Chapter 3 Demand-side Optimal Power Supply
3.2.2. Approach to Recognize Optimal Location and Sizing of DG
In this subsection, a new approach using an exact solution method for obtaining an optimal location and size of DG in a simple distribution system model is provided.
Utilization of the Enumeration Method as an Exact Solution Method (1)
calculation, the enumeration method which enumerates solutions for every possible combinations is used in this study. If all possible combinations of the problem would be enumerated, the optimized solution was able to be selected from them. This is a very simple and reliable approach and versatile. However possible combinations of location and size for DG should be enormous in the problem (called Optimal DG installation ) generally and the number of calculations would increase dramatically if additional variables such as attributes of DG would be added. Therefore, it is generally Optimal DG installation
possible if the numbers of possible combinations for calculation would be reduced by using a simple power system model and adding constraints for the DG location bus and the injecting power type.
Therefore, this research Optimal DG
installation distribution system
model with some constraints so that possible combination numbers of DG installation are limited to around several dozes.
Definition of the Simple Distribution System Model (2)
In this research, 6-buses and no branched simple system is used as the simple distribution system model illustrated in Figure 3-1. In this figure, Bus1 is the slack bus and resistanceriand reactancexi(i=1, 2, 5) are considered in each branch.
Figure 3-1 Simple Distribution System Model Formulation of Optimal DG Allocation Problem
(3)
In this research, minimization of active power loss in whole targeted distribution networks is defined as an objective function, and firstly the formulation of optimal DG allocation problem is implemented. In the formulation, allowable DG locations are buses in
r1,x1
1 2 r2,x2 3 r3,x3 4 r4,x4 5 r5,x5 6
the targeted system model and 0-1 variable y is utilized for showing whether DG is installed or not at the targeted bus. As constraints, power flow law and cardinalities which reduce the number of combinations for DG location candidates are considered but voltage and apparent current constraints are not considered. If simulation results would excess these unconsidered limits, new constraints would be considered. Based on these conditions, the problem for optimal allocation of DGs is formulated as follows.
M i Plossi
min (3-3)
Subject to
) (
* j N
I V G
Sj j j j (3-4)
) ( 1 ,
0 k K
yk (3-5)
) ( ,...,
1 ,
0 n k K
xk k (3-6)
Smax
x
Ksize
k k (3-7)
max
min y K
K k K k (3-8)
where
K : Set of candidate DG locations
nk : Number of allowable DG sizes at candidate DG locationk(k K) []
size : Allowable DG sizes for each candidate DG location (P- or Q-values)
yk : 1 if candidate DG locationkis selected, 0 otherwise (k K) xk : Decision variable for DG size at candidate DG locationk(k K), 0
ifyk=0
Smax : Total allowable capacity of installed DG's (P- or Q-value) Kmin : Minimum cardinality for the number of installed DG's Kmax : Maximum cardinality for the number of installed DG's
M :
N : Set of grid buses : { 1,2, , n}
Sj : Complex power load at grid bus-j(nonpositive value) Gj :
Complex power generation at grid bus-j(nonnegative value) See Table.1 how to specify Gjat the bus corr. to the candidate DG locationk.
Assume that Gj= 0 at all load buses before placing the DGs.
Vj : complex voltage obtained by power flow at grid bus-j
I
j : Complex injected current obtained by power flow at grid bus-j [[Ij]=[Y][Vj] ([Y]: Admittance matrix)Plossi : Active power loss obtained by power flow at grid branch-i
(3-3) is the objective function which represents overall active power loss minimization in the targeted system. (3-4)-(3-8) are constraints. (3-4) is the power flow constraint. (3-5) is 0-1 variable which represents status whether DG is installed or not at
the targeted bus. (3-6) is the integer constraint for discrete variable x which decides discrete capacity in DG in each location. As input data for DG location candidates, bus number, group number, DG type and the number of considered capacity and its discrete values are defined. (3-7) is the capacity constraint for total DGs so as not to exceed the value ofSmax. (3-8) is the constraints for total number of DGs.
The DG type considered at each candidate location bus is PQ type injecting both active and reactive power, and capacity of each installed DG is defined. Table 3-1 shows DG type and defined discrete capacity values for the DG.
Table 3-1 Types and Capacity of DGs
DG type Bus Specification Input Values Used in Power flow PQType:
Both active and reactive power injecting DG
PQBus P1~Pnk,P/Qratio or
Power Factor Re( ), Im( )
Solution Procedure Optimal DG Installation P (4)
In this research, an optimal DG installation is defined as the DG installation which minimizes the power loss of a targeted distribution system and its decision procedure is provided. In the procedure, one DG installation into any one bus from Bus2 to Bus5 in the simple distribution system model is considered (Bus1 is the slack bus). In order to find the DG location and its size to minimize power loss in the simple system model, the amount of power loss in case of DG installation into every one bus is calculated by a power flow calculation method. As the method of power flow calculation, the backward and forward (B/F) method is used in the research. The reason is that the B/F method is considered as a suitable power calculation method for a radial distribution system which is the Japanese typical distribution system style and its computational speed is fast compared with the Newton-Raphson method which is used for power flow calculation commonly [3-13].
