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Secured Economic Dispatch Algorithm using GSDF Matrix

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 Leonardo Journal of Sciences Issue 24, January-June 2014
ISSN 1583-0233 p. 1-14

Secured Economic Dispatch Algorithm using GSDF Matrix

Slimane SOUAG, Farid BENHAMIDA*

IRECOM Laboratory, Department of Electrical Engineering University of Djilali Liabes

22000, Sidi Bel Abbes, Algeria
E-mails: slimane.souag@gmail.com, farid.benhamida@yahoo.fr
*
Corresponding author: Phone: +213666598556

Abstract
In this paper we present a new method for solving the secured power flow
problem by the economic dispatch using DC power flow method and
Generation Shift Distribution Factor (GSDF). A graphical interface in
LabVIEW has been created as a virtual instrument. Hence the DC power flow
reduces the power flow problem to a set of linear equations, which make the
iterative calculation very fast and the GSFD matrix present the effects of
single and multiple generator MW change on the transmission line. The
effectiveness of the method developed is identified through its application to
an IEEE-14 bus test system. The calculation results show excellent
performance of the proposed method, in regard to computation time and
quality of results.
Keywords
Economic Dispatch; Sensitivity Matrix; Power System; Virtual Instrument;
Network Security; LabVIEW.

Introduction

The most accurate approach for modeling the steady state behaviour of balanced, three
phase, electric power transmission networks is through the solution of the power flow [1].
With modern computers the power flow for even a fairly large system, such as the NERC

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 Secured Economic Dispatch Algorithm using GSDF Matrix
Slimane SOUAG, Farid BENHAMIDA

43,000 bus model of the North American Eastern Interconnect, can often be solved in
seconds.
Solving this problem has led many researchers to find ways easier and faster to
improve their convergence, reducing the execution time and save a lot of computer memory
by using usually digital processes that can be classified into two groups:
(1) Iterative process: Gauss, Gauss Seidel, etc.
(2) Variationel process: Newton-Raphson method, or Jacobian.
However, the power flow solution can often be maddeningly difficult to obtain
particularly when a good initial guess of the solution is not available. The “flat start” starting
point taught to undergraduates for small systems not often works when solving realistic
(large) systems. These convergence problems are especially troublesome when one tries to
significantly change the operating point for a previously solved case, such as by scaling the
load/generation levels.
The calculation of the power flow [2] is used to determine: (1) the complex tensions at
different buses, (2) the transmitted power from one bus to another, (3) the powers injected in a
bus and (4) real and reactive losses in the power system.
In this work, we were interested in monitoring the transmission line while working in
economic dispatch mode; the power flow in the network can be estimated just using the DC
(linearized) power flow method. But it is just a result; it is important to know what the value
of the generator MW output is, and if a secured power flow in all the transmission line of the
network can be make. It is also important to meet load demand at minimum operating total
fuel cost, subject to equality constraints on power balance and inequality constraints on power
outputs. This makes the ED problem a large-scale highly nonlinear constrained optimization
problem. Improvements in scheduling of the generator power outputs can lead to very
important fuel cost savings.

Material and Method

Economic dispatch
The basic economic dispatch can described mathematically as a minimization of
problem [3].

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 Leonardo Journal of Sciences Issue 24, January-June 2014
ISSN 1583-0233 p. 1-14

N
min ∑ Fi ( Pi ) (1)
i =1

where Fi (Pi) is the fuel cost equation of the ith plant. It is the variation of fuel cost ($) with
generated power (MW).

F ( Pi ) = ai Pi 2 + bi Pi + ci (2)
If ai > 0 then the quadratic fuel cost function is monotonic. The total fuel cost is to be
minimized subject to the following constraints.
N

∑P = D+ P
i =1
i L (3)

N N N
PL = ∑∑ PB
i ij Pj + ∑ B0i Pi + B00 (4)
i =1 j =1 i =1

Pi min ≤ Pi ≤ Pi max (5)

where, D is The real power load, Pi is the real power output at generator bus i, Bij, B0j, B00 are
the B-coefficients of the transmission loss formula [4], Pimin is the minimal real power output
at generator i, Pimax is the maximal real power output at generator i, PL is the transmission line
losses, Fi is the fuel cost function of the generator i and N is the number of generators.

DC Power Flow Formulation

The simplification on fast decoupled Newton Raphson power flow algorithm [5] can
be performed by neglecting simply any QV equation. This gives as result a linear and non-
iterative power flow algorithm. To achieve these simplifications, we simply assume that |Vi| =
1 pu for every bus i.
And we have:

⎡ ΔP1 ⎤ ⎡ Δδ1 ⎤
⎢ΔP ⎥ = B ' ⎢Δδ ⎥
⎢ 2 ⎥ [ ]⎢ 2⎥ (6)
⎣⎢ ... ⎦⎥ ⎣⎢ ... ⎦⎥
The elements of matrices B’ are:

Bik' = −1 xik (i connected to k ) (