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dc shipboard power systems including distributed generation and islanding techniques
Available online at www.sciencedirect.com
Electric Power Systems Research 78 (2008) 1528–1536
Optimized restoration of combined ac/dc shipboard power systems including distributed generation and islanding techniques Sarika Khushalani, Jignesh Solanki, Noel Schulz ∗ Department of Electrical & Computer Engineering, Mississippi State University, 216 Simrall Engr Building, Hardy Road, Box 9571, MS State, MS 39762, United States Received 7 June 2007; accepted 24 January 2008 Available online 26 March 2008
Abstract Reconfiguration involves changing the status (OFF/ON) of switches, and reconfiguration for restoration involves changing the switch status to maximize the supply to loads that are left unsupplied after fault removal. Shipboard Power Systems (SPS) need automated reconfiguration for restoration schemes to restore vital loads quickly and efficiently in order to improve fight-through and survivability capabilities. The restoration in this paper is achieved using optimization with multiple objectives—maximizing the restored load and giving priority to vital loads. A restoration scheme for SPS with an integrated power system (IPS) and distributed generation (DG) involving islanding has been developed. This formulation includes a hybrid power system that has both ac and dc parts. The restoration formulation in this paper also considers the unbalanced nature of SPS operation with mutual coupling. © 2008 Elsevier B.V. All rights reserved. Keywords: Restoration; Distributed generation; Islanding; Shipboard power systems
1. Introduction Reconfiguration for restoration involves restoring power to outaged portions of the feeder, which improves service to loads by reducing outage time. The manual process of restoration for a shipboard power system (SPS) leave many loads without supply, especially if they were downstream of the fault on a radial system. Thus, a need exists for automated restoration. The integrated power system (IPS) of the SPS presents a better survivability solution in a battle situation since multiple generators can be scattered in various locations throughout the ship. Taking advantage of this survivability requires reconfiguring the power system to minimize the amount of service interruption when a portion of the system is suddenly taken out of service due to battle damage or other faults. Restoration is a combinatorial problem where the state space to search for the solution is huge and a complete listing of all possible states is very difficult. The problems with integer variables are non-deterministic polynomial-time (NP) hard, meaning no
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0378-7796/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2008.01.019
known algorithm exists to solve these problems in polynomial time. The basic objective of the restoration in this paper is to maximize the number of loads supplied, giving priority to vital loads. Designers are considering the inclusion of both centralized power sources as well as localized DG for SPS. Energy storage devices like flywheels and batteries are already on ships. DG creates a new set of constraints for shipboard system analysis relating to restoration optimization. When a DG or several DGs energize a portion of the system that has been separated from the main generation system, it is called islanding. Islanding can be either intentional or unintentional. Intentional islanding increases reliability and helps to maintain the continuity of supply to the important loads. If the DG cannot carry the entire load of the island, then part of the load needs to be shed. The load to be shed should be decided in an optimum manner, considering all the priorities. Restoration for balanced terrestrial distribution systems has been approached using heuristics [1–3], mathematical programming [4,5], meta-heuristics [6,7] and expert systems [8,9]. Additionally, some combination approaches [10,11] have been formulated. However, most of the approaches use a resistive model of loads and lines and simplify the distribution system.
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Most of the methods require running a complete power flow after each switching step to determine if the constraints are satisfied. Reference [12] provides a solution to unbalanced ac distribution systems using optimization. Butler et al. [13] use a novel fixed charge network flow method for restoration of SPS, which is essentially linear optimization, performed using the software CPLEX. However, they consider distribution system loads as constant current and cables as three phase with no mutual couplings so that the three phases could be decomposed. After decomposition, three separate independent equations are formulated for each phase, which simplifies the optimization process. Only the magnitude of current is considered for calculations, and bi-directional flow of current was not allowed during restoration. However, with several fault scenarios and the introduction of DG, this assumption is not valid. Also, with the IPS, the SPS has ac as well as dc components, and a need for strategic change clearly arises.
semi-vital loads are those loads required for combat systems, fire systems, etc. Non-vital loads provided from the nearest load center can be shed for survivability. Both load centers provide power through automatic bus transfers (ABT) for vital and semivital loads in the zone. SPS have a tightly coupled structure, due to the low impedance of the cables. The power flow analysis of a SPS shows that the voltages at the nodes are approximately equal with similar voltage angles. Because of this nature, the fault currents are very high, thus necessitating a fast and efficient restoration scheme. For the unbalanced SPS, the problem remains the same: maximizing the supply to out-of-service loads giving priority to vital loads. The power flow equations of balanced SPS when applied to unbalanced SPS fail to converge. Unbalanced SPS have mutually coupled cables and different loadings in all three phases, leading to unbalanced voltages and currents, unlike the balanced SPS, thus requiring a different forms of analysis.
2. Shipboard Power Systems
3. Introduction to LINGO
SPS have radial distribution architecture, but currently researchers are contrasting the radial distribution architecture with a zonal approach. The zonal approach employs a starboard bus and a port bus and partitions the ship into a number of electrical zones. The zonal architecture minimizes switchboard feeder cables length and hence the weight of ship. In the ac distribution system, the distributed three-phase ac must be rectified, converted to 400 Hz with an inverter, shifted to an appropriate voltage level with a transformer, and then once again rectified to provide the required dc power. The dc zonal electric distribution system (ZEDS) does not need to have an intermediate 60 Hz step. The power is converted to dc at the output of the generator, and is reconverted to the form required at the point of use, so fewer distribution transformers and ac switchgears are required, and thus it beneficially reduces the weight and size of the ship. The ZEDS, as shown in Fig. 1, is a zonal architecture where the ship is divided into electrical zones; it shows the interconnectivity and location of generators, switchboards and bus tiebreakers. The power is radially distributed from the generator switchboards to load. Each zone has two load centers; one fed from the port bus and the other from the starboard bus. The loads are classified as non-vital, semi-vital and vital loads. Vital and
The LINGO commercial optimization software package from LINDO Systems Inc. solves the constrained optimization problem [15]. LINGO is a tool for solving both linear and non-linear optimization problems. Branch-and-bound type techniques cannot be directly applied unless the problems are convex. LINGO has a direct solver, a linear solver, a non-linear solver and a branch-and-bound manager. LINGO uses the revised simplex method for its linear solver, and successive linear programming, as well as a generalized reduced gradient for its non-linear solver. LINGO can solve problems with unlimited constraints and variables but cannot handle complex numbers. The formulation is input in the format desired by the software. The direct solver first computes the values for as many unknown variables as possible, and if, at that stage all unknown variables are calculated, then the solution report is displayed. If unknown variables still exist, then LINGO calls other solvers based on the model equations. If the model is continuous and linear, LINGO calls the linear solver. If the problem involves non-linear constraints, LINGO calls the non-linear solver. In the case of integers, LINGO uses the branch-and-bound manager. LINGO’s solver status window gives a count of the linear and non-linear variables and constraints in a model. If there
Fig. 1. Shipboard power system (from [14]).
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are any non-linear variables in the model, the non-linear solver runs, which is slower. However, upper and lower bounds on the variables can be provided. LINGO provides several mathematical functions and also supports links to database management systems (DBMS) for reading and writing data that has an open database connectivity (ODBC) driver. The constrained optimization problem formulated here uses non-linear and integer solvers of the LINGO software. Using branch and bound guarantees an optimal solution.