The procedure of the B/F method is provided as follows. A dot on the top of the character denotes complex numbers and an asterisk on the right hand side denotes complex conjugate.
1) Predefine the voltage Vi(i 1,2,...,n( 6in thesystemmodel)) for each bus.
2) Calculate the injecting current Ii(i 1,2,...,n) for each bus using the following formula.
i S
V S
Ii i/ i * i:Loadof bus (3-9)
G G
3) Sum up injecting currents for each bus and set the each branch current by the following manner. (Backward sweep)
4) Calculate the sum of currents for each bus
(KCL) starting from every end bus (Bus6 in the pilot system) to the slack bus (Bus1 in the simple distribution system) sequentially (Bus6 Bus5 Bus1 in the pilot system).
r
sr I
I (s: sending end bus, r: receiving end bus) (3-10)
r s
s I I
I (3-11)
5) Calculate the voltage drop for each bus by the following manner. (Forward sweep) 6)
using the calculated current Iiby the Backward Sweep, starting from the Slack bus (Bus1) to the every end bus (Bus6) sequentially (Bus1 Bus6).
sr sr s
r V Z I
V (Z:Impedance,s:sending end bus,r: receiving end bus) (3-12) 7) Calculate the voltage difference between the calculated voltage Vi and the
previous voltage Violdfor each bus and execute the following convergence test.
criterion e
Convergenc :
voltage Previous
iold :
iold
i V V
V (3-13)
8) Repeat 2) to 5) using the voltage Vi(i 1,2,...,n)obtained by the Forward Sweep until meeting the condition (6).
9) Calculate the power loss of the targeted system in comparison of the total injected power with the total load. (The difference is the power loss.)
Preparation Works (5)
Here, preparation works for optimal allocation of DG are described such as definition of assumption, initial settings and base calculation.
a. Assumptions and Initial Settings
Assumptions in the calculation are as follows.
Only one DG is installed into any one bus of the simple distribution system model except for the slack bus.
Three types of DGs capable of injecting active power (P) only, reactive power (Q) only and both active and reactive power (P & Q
power is set from 0 to 1 with 0.05 increments. In case of injecting both active and
reactive power, the size of reactive power is set at 1/2 of active power considering a general power factor (0.89).
Load of each bus is uniform (except for the slack bus) No susceptance is considered.
Initial values of voltage V0, active power P0, reactive power Q0 for each bus are showed in Table 3-2, and resistance R and reactance x values of each branch are showed in Table 3-3. The data of amplitude are all per unit (p.u.)
Table 3-2 Bus Data
Bus P0 Q0 V0 Remarks
1 - - 1.0 Slack bus
2 -0.1 -0.05 1.0
3 -0.1 -0.05 1.0
4 -0.1 -0.05 1.0
5 -0.1 -0.05 1.0
6 -0.1 -0.05 1.0
The data of amplitude are all p.u.
Table 3-3 Branch Data
Branch R x Remarks
Bus 1-2 0.02 0.01 r1, x1
Bus 2-3 0.02 0.01 r2, x2
Bus 3-4 0.02 0.01 r3, x3
Bus 4-5 0.02 0.01 r4, x4
Bus 5-6 0.02 0.01 r5, x5
The data of amplitude are all p.u.
b. Base Calculation
Table 3-4 shows the power flow calculation results for the simple distribution system model without DG to obtain the power loss without DG placement.
Table 3-4 Power Flow Calculation Result without DG
Bus P(Load) Q(Load) ReV ImV ReI ImI P Q
1 1.0000 +0.0000 0.5147 -0.2573 0.5147 0.2573
2 -0.1000 -0.0500 0.9871 +0.0000 -0.1013 0.0507 -0.1000 -0.0500 3 -0.1000 -0.0500 0.9768 +0.0000 -0.1024 0.0512 -0.1000 -0.0500 4 -0.1000 -0.0500 0.9690 +0.0000 -0.1032 0.0516 -0.1000 -0.0500 5 -0.1000 -0.0500 0.9638 +0.0000 -0.1038 0.0519 -0.1000 -0.0500 6 -0.1000 -0.0500 0.9612 +0.0000 -0.1040 0.0520 -0.1000 -0.0500
Power Loss 0.0147 0.0073
The data of amplitude are all p.u.
The result shows the active power loss of the system model without DG is 0.0147. The accuracy of this calculation result is confirmed by the power mismatch between the
defined and the calculated active and reactive power of each bus is sufficiently small. (The max mismatch is 4.637e-009 in this calculation).