.1469 a 0.1469 -61.152 0.1461 -61.307 0.5446 0.253 I3b b 0.1468 178.880 0.1460 178.722 0.5450 0.088 58.927 0.1461 58.757 0.5446 0.291 c 0.1469 a 0.2786 -40.118 0.2841 -40.039 1.9742 0.195 I4 b 0.246 -167.301 0.2502 -167.377 1.7073 0.045 83.416 0.2389 83.589 1.6163 0.207 c 0.2351 a 0.0723 -23.760 0.0725 -23.758 0.2766 0.009 IM2 b 0.0718 -144.720 0.0719 -144.717 0.1393 0.002 c 0.0709 96.480 0.0709 96.486 0.0000 0.007 a 0.0923 0.000 0.0956 0.153 3.5753 100.000 ISL2 b 0.0476 -118.800 0.0484 -118.899 1.6807 0.083 c 0.0925 120.240 0.0958 119.769 3.5676 0.393

V. CONCLUSIONS Due to the distinct features of SPS, conventional load flow methods and existing three phase distribution methods cannot be applied for load flow analysis of SPS. This paper proposed a three phase load flow algorithm developed for SPS. The

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three methods of the Backward-Forward Sweep, Breakpoint Compensation, and collapsed multiple sources were incorporated in this algorithm. Multiple sources in the system are assumed to be in the same proximity and with nearly the same voltage. Therefore, multiple sources are collapsed into one source. The Breakpoint Compensation method is applied to the system to convert it to a pseudo radial system. Then the Backward-Forward Sweep is applied to solve the radial load flow. The load flow algorithm was tested on two different test systems. The results produced small percent errors, which proves that this is a valid load flow method for SPS. The PV node compensation method will be included and the multiple swing bus concept to address the multiple sources will be further explored. The proposed three phase load flow algorithm will be used by the Power Systems Automation Laboratory at Texas A&M University as support for a reconfiguration/restoration method for SPS. VI. REFERENCES [1]

K. L. Butler, N.D. R. Sarma, C. Whitcomb, H. D. Carmo, H. Zhang, “Shipboard Systems Deploy Automated Protection”, IEEE Computer Application on Power Systems, April 1998, pp. 31-36. [2] W. H. Kersting, Distribution System Modeling and Analysis, New York: CRC Press LLC, 2002. [3] R. Zimmerman, “Comprehensive Distribution Power Flow: Modeling, Formulation, Solution Algorithms and Analysis,” Ph.D. dissertation, Dept. of Elec. Eng., Cornell Univ., Jan. 1995. [4] C.S. Cheng and D. Shirmohammadi, “A Three-Phase Power Flow Method for Real-Time Distribution System Analysis,” IEEE Trans. Of Power Systems, vol. 10, no. 2, May 1995, pp. 671-679. [5] S. Ghosh, D. Das, “Method for load flow solution of radial distribution networks,” IEE Proc. Generation Transmission Distribution, vol. 146, no. 6, Nov. 1999. [6] M.H. Haque, “Efficient load flow method for distribution systems with radial or mesh configuration,” IEE Proc. Generation, Transmission, and Distribution, vol. 143, no. 1, Jan. 1996. [7] M. Medina, "A Three Phase Load Flow for Shipboard Power Systems," Masters Thesis, Dept. Elec. Eng., Texas A&M Univ., College Station, 2003. [8] MATLAB, Version 6.5.0 18091 3a, Release 13, The Mathworks, Inc., 1984-2002. [9] M. Chen and W.E. Dillon, “Power System Modeling,” Proc. Of IEEE, vol. 62, no. 7, July 1974, pp. 901-915. [10] IEEE Distribution System Analysis Subcommittee, “Radial Distribution Test Feeders,” IEEE Trans. on Power Systems, vol. 6, no. 3, Aug. 1991, pp.975-985.

VII. BIOGRAPHIES Monica M. Medina was born San Juan, Puerto Rico. She received her B.S.E.E from Texas A&M University. She is now pursuing her M.S.E.E at Texas A&M University. She is a research assistant in Power System Automation Lab of Texas A&M University. Li Qi was born Xi’an, China P.R., received her B.S.E.E from Xi’an Jiaotong University and M.S.E.E from Zhejiang University. She is now pursuing her Ph.D. in Texas A&M University. She is a research assistant in Power System Automation Lab of Texas A&M University. Karen Butler-Purry is an associate professor in the department of electrical engineering at Texas A&M University. She received the B.S. degree from Southern University-Baton Rouge in 1985, the M.S. degree from the University of Texas at Austin in 1987, and the Ph.D. degree from Howard University in 1994, all in electrical engineering. Dr. Butler-Purry holds memberships in IEEE, IEEE Power Engineering Society (PES), and the Louisiana Engineering Society.