◦ Inequality constraints ≤ PGi ≤ PGmax PGmin i
(5)
QGmin ≤ QGG ≤ QGmax i Gi
(6)
× SWi , PLi ≤ PLmax i
(7)
PLi =
Ti × PLmax i
Iij ≤ Iijmax ,
This section presents the formulations for balanced and unbalanced SPS. Some initial discussions on the balanced SPS formulation can be found in reference [16]. This paper extends this work to include examples of a small balanced and larger unbalanced SPS system. 4.1. Balanced SPS The problem is formulated as a mixed integer non-linear optimization problem with an objective that is subject to several constraints. A weighting factor is introduced in the objective function so that the contribution of high priority loads is greater than the contribution of low priority loads. WNVL =1 > W WVL SVL
(1)
where a NVL is a non-vital Load, a VL is a vital load and a SVL is a semi-vital load. The weighting factor is selected so that the vital and semi-vital load contribution is greater than the largest non-vital load contribution. The constraints enforced on the solution are the power flow, generator limiting, load limiting, line limiting and voltage limiting constraints. A separate set of equality constraints occur for the dc side of the power system. The real power drawn from the dc side of the bus is taken to be equal to the real power delivered from the ac side. In this formulation, some of the constraints have been formulated as binary variables, and the objective uses continuous variables. • Objective
WVL SVLI + WSVL SSVLI + WNVL SNVLI
(2)
for fixed load
line limits
δmin ≤ δi ≤ δmax i i ,
(8) (9)
voltage limits
(10)
angle limits
• DC constraints ◦ Equality constraints Iini = Iouti + ILi , i
Max
× SW,
Vimin ≤ Vi ≤ Vimax ,
4. Formulations
for variable load
i ∈ FN,
(11)
j ∈ TN
(12)
i
Vi = Vj + Iij × Zij ◦ Inequality constraints × SWi , PLi ≤ PLmax i
for variable load
PLi = Ti × PLmax × SWi , i Iij ≤ Iijmax ,
for fixed load
line limits
Vimin ≤ Vi ≤ Vimax ,
voltage limits
(13) (14) (15) (16)
• Switching constraints SWi = 1, if switch ‘i’ is closed SWi = 0, if switch ‘i’ is open i ∈ PS, j ∈ SS SWi + SWj = 1,
(17)
where PG and QG are the real and reactive power generation, PD and QD are the real and reactive power demand, Vi is the voltage at bus i, δi is the angle associated with the voltage at bus i, Yij is the element of the bus admittance matrix, θ is the angle associated with Yij , PLi is the load at bus i, Ti is the binary variable, 0 or 1, Iij is the current flow from bus i to bus j, Iini and Iouti are the currents entering and leaving bus i, FN is a set of from buses, TN is a set of to buses, Zij is the impedance of branch ij, PS is the set of port side switches, SS is the set of starboard side switches, Np is the set of priority loads.
i∈L
4.2. Formulation for unbalanced SPS
subject to • AC constraints ◦ Equality constraints PGi − PDi = Vi Vj Yij cos(θij + δj − δi )
Initially the three-phase Newton equations shown in (18) and (19) were used as equality constraints for unbalanced SPS: (3)
j
QGi − QDi =
j
Vi Vj Yij sin(θij + δj − δi )
(4)
n c p m pm p V G cos θ pm + Bpm sin θ pm Pi = Vi k ik ik ik ik
(18)
n c m pm p p V G sin θ pm − Bpm cos θ pm Qi = Vi k ik ik ik ik
(19)
k=1 m=a
k=1 m=a
S. Khushalani et al. / Electric Power Systems Research 78 (2008) 1528–1536 p
where p ∈ set of phases a, b and c and L ∈ set of load nodes. Pi p and Qi are the net active and reactive power injections in phase pm pm p of node i. Gik and Bik are the real and reactive parts of the 3 × 3 admittance matrix (Y) of branch between node i and node pm k, θik is the difference in voltage angle between phases p and m of nodes i and k. These equations work for small unbalanced systems. However, when these equations are used for restoration of systems with number of nodes larger than four, the solution failed to converge. This failure to converge is attributed to the fact that unbalanced SPS are ill-conditioned systems, and thus the Jacobian became singular. A decoupling procedure was also tried, which split the problem into two sub problems P–θ and Q–V, but due to the high R/X ratio of the lines, it still failed to converge. To counter the issues of equality constraints in (18) and (19), a new formulation using a different set of equality constraints is developed. • Objective Max
WVL SVLI + WSVL SSVLI + WNVL SNVLI +
i∈L
p
ILi
i∈L
(20)
subject to • AC constraints ◦ Equality constraints mp p p p Zij Iij Vj = Vi − m
N p Iij
p
Iij −
=
O
(21)
p
p
p
Ijr − ILj = 0
(22)
p SWij × slackcurrij p
p
ILp = Ti × ILi,max ,
(23)
for fixed load
(24)
◦ Inequality constraints p
p
Vi,min ≤ Vi ≤ Vi,max , p
p
ILi ≤ ILi,max , p
p
Iij ≤ Iij,max ,
voltage magnitude limits
for variable load
(25) (26)
line limits
(27)
i ∈ FN,
j ∈ TN
(28)
i
Vimin ≤ Vi ≤ Vimax ,
(32)
voltage limits
(33)
• Switching constraints SWi = 1, SWi = 0,
if switch ‘i’ is closed if switch ‘i’ is open
SWi + SWj = 1,
i ∈ PS,
(34) j ∈ SS p
where CS is the closed switch, OS is the open switch, ILi is the load current flowing in phase p of node i, WS is the weighting factor which is less than 1, SWij is the switch between nodes i mp and j. Zij is mutually coupled 3 × 3 impedance matrix of the p branch between nodes i and j. Ti is a binary variable. N is the set of branches with currents going into the node j, O is the set of branches with currents coming out of the node j. PLi is the load at bus i, Ti is the binary variable, 0 or 1, Iij is the current flow from bus i to bus j, Iini and Iouti are the currents entering and leaving bus i, FN is a set of from buses, TN is a set of to buses, Zij is the impedance of branch ij, PS is the set of port side switches, SS is the set of starboard side switches, Np is the set of priority loads. The fault is detected and isolated before performing restoration. This action is simulated by setting the corresponding SWij = 0 and thereby Iij = 0. The variable slackcurr ensures that (22) is valid when SWij = 0. Eqs. (20)–(34) in LINGO syntax are input to the software, along with the system data. LINGO does not have a facility for complex numbers, and hence the objective and ac constraints are subdivided into real and imaginary sub problems. p p p Max ILk R + ILk I + 1 − ILk R k ∈ PS
+
p 1 − ILk I +
k ∈ NG
WVL SVLI i∈L
+ WSVL SSVLI
+ WNVL SNVLI
(35)
• Equality constraints p Vj R
(29)
= SWij
p
i
◦ Inequality constraints for variable load
× SWi , PLi = Ti × PLmax i
for fixed load
p Vj R −
= SWij
p Vj I
i
Vi = Vj + Iij × Zij
× SWi , PLi ≤ PLmax i
line limits
subject to
• DC constraints ◦ Equality constraints Iini = Iouti + ILi , i
Iij ≤ Iijmax ,
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Iij R − p
Iij I −
p Vj I
r
r
p
−
m=c m=a
m=c m=a
mp mp Zij R × Iij R
mp Zij I
p
Ijr R − ILj R = 0 p
p
Ijr I − ILj I = 0
p
p
p
p
(30)
Iij R = SWij × slackcurrij R
(31)
Iij I = SWij × slackcurrij I
(36)
mp × Iij I
(37) (38) (39) (40) (41)
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p p p p p 0.5 Vi − V i R × V i R + V i I × V i I =0 p p p p p 0.5 Ii − Ii R × Ii R + Ii I × Ii I =0
p p p p p 0.5 Iij − Iij R × Iij R + Iij IIij I =0 p
p
(42)
5.1. Balanced ship system
(43)
The model of the SPS shown in Fig. 2 has a 4-zone topology with two Ship Service Converter Modules serving loads in each zone. The system is essentially radial, as radial distribution networks have a significant advantage over meshed networks. These advantages are lower short circuit currents, simpler switching and easy design of protective equipment and its coordination. However, radial structure provides lower reliability. In order to increase reliability, DG can be employed. The ac generator is in zone 4, and the DG is in zone 1. AC to dc rectifiers are in zones 1 and 4. Vital and semi-vital loads are provided with an alternative path using ABT, which switches between the port and starboard buses. Fig. 3 shows details of each load center. Each load center has a vital, semi-vital and a non-vital load. The ABT uses two switches in this model; only one of these switches is in the closed position at a certain time. The total load of the balanced SPS is 10 MW. The generator can generate up to 8 MW. The DG has, however, less capacity and can only generate up to 4 MW. The optimization formulation in Section 4.1 was used to reconfigure the SPS under fault conditions. The pre-fault system conditions are as shown in Table 1. To simulate a fault, the corresponding variable is made zero, due to unavailability of that component. The variable is equated to zero because if there is a fault in any part, that part of the ship is not available. Because of the fault and the isolation of the fault, some loads will be left without any supply. Affected loads need to have their power restored. Consider Scenario 1, where faults occur on lines 2–10 and 4–12, as shown in Fig. 4. This fault leads to loss of non-vital loads PLN10 and PLN12. No restoration path is available for these loads, so no switching occurs, and the DG ramps down
(44)
p
ILi R = Ti R × ILi,max R
(45)
p ILi I
(46)
=
p Ti I
p × ILi,max I
• Inequality constraints p
p
p
Vi,min ≤ Vi ≤ Vi,max , p
p
p
p
ILi R ≤ ILi,max R ILi I ≤ ILi,max I for
voltage magnitude limits
(47)
(ILPh i R)
and
(ILPh i I) > 0,
(48)
(ILPh i R)
and
(ILPh i I) < 0,
(49)
variable load
p
p
ILi R ≥ ILi,max R p
p
ILi I ≥ ILi,max I for variable load p
p
Iij ≤ Iij,max ,
line limits
(50)
where PS are the set of loads with positive real and imaginary parts, NG are those sets with negative real and imaginary parts. The results in the next section will validate this formulation. 5. Results This section considers two systems: a balanced ship system and an unbalanced ship system. Three cases are considered for each system.
Fig. 2. Balanced SPS model.
Fig. 3. Detailed topology of balanced SPS.
S. Khushalani et al. / Electric Power Systems Research 78 (2008) 1528–1536
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Table 1 After restoration configuration Scenario
Load PL
Vital (MW)
Semi-vital (MW)
Non-vital at PB (MW)
Non-vital at SB (MW)
Switch position (closed)
Time (s)
Total generation gen DG (MW)
Pre-fault
Load 1/2 Load 3/4 Load 9/10 Load 11/12
0.5 0.5 0.5 0.5
1 1 1 1
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
SW1 SW4 SW9 SW11
1
7.33
2.95
1
Load 1/2 Load 3/4 Load 9/10 Load 11/12
0.5 0.5 0.5 0.5
1 1 1 1
0.5 0.5 0.0 0.0
0.5 0.5 0.5 0.5
SW1 SW4 SW9 SW11
1
7.33
1.38
2
Load 1/2 Load 3/4 Load 9/10 Load 11/12
0.5 0.5 0.5 0.5
1 1 1 1
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
SW1 SW4 SW10 SW12
1
6.87
3.6
3
Load 1/2 Load 3/4 Load 9/10 Load 11/12
0.5 0.5 0.5 0.5
1 1 0.76 0.74
0.0 0.5 0.0 0.0
0.0 0.5 0.0 0.0
SW1 SW3 SW10 SW11
1
2.54
4.00
Fig. 4. Balanced SPS Fault-Scenario 1
to 1.38 MW due to loss of two non-vital loads. Table 1 shows the CPU time to find a global optimum with the switch configuration. Consider Scenario 2, where faults occur on lines 2–10 and 3–11. An alternate path is available to the vital and semivital loads left unrestored after the clearance of the fault. Now consider Scenario 3, where faults occur on lines 4–12 and 3–11.
Many priority loads are lost, and no other alternative path is available from the generator, under such a fault scenario. Now the DG, which was only supplying some part of the load in normal conditions (without fault), ramps up. Since islanding is taking place, only priority loads are restored, and the rest of the loads are shed. The DG ramps up to maximum output power,
Fig. 5. After restoration of balanced SPS-Scenario 3.
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Fig. 6. Unbalanced SPS Table 2 Unbalanced ac voltages (magnitude in pu, angle in ◦ )
Table 3 DC voltages (magnitude in pu)
Node ID
Van
∠Van
Vbn
∠Vbn
Vcn
∠Vcn
Node ID
VDC
13 14 15 16 17 18 19 20
0.999 0.997 0.996 0.996 0.999 0.996 0.995 0.994
−29.95 −30.02 −30.06 −30.07 −29.95 −30.07 −30.15 −30.21
0.999 0.998 0.997 0.997 0.999 0.998 0.997 0.997
−149.95 −150.02 −150.06 −150.06 −150.05 −150.16 −150.24 −150.29
1.001 0.996 0.994 0.993 0.999 0.991 0.985 0.981
89.95 90.52 90.82 90.84 90.05 90.77 91.21 91.38
1 2 3 4 5 7 9 10 11 12
0.964845 0.95 0.964969 0.950095 1.0 0.963351 0.964858 0.950012 0.964909 0.950034
Table 4 Restoration of unbalanced ship system Scenario
Load PL
Vital (MW)
Semi-vital (MW)
Non-vital at PB (MW)
Non-vital at SB (MW)
Switch position (closed)
Time (s)
Total generation G1 G2 (MW)
Pre-fault
Load 1/2 Load 3/4 Load 9/10 Load 11/12 Load 14 Load 15 Load 18 Load 19
0.5 0.5 0.5 0.5 8.12 8.11 8.11 8.08
1 1 1 1 – – – –
0.5 0.5 0.5 0.5 – – – –
0.5 0.5 0.5 0.5 – – – –
SW2 SW4 SW9 SW12 SW1314 SW1415 SW1718 SW1819
4
25.57
17.56
1
Load 1/2 Load 3/4 Load 9/10 Load 11/12 Load 14 Load 15 Load 18 Load 19
0.5 0.5 0.5 0.5 8.14 8.12 8.13 8.10
1 1 1 1 – – – –
0.5 0.5 0.5 0.5 – – – –
0.5 0.5 0.5 0.5 – – – –
SW1 SW4 SW9 SW12 SW1314 SW1415 SW1718 SW1819
30
21.57
21.59
2
Load 1/2 Load 3/4 Load 9/10 Load 11/12 Load 14 Load 15 Load 18 Load 19
0.5 0.5 0.5 0.5 8.13 8.11 8.13 8.11
1 1 1 1 – – – –
0.5 0.5 0.5 0.5 – – – –
0.5 0.5 0.5 0.5 – – – –
SW1 SW4 SW9 SW12 SW1314 SW1415 SW1718 SW1819
5
18.86
24.20
3
Load 1/2 Load 3/4 Load 9/10 Load 11/12 Load 14 Load 15 Load 18 Load 19
0.5 0.5 0.5 0.5 8.15 8.11 8.10 8.07
1 1 1 1 – – – –
0.5 0.5 0.5 0.5 – – – –
0.5 0.5 0.5 0.5 –– – – –
SW2 SW4 SW9 SW12 SW1314 SW1415 SW1718 SW1819
50
35.34
8.16
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SPS. The speed of solution of the problem depends on how well it is formulated. 6. Conclusions
Fig. 7. Voltages after restoration for unbalanced SPS.
i.e., 4 MW and supplies priority loads in zones 1–3. This action creates an island supplied by DG, as shown in Fig. 5. However, the main generator continues to supply the load in zone 4. 5.2. Unbalanced SPS The model of the SPS shown in Fig. 6 has a 4-zone topology similar to balanced SPS. However, the threephase ac system is unbalanced with un-transposed mutually coupled 3 × 3 impedance matrix and unbalanced loads in three phases. The optimization formulation in Section 4.2 was used to reconfigure the system under fault conditions. Tables 2 and 3 show the pre-fault phase unbalanced ac voltages and dc voltages obtained by the optimization software with SW1314, SW1415, SW1516, SW1920, SW1819 and SW1718 closed. The pre-fault system conditions are shown in Table 4. Consider Scenario 1, where faults occur on lines 2–10 and 3–11. An alternate path is available to vital and semi-vital loads left un-restored after the clearance of the fault. The CPU time to find a global optimal with the switch configuration is shown in Table 4. In Scenario 2, a fault occurs on lines 4–12 and 3–11. All loads are restored as limits imposed on the generators are not violated. Consider Scenario 3, where a fault occurs on line 14–15. The de-energized Load 15 is now restored by G1, and G2 supplies Load 14 only. A plot of voltages for some critical nodes, for pre-fault and fault scenarios is shown in Fig. 7. The voltages are within the prescribed tolerances of 0.95 and 1.05. The simulation results for balanced and unbalanced SPS restoration were obtained by using a 2.0-GHz Pentium® 4 PC. Table 5 contains the list of number of integers, linear and nonlinear variables and constraints for the unbalanced and balanced Table 5 Size of optimization problem Elements
Type
Balanced SPS
Unbalanced SPS
Variables
Integer Linear Non-linear
8 41 24
8 179 78
Constraints
Linear Non-linear
84 19
245 45
This paper proposes an optimal method for restoring power to loads, including islanding, if necessary, on a shipboard power system. The service restoration problem is formulated for both balanced and unbalanced SPS. The balanced SPS has IPS, which is essentially an ac–dc zonal architecture with embedded DG. The unbalanced SPS, though similar to the balanced IPS, has an unbalanced three-phase ac system. The equations are non-linear with integer variables. The optimization techniques are demonstrated on two different shipboard power systems with several different scenarios and fault cases. For the unbalanced SPS the equality constraints were changed to ensure convergence as the equality constraints based on the Newton method failed to converge for some larger systems. A comparison of solution times for both balanced and unbalanced SPS demonstrates that since the unbalanced SPS has more non-linear variables and constraints it takes longer to find a solution. The longer solution time is also attributed to the fact that cables are mutually coupled and the three phases cannot be decoupled. These optimization techniques provide a tool for investigating various design and operational possibilities of shipboard systems to determine the best topologies and distributed generation configurations to increase survivability and fight through features. Acknowledgements This work was supported by the United States Office of Naval Research under Grants N00014-02-1-0623 and N00014-03-10744 References [1] A.L. Morelato, A.J. Monticelli, Heuristic search approach to distribution system restoration, IEEE Trans. Power Deliv. 4 (October (4)) (1989) 223–241. [2] J.S. Wu, K.L. Tomsovic, C.S. Chen, A heuristic search approach to feeder switching operations for overload, faults, unbalanced flow and maintenance, IEEE Trans. Power Deliv. 6 (October (4)) (1991) 1579– 1586. [3] D. Shirmohammadi, Service restoration in distribution networks via network reconfiguration, IEEE Trans. Power Deliv. 7 (April (2)) (1992) 952–958. [4] K.L. Butler, N.D.R. Sarma, V. Ragendra Prasad, Network reconfiguration for service restoration in shipboard power distribution systems, IEEE Trans. Power Syst. 16 (November (4)) (2001) p653–p661. [5] T. Nagata, H. Sasaki, An efficient algorithm for distribution network restoration, IEEE Power Eng. Soc. Summer Meet. 1 (2001) 54–59. [6] S. Toune, H. Fudo, T. Genji, Y. Fukuyama, Y. Nakanishi, A reactive tabu search for service restoration in electric power distribution systems, in: Proceedings of the IEEE International Conference on Evolutionary Computation, May 4–9, 1998, pp. 763–768. [7] Y. Hsiao, C. Chien, Enhancement of restoration service in distribution systems using a combination fuzzy-GA method, IEEE Trans. Power Syst. 15 (November (4)) (2000) 1394–1400. [8] C.-S. Chen, C.-H. Lin, H.-Y. Tsai, A rule-based expert system with colored Petri net models for distribution system service restoration”, IEEE Trans. Power Syst. 17 (November (4)) (2002) 1073–1080.
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[9] K.L. Butler, J.A. Momoh, L.G. Dias, Expert system assisted identification of line faults on delta-delta distribution systems, in: Proceedings of the 35th Midwest Symposium on Circuits and Systems, vol. 2, August 9–12, 1992, pp. 1208–1213. [10] Y.T. Hsiao, C.Y. Chien, Enhancement of restoration service in distribution systems using a combination fuzzy-GA method, IEEE Trans. Power Syst. 15 (November (4)) (2000) 1394–1400. [11] M.C. Mendiola, C.S. Chang, S. Elangoyan, Fuzzy expert system for distribution system restoration and contingency operation, International Conference on Energy Management and Power Delivery, vol. 1, 1995, pp. 73–79. [12] S. Khushalani, J. Solanki, N.S. Noel, Optimized restoration of unbalanced distribution systems, IEEE Trans. Power Syst. 22 (May (2)) (2007) 624–630. [13] K.L. Butler-Purry, N.D.R. Sarma, I.V. Hicks, Service restoration in naval shipboard power systems, IEE Proc. Gener. Transm. Distrib. 151 (January (1)) (2004) 95–102. [14] P.R. Chester, W. Rumburg Jay, Zonal electrical distribution systems: an affordable architecture for future, Naval Eng. J. (1993) 45–51. [15] LINDO Systems Inc., 1415 North Dayton Street, Chicago IL, 60622, Release-8, 2003, http://www.lindo.com/. [16] S. Khushalani, N.N. Schulz, Restoration optimization with distributed generation considering islanding, Proceedings of the 2005 IEEE Power Engineering Society General Meeting, San Francisco, CA, June, 2005. Sarika Khushalani completed her Ph.D. degree from the Electrical and Computer Engineering Department of Mississippi State University in the fall of 2006.
She was a Honda Fellowship Award recipient at MSU. She received her B.E. degree from Nagpur University and her M.E. degree from Mumbai University, India in 1998 and 2000, respectively. She was involved in research activities at IIT Bombay, India. Her research interests are computer applications in power system analysis and power system control. She is now working for Open Systems International in Minneapolis, MN. Jignesh M. Solanki completed his Ph.D. degree from the Electrical and Computer Engineering Department of Mississippi State University in the fall of 2006. He received his B.E. degree from V.N.I.T., Nagpur and his M.E. degree from Mumbai University, India, in 1998 and 2000, respectively. He was involved in research activities at IIT Bombay, India. His research interests are power system analysis and its control. He is now working for Open Systems International in Minneapolis, MN. Noel N. Schulz received her B.S.E.E. and M.S.E.E. degrees from Virginia Polytechnic Institute and State University in 1988 and 1990, respectively. She received her Ph.D. in EE from the University of Minnesota in 1995. She has been an associate professor in the ECE department at Mississippi State University since July 2001 and holds the TVA Endowed Professorship in Power Systems Engineering. Prior to that she spent 6 years on the faculty of Michigan Tech. Her research interests are in computer applications in power system operations including artificial intelligence techniques. She is a NSF CAREER award recipient. She has been active in the IEEE Power Engineering Society and is serving as Secretary for 2004–2007. She was the 2002 recipient of the IEEE/PES Walter Fee Outstanding Young Power Engineer Award. Dr. Schulz is a member of Eta Kappa Nu and Tau Beta Pi.
Electric Power Systems Research 78 (2008) 1528–1536
Optimized restoration of combined ac/dc shipboard power systems including distributed generation and islanding techniques Sarika Khushalani, Jignesh Solanki, Noel Schulz ∗ Department of Electrical & Computer Engineering, Mississippi State University, 216 Simrall Engr Building, Hardy Road, Box 9571, MS State, MS 39762, United States Received 7 June 2007; accepted 24 January 2008 Available online 26 March 2008
Abstract Reconfiguration involves changing the status (OFF/ON) of switches, and reconfiguration for restoration involves changing the switch status to maximize the supply to loads that are left unsupplied after fault removal. Shipboard Power Systems (SPS) need automated reconfiguration for restoration schemes to restore vital loads quickly and efficiently in order to improve fight-through and survivability capabilities. The restoration in this paper is achieved using optimization with multiple objectives—maximizing the restored load and giving priority to vital loads. A restoration scheme for SPS with an integrated power system (IPS) and distributed generation (DG) involving islanding has been developed. This formulation includes a hybrid power system that has both ac and dc parts. The restoration formulation in this paper also considers the unbalanced nature of SPS operation with mutual coupling. © 2008 Elsevier B.V. All rights reserved. Keywords: Restoration; Distributed generation; Islanding; Shipboard power systems
1. Introduction Reconfiguration for restoration involves restoring power to outaged portions of the feeder, which improves service to loads by reducing outage time. The manual process of restoration for a shipboard power system (SPS) leave many loads without supply, especially if they were downstream of the fault on a radial system. Thus, a need exists for automated restoration. The integrated power system (IPS) of the SPS presents a better survivability solution in a battle situation since multiple generators can be scattered in various locations throughout the ship. Taking advantage of this survivability requires reconfiguring the power system to minimize the amount of service interruption when a portion of the system is suddenly taken out of service due to battle damage or other faults. Restoration is a combinatorial problem where the state space to search for the solution is huge and a complete listing of all possible states is very difficult. The problems with integer variables are non-deterministic polynomial-time (NP) hard, meaning no
∗
Corresponding author. Tel.: +1 6623252020; fax: +1 6623252298. E-mail address: [email protected] (N. Schulz).
0378-7796/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2008.01.019
known algorithm exists to solve these problems in polynomial time. The basic objective of the restoration in this paper is to maximize the number of loads supplied, giving priority to vital loads. Designers are considering the inclusion of both centralized power sources as well as localized DG for SPS. Energy storage devices like flywheels and batteries are already on ships. DG creates a new set of constraints for shipboard system analysis relating to restoration optimization. When a DG or several DGs energize a portion of the system that has been separated from the main generation system, it is called islanding. Islanding can be either intentional or unintentional. Intentional islanding increases reliability and helps to maintain the continuity of supply to the important loads. If the DG cannot carry the entire load of the island, then part of the load needs to be shed. The load to be shed should be decided in an optimum manner, considering all the priorities. Restoration for balanced terrestrial distribution systems has been approached using heuristics [1–3], mathematical programming [4,5], meta-heuristics [6,7] and expert systems [8,9]. Additionally, some combination approaches [10,11] have been formulated. However, most of the approaches use a resistive model of loads and lines and simplify the distribution system.
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Most of the methods require running a complete power flow after each switching step to determine if the constraints are satisfied. Reference [12] provides a solution to unbalanced ac distribution systems using optimization. Butler et al. [13] use a novel fixed charge network flow method for restoration of SPS, which is essentially linear optimization, performed using the software CPLEX. However, they consider distribution system loads as constant current and cables as three phase with no mutual couplings so that the three phases could be decomposed. After decomposition, three separate independent equations are formulated for each phase, which simplifies the optimization process. Only the magnitude of current is considered for calculations, and bi-directional flow of current was not allowed during restoration. However, with several fault scenarios and the introduction of DG, this assumption is not valid. Also, with the IPS, the SPS has ac as well as dc components, and a need for strategic change clearly arises.
semi-vital loads are those loads required for combat systems, fire systems, etc. Non-vital loads provided from the nearest load center can be shed for survivability. Both load centers provide power through automatic bus transfers (ABT) for vital and semivital loads in the zone. SPS have a tightly coupled structure, due to the low impedance of the cables. The power flow analysis of a SPS shows that the voltages at the nodes are approximately equal with similar voltage angles. Because of this nature, the fault currents are very high, thus necessitating a fast and efficient restoration scheme. For the unbalanced SPS, the problem remains the same: maximizing the supply to out-of-service loads giving priority to vital loads. The power flow equations of balanced SPS when applied to unbalanced SPS fail to converge. Unbalanced SPS have mutually coupled cables and different loadings in all three phases, leading to unbalanced voltages and currents, unlike the balanced SPS, thus requiring a different forms of analysis.
2. Shipboard Power Systems
3. Introduction to LINGO
SPS have radial distribution architecture, but currently researchers are contrasting the radial distribution architecture with a zonal approach. The zonal approach employs a starboard bus and a port bus and partitions the ship into a number of electrical zones. The zonal architecture minimizes switchboard feeder cables length and hence the weight of ship. In the ac distribution system, the distributed three-phase ac must be rectified, converted to 400 Hz with an inverter, shifted to an appropriate voltage level with a transformer, and then once again rectified to provide the required dc power. The dc zonal electric distribution system (ZEDS) does not need to have an intermediate 60 Hz step. The power is converted to dc at the output of the generator, and is reconverted to the form required at the point of use, so fewer distribution transformers and ac switchgears are required, and thus it beneficially reduces the weight and size of the ship. The ZEDS, as shown in Fig. 1, is a zonal architecture where the ship is divided into electrical zones; it shows the interconnectivity and location of generators, switchboards and bus tiebreakers. The power is radially distributed from the generator switchboards to load. Each zone has two load centers; one fed from the port bus and the other from the starboard bus. The loads are classified as non-vital, semi-vital and vital loads. Vital and
The LINGO commercial optimization software package from LINDO Systems Inc. solves the constrained optimization problem [15]. LINGO is a tool for solving both linear and non-linear optimization problems. Branch-and-bound type techniques cannot be directly applied unless the problems are convex. LINGO has a direct solver, a linear solver, a non-linear solver and a branch-and-bound manager. LINGO uses the revised simplex method for its linear solver, and successive linear programming, as well as a generalized reduced gradient for its non-linear solver. LINGO can solve problems with unlimited constraints and variables but cannot handle complex numbers. The formulation is input in the format desired by the software. The direct solver first computes the values for as many unknown variables as possible, and if, at that stage all unknown variables are calculated, then the solution report is displayed. If unknown variables still exist, then LINGO calls other solvers based on the model equations. If the model is continuous and linear, LINGO calls the linear solver. If the problem involves non-linear constraints, LINGO calls the non-linear solver. In the case of integers, LINGO uses the branch-and-bound manager. LINGO’s solver status window gives a count of the linear and non-linear variables and constraints in a model. If there
Fig. 1. Shipboard power system (from [14]).
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are any non-linear variables in the model, the non-linear solver runs, which is slower. However, upper and lower bounds on the variables can be provided. LINGO provides several mathematical functions and also supports links to database management systems (DBMS) for reading and writing data that has an open database connectivity (ODBC) driver. The constrained optimization problem formulated here uses non-linear and integer solvers of the LINGO software. Using branch and bound guarantees an optimal solution.
◦ Inequality constraints ≤ PGi ≤ PGmax PGmin i
(5)
QGmin ≤ QGG ≤ QGmax i Gi
(6)
× SWi , PLi ≤ PLmax i
(7)
PLi =
Ti × PLmax i
Iij ≤ Iijmax ,
This section presents the formulations for balanced and unbalanced SPS. Some initial discussions on the balanced SPS formulation can be found in reference [16]. This paper extends this work to include examples of a small balanced and larger unbalanced SPS system. 4.1. Balanced SPS The problem is formulated as a mixed integer non-linear optimization problem with an objective that is subject to several constraints. A weighting factor is introduced in the objective function so that the contribution of high priority loads is greater than the contribution of low priority loads. WNVL =1 > W WVL SVL
(1)
where a NVL is a non-vital Load, a VL is a vital load and a SVL is a semi-vital load. The weighting factor is selected so that the vital and semi-vital load contribution is greater than the largest non-vital load contribution. The constraints enforced on the solution are the power flow, generator limiting, load limiting, line limiting and voltage limiting constraints. A separate set of equality constraints occur for the dc side of the power system. The real power drawn from the dc side of the bus is taken to be equal to the real power delivered from the ac side. In this formulation, some of the constraints have been formulated as binary variables, and the objective uses continuous variables. • Objective
WVL SVLI + WSVL SSVLI + WNVL SNVLI
(2)
for fixed load
line limits
δmin ≤ δi ≤ δmax i i ,
(8) (9)
voltage limits
(10)
angle limits
• DC constraints ◦ Equality constraints Iini = Iouti + ILi , i
Max
× SW,
Vimin ≤ Vi ≤ Vimax ,
4. Formulations
for variable load
i ∈ FN,
(11)
j ∈ TN
(12)
i
Vi = Vj + Iij × Zij ◦ Inequality constraints × SWi , PLi ≤ PLmax i
for variable load
PLi = Ti × PLmax × SWi , i Iij ≤ Iijmax ,
for fixed load
line limits
Vimin ≤ Vi ≤ Vimax ,
voltage limits
(13) (14) (15) (16)
• Switching constraints SWi = 1, if switch ‘i’ is closed SWi = 0, if switch ‘i’ is open i ∈ PS, j ∈ SS SWi + SWj = 1,
(17)
where PG and QG are the real and reactive power generation, PD and QD are the real and reactive power demand, Vi is the voltage at bus i, δi is the angle associated with the voltage at bus i, Yij is the element of the bus admittance matrix, θ is the angle associated with Yij , PLi is the load at bus i, Ti is the binary variable, 0 or 1, Iij is the current flow from bus i to bus j, Iini and Iouti are the currents entering and leaving bus i, FN is a set of from buses, TN is a set of to buses, Zij is the impedance of branch ij, PS is the set of port side switches, SS is the set of starboard side switches, Np is the set of priority loads.
i∈L
4.2. Formulation for unbalanced SPS
subject to • AC constraints ◦ Equality constraints PGi − PDi = Vi Vj Yij cos(θij + δj − δi )
Initially the three-phase Newton equations shown in (18) and (19) were used as equality constraints for unbalanced SPS: (3)
j
QGi − QDi =
j
Vi Vj Yij sin(θij + δj − δi )
(4)
n c p m pm p V G cos θ pm + Bpm sin θ pm Pi = Vi k ik ik ik ik
(18)
n c m pm p p V G sin θ pm − Bpm cos θ pm Qi = Vi k ik ik ik ik
(19)
k=1 m=a
k=1 m=a
S. Khushalani et al. / Electric Power Systems Research 78 (2008) 1528–1536 p
where p ∈ set of phases a, b and c and L ∈ set of load nodes. Pi p and Qi are the net active and reactive power injections in phase pm pm p of node i. Gik and Bik are the real and reactive parts of the 3 × 3 admittance matrix (Y) of branch between node i and node pm k, θik is the difference in voltage angle between phases p and m of nodes i and k. These equations work for small unbalanced systems. However, when these equations are used for restoration of systems with number of nodes larger than four, the solution failed to converge. This failure to converge is attributed to the fact that unbalanced SPS are ill-conditioned systems, and thus the Jacobian became singular. A decoupling procedure was also tried, which split the problem into two sub problems P–θ and Q–V, but due to the high R/X ratio of the lines, it still failed to converge. To counter the issues of equality constraints in (18) and (19), a new formulation using a different set of equality constraints is developed. • Objective Max
WVL SVLI + WSVL SSVLI + WNVL SNVLI +
i∈L
p
ILi
i∈L
(20)
subject to • AC constraints ◦ Equality constraints mp p p p Zij Iij Vj = Vi − m
N p Iij
p
Iij −
=
O
(21)
p
p
p
Ijr − ILj = 0
(22)
p SWij × slackcurrij p
p
ILp = Ti × ILi,max ,
(23)
for fixed load
(24)
◦ Inequality constraints p
p
Vi,min ≤ Vi ≤ Vi,max , p
p
ILi ≤ ILi,max , p
p
Iij ≤ Iij,max ,
voltage magnitude limits
for variable load
(25) (26)
line limits
(27)
i ∈ FN,
j ∈ TN
(28)
i
Vimin ≤ Vi ≤ Vimax ,
(32)
voltage limits
(33)
• Switching constraints SWi = 1, SWi = 0,
if switch ‘i’ is closed if switch ‘i’ is open
SWi + SWj = 1,
i ∈ PS,
(34) j ∈ SS p
where CS is the closed switch, OS is the open switch, ILi is the load current flowing in phase p of node i, WS is the weighting factor which is less than 1, SWij is the switch between nodes i mp and j. Zij is mutually coupled 3 × 3 impedance matrix of the p branch between nodes i and j. Ti is a binary variable. N is the set of branches with currents going into the node j, O is the set of branches with currents coming out of the node j. PLi is the load at bus i, Ti is the binary variable, 0 or 1, Iij is the current flow from bus i to bus j, Iini and Iouti are the currents entering and leaving bus i, FN is a set of from buses, TN is a set of to buses, Zij is the impedance of branch ij, PS is the set of port side switches, SS is the set of starboard side switches, Np is the set of priority loads. The fault is detected and isolated before performing restoration. This action is simulated by setting the corresponding SWij = 0 and thereby Iij = 0. The variable slackcurr ensures that (22) is valid when SWij = 0. Eqs. (20)–(34) in LINGO syntax are input to the software, along with the system data. LINGO does not have a facility for complex numbers, and hence the objective and ac constraints are subdivided into real and imaginary sub problems. p p p Max ILk R + ILk I + 1 − ILk R k ∈ PS
+
p 1 − ILk I +
k ∈ NG
WVL SVLI i∈L
+ WSVL SSVLI
+ WNVL SNVLI
(35)
• Equality constraints p Vj R
(29)
= SWij
p
i
◦ Inequality constraints for variable load
× SWi , PLi = Ti × PLmax i
for fixed load
p Vj R −
= SWij
p Vj I
i
Vi = Vj + Iij × Zij
× SWi , PLi ≤ PLmax i
line limits
subject to
• DC constraints ◦ Equality constraints Iini = Iouti + ILi , i
Iij ≤ Iijmax ,
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Iij R − p
Iij I −
p Vj I
r
r
p
−
m=c m=a
m=c m=a
mp mp Zij R × Iij R
mp Zij I
p
Ijr R − ILj R = 0 p
p
Ijr I − ILj I = 0
p
p
p
p
(30)
Iij R = SWij × slackcurrij R
(31)
Iij I = SWij × slackcurrij I
(36)
mp × Iij I
(37) (38) (39) (40) (41)
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p p p p p 0.5 Vi − V i R × V i R + V i I × V i I =0 p p p p p 0.5 Ii − Ii R × Ii R + Ii I × Ii I =0
p p p p p 0.5 Iij − Iij R × Iij R + Iij IIij I =0 p
p
(42)
5.1. Balanced ship system
(43)
The model of the SPS shown in Fig. 2 has a 4-zone topology with two Ship Service Converter Modules serving loads in each zone. The system is essentially radial, as radial distribution networks have a significant advantage over meshed networks. These advantages are lower short circuit currents, simpler switching and easy design of protective equipment and its coordination. However, radial structure provides lower reliability. In order to increase reliability, DG can be employed. The ac generator is in zone 4, and the DG is in zone 1. AC to dc rectifiers are in zones 1 and 4. Vital and semi-vital loads are provided with an alternative path using ABT, which switches between the port and starboard buses. Fig. 3 shows details of each load center. Each load center has a vital, semi-vital and a non-vital load. The ABT uses two switches in this model; only one of these switches is in the closed position at a certain time. The total load of the balanced SPS is 10 MW. The generator can generate up to 8 MW. The DG has, however, less capacity and can only generate up to 4 MW. The optimization formulation in Section 4.1 was used to reconfigure the SPS under fault conditions. The pre-fault system conditions are as shown in Table 1. To simulate a fault, the corresponding variable is made zero, due to unavailability of that component. The variable is equated to zero because if there is a fault in any part, that part of the ship is not available. Because of the fault and the isolation of the fault, some loads will be left without any supply. Affected loads need to have their power restored. Consider Scenario 1, where faults occur on lines 2–10 and 4–12, as shown in Fig. 4. This fault leads to loss of non-vital loads PLN10 and PLN12. No restoration path is available for these loads, so no switching occurs, and the DG ramps down
(44)
p
ILi R = Ti R × ILi,max R
(45)
p ILi I
(46)
=
p Ti I
p × ILi,max I
• Inequality constraints p
p
p
Vi,min ≤ Vi ≤ Vi,max , p
p
p
p
ILi R ≤ ILi,max R ILi I ≤ ILi,max I for
voltage magnitude limits
(47)
(ILPh i R)
and
(ILPh i I) > 0,
(48)
(ILPh i R)
and
(ILPh i I) < 0,
(49)
variable load
p
p
ILi R ≥ ILi,max R p
p
ILi I ≥ ILi,max I for variable load p
p
Iij ≤ Iij,max ,
line limits
(50)
where PS are the set of loads with positive real and imaginary parts, NG are those sets with negative real and imaginary parts. The results in the next section will validate this formulation. 5. Results This section considers two systems: a balanced ship system and an unbalanced ship system. Three cases are considered for each system.
Fig. 2. Balanced SPS model.
Fig. 3. Detailed topology of balanced SPS.
S. Khushalani et al. / Electric Power Systems Research 78 (2008) 1528–1536
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Table 1 After restoration configuration Scenario
Load PL
Vital (MW)
Semi-vital (MW)
Non-vital at PB (MW)
Non-vital at SB (MW)
Switch position (closed)
Time (s)
Total generation gen DG (MW)
Pre-fault
Load 1/2 Load 3/4 Load 9/10 Load 11/12
0.5 0.5 0.5 0.5
1 1 1 1
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
SW1 SW4 SW9 SW11
1
7.33
2.95
1
Load 1/2 Load 3/4 Load 9/10 Load 11/12
0.5 0.5 0.5 0.5
1 1 1 1
0.5 0.5 0.0 0.0
0.5 0.5 0.5 0.5
SW1 SW4 SW9 SW11
1
7.33
1.38
2
Load 1/2 Load 3/4 Load 9/10 Load 11/12
0.5 0.5 0.5 0.5
1 1 1 1
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
SW1 SW4 SW10 SW12
1
6.87
3.6
3
Load 1/2 Load 3/4 Load 9/10 Load 11/12
0.5 0.5 0.5 0.5
1 1 0.76 0.74
0.0 0.5 0.0 0.0
0.0 0.5 0.0 0.0
SW1 SW3 SW10 SW11
1
2.54
4.00
Fig. 4. Balanced SPS Fault-Scenario 1
to 1.38 MW due to loss of two non-vital loads. Table 1 shows the CPU time to find a global optimum with the switch configuration. Consider Scenario 2, where faults occur on lines 2–10 and 3–11. An alternate path is available to the vital and semivital loads left unrestored after the clearance of the fault. Now consider Scenario 3, where faults occur on lines 4–12 and 3–11.
Many priority loads are lost, and no other alternative path is available from the generator, under such a fault scenario. Now the DG, which was only supplying some part of the load in normal conditions (without fault), ramps up. Since islanding is taking place, only priority loads are restored, and the rest of the loads are shed. The DG ramps up to maximum output power,
Fig. 5. After restoration of balanced SPS-Scenario 3.
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S. Khushalani et al. / Electric Power Systems Research 78 (2008) 1528–1536
Fig. 6. Unbalanced SPS Table 2 Unbalanced ac voltages (magnitude in pu, angle in ◦ )
Table 3 DC voltages (magnitude in pu)
Node ID
Van
∠Van
Vbn
∠Vbn
Vcn
∠Vcn
Node ID
VDC
13 14 15 16 17 18 19 20
0.999 0.997 0.996 0.996 0.999 0.996 0.995 0.994
−29.95 −30.02 −30.06 −30.07 −29.95 −30.07 −30.15 −30.21
0.999 0.998 0.997 0.997 0.999 0.998 0.997 0.997
−149.95 −150.02 −150.06 −150.06 −150.05 −150.16 −150.24 −150.29
1.001 0.996 0.994 0.993 0.999 0.991 0.985 0.981
89.95 90.52 90.82 90.84 90.05 90.77 91.21 91.38
1 2 3 4 5 7 9 10 11 12
0.964845 0.95 0.964969 0.950095 1.0 0.963351 0.964858 0.950012 0.964909 0.950034
Table 4 Restoration of unbalanced ship system Scenario
Load PL
Vital (MW)
Semi-vital (MW)
Non-vital at PB (MW)
Non-vital at SB (MW)
Switch position (closed)
Time (s)
Total generation G1 G2 (MW)
Pre-fault
Load 1/2 Load 3/4 Load 9/10 Load 11/12 Load 14 Load 15 Load 18 Load 19
0.5 0.5 0.5 0.5 8.12 8.11 8.11 8.08
1 1 1 1 – – – –
0.5 0.5 0.5 0.5 – – – –
0.5 0.5 0.5 0.5 – – – –
SW2 SW4 SW9 SW12 SW1314 SW1415 SW1718 SW1819
4
25.57
17.56
1
Load 1/2 Load 3/4 Load 9/10 Load 11/12 Load 14 Load 15 Load 18 Load 19
0.5 0.5 0.5 0.5 8.14 8.12 8.13 8.10
1 1 1 1 – – – –
0.5 0.5 0.5 0.5 – – – –
0.5 0.5 0.5 0.5 – – – –
SW1 SW4 SW9 SW12 SW1314 SW1415 SW1718 SW1819
30
21.57
21.59
2
Load 1/2 Load 3/4 Load 9/10 Load 11/12 Load 14 Load 15 Load 18 Load 19
0.5 0.5 0.5 0.5 8.13 8.11 8.13 8.11
1 1 1 1 – – – –
0.5 0.5 0.5 0.5 – – – –
0.5 0.5 0.5 0.5 – – – –
SW1 SW4 SW9 SW12 SW1314 SW1415 SW1718 SW1819
5
18.86
24.20
3
Load 1/2 Load 3/4 Load 9/10 Load 11/12 Load 14 Load 15 Load 18 Load 19
0.5 0.5 0.5 0.5 8.15 8.11 8.10 8.07
1 1 1 1 – – – –
0.5 0.5 0.5 0.5 – – – –
0.5 0.5 0.5 0.5 –– – – –
SW2 SW4 SW9 SW12 SW1314 SW1415 SW1718 SW1819
50
35.34
8.16
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SPS. The speed of solution of the problem depends on how well it is formulated. 6. Conclusions
Fig. 7. Voltages after restoration for unbalanced SPS.
i.e., 4 MW and supplies priority loads in zones 1–3. This action creates an island supplied by DG, as shown in Fig. 5. However, the main generator continues to supply the load in zone 4. 5.2. Unbalanced SPS The model of the SPS shown in Fig. 6 has a 4-zone topology similar to balanced SPS. However, the threephase ac system is unbalanced with un-transposed mutually coupled 3 × 3 impedance matrix and unbalanced loads in three phases. The optimization formulation in Section 4.2 was used to reconfigure the system under fault conditions. Tables 2 and 3 show the pre-fault phase unbalanced ac voltages and dc voltages obtained by the optimization software with SW1314, SW1415, SW1516, SW1920, SW1819 and SW1718 closed. The pre-fault system conditions are shown in Table 4. Consider Scenario 1, where faults occur on lines 2–10 and 3–11. An alternate path is available to vital and semi-vital loads left un-restored after the clearance of the fault. The CPU time to find a global optimal with the switch configuration is shown in Table 4. In Scenario 2, a fault occurs on lines 4–12 and 3–11. All loads are restored as limits imposed on the generators are not violated. Consider Scenario 3, where a fault occurs on line 14–15. The de-energized Load 15 is now restored by G1, and G2 supplies Load 14 only. A plot of voltages for some critical nodes, for pre-fault and fault scenarios is shown in Fig. 7. The voltages are within the prescribed tolerances of 0.95 and 1.05. The simulation results for balanced and unbalanced SPS restoration were obtained by using a 2.0-GHz Pentium® 4 PC. Table 5 contains the list of number of integers, linear and nonlinear variables and constraints for the unbalanced and balanced Table 5 Size of optimization problem Elements
Type
Balanced SPS
Unbalanced SPS
Variables
Integer Linear Non-linear
8 41 24
8 179 78
Constraints
Linear Non-linear
84 19
245 45
This paper proposes an optimal method for restoring power to loads, including islanding, if necessary, on a shipboard power system. The service restoration problem is formulated for both balanced and unbalanced SPS. The balanced SPS has IPS, which is essentially an ac–dc zonal architecture with embedded DG. The unbalanced SPS, though similar to the balanced IPS, has an unbalanced three-phase ac system. The equations are non-linear with integer variables. The optimization techniques are demonstrated on two different shipboard power systems with several different scenarios and fault cases. For the unbalanced SPS the equality constraints were changed to ensure convergence as the equality constraints based on the Newton method failed to converge for some larger systems. A comparison of solution times for both balanced and unbalanced SPS demonstrates that since the unbalanced SPS has more non-linear variables and constraints it takes longer to find a solution. The longer solution time is also attributed to the fact that cables are mutually coupled and the three phases cannot be decoupled. These optimization techniques provide a tool for investigating various design and operational possibilities of shipboard systems to determine the best topologies and distributed generation configurations to increase survivability and fight through features. Acknowledgements This work was supported by the United States Office of Naval Research under Grants N00014-02-1-0623 and N00014-03-10744 References [1] A.L. Morelato, A.J. Monticelli, Heuristic search approach to distribution system restoration, IEEE Trans. Power Deliv. 4 (October (4)) (1989) 223–241. [2] J.S. Wu, K.L. Tomsovic, C.S. Chen, A heuristic search approach to feeder switching operations for overload, faults, unbalanced flow and maintenance, IEEE Trans. Power Deliv. 6 (October (4)) (1991) 1579– 1586. [3] D. Shirmohammadi, Service restoration in distribution networks via network reconfiguration, IEEE Trans. Power Deliv. 7 (April (2)) (1992) 952–958. [4] K.L. Butler, N.D.R. Sarma, V. Ragendra Prasad, Network reconfiguration for service restoration in shipboard power distribution systems, IEEE Trans. Power Syst. 16 (November (4)) (2001) p653–p661. [5] T. Nagata, H. Sasaki, An efficient algorithm for distribution network restoration, IEEE Power Eng. Soc. Summer Meet. 1 (2001) 54–59. [6] S. Toune, H. Fudo, T. Genji, Y. Fukuyama, Y. Nakanishi, A reactive tabu search for service restoration in electric power distribution systems, in: Proceedings of the IEEE International Conference on Evolutionary Computation, May 4–9, 1998, pp. 763–768. [7] Y. Hsiao, C. Chien, Enhancement of restoration service in distribution systems using a combination fuzzy-GA method, IEEE Trans. Power Syst. 15 (November (4)) (2000) 1394–1400. [8] C.-S. Chen, C.-H. Lin, H.-Y. Tsai, A rule-based expert system with colored Petri net models for distribution system service restoration”, IEEE Trans. Power Syst. 17 (November (4)) (2002) 1073–1080.
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[9] K.L. Butler, J.A. Momoh, L.G. Dias, Expert system assisted identification of line faults on delta-delta distribution systems, in: Proceedings of the 35th Midwest Symposium on Circuits and Systems, vol. 2, August 9–12, 1992, pp. 1208–1213. [10] Y.T. Hsiao, C.Y. Chien, Enhancement of restoration service in distribution systems using a combination fuzzy-GA method, IEEE Trans. Power Syst. 15 (November (4)) (2000) 1394–1400. [11] M.C. Mendiola, C.S. Chang, S. Elangoyan, Fuzzy expert system for distribution system restoration and contingency operation, International Conference on Energy Management and Power Delivery, vol. 1, 1995, pp. 73–79. [12] S. Khushalani, J. Solanki, N.S. Noel, Optimized restoration of unbalanced distribution systems, IEEE Trans. Power Syst. 22 (May (2)) (2007) 624–630. [13] K.L. Butler-Purry, N.D.R. Sarma, I.V. Hicks, Service restoration in naval shipboard power systems, IEE Proc. Gener. Transm. Distrib. 151 (January (1)) (2004) 95–102. [14] P.R. Chester, W. Rumburg Jay, Zonal electrical distribution systems: an affordable architecture for future, Naval Eng. J. (1993) 45–51. [15] LINDO Systems Inc., 1415 North Dayton Street, Chicago IL, 60622, Release-8, 2003, http://www.lindo.com/. [16] S. Khushalani, N.N. Schulz, Restoration optimization with distributed generation considering islanding, Proceedings of the 2005 IEEE Power Engineering Society General Meeting, San Francisco, CA, June, 2005. Sarika Khushalani completed her Ph.D. degree from the Electrical and Computer Engineering Department of Mississippi State University in the fall of 2006.
She was a Honda Fellowship Award recipient at MSU. She received her B.E. degree from Nagpur University and her M.E. degree from Mumbai University, India in 1998 and 2000, respectively. She was involved in research activities at IIT Bombay, India. Her research interests are computer applications in power system analysis and power system control. She is now working for Open Systems International in Minneapolis, MN. Jignesh M. Solanki completed his Ph.D. degree from the Electrical and Computer Engineering Department of Mississippi State University in the fall of 2006. He received his B.E. degree from V.N.I.T., Nagpur and his M.E. degree from Mumbai University, India, in 1998 and 2000, respectively. He was involved in research activities at IIT Bombay, India. His research interests are power system analysis and its control. He is now working for Open Systems International in Minneapolis, MN. Noel N. Schulz received her B.S.E.E. and M.S.E.E. degrees from Virginia Polytechnic Institute and State University in 1988 and 1990, respectively. She received her Ph.D. in EE from the University of Minnesota in 1995. She has been an associate professor in the ECE department at Mississippi State University since July 2001 and holds the TVA Endowed Professorship in Power Systems Engineering. Prior to that she spent 6 years on the faculty of Michigan Tech. Her research interests are in computer applications in power system operations including artificial intelligence techniques. She is a NSF CAREER award recipient. She has been active in the IEEE Power Engineering Society and is serving as Secretary for 2004–2007. She was the 2002 recipient of the IEEE/PES Walter Fee Outstanding Young Power Engineer Award. Dr. Schulz is a member of Eta Kappa Nu and Tau Beta Pi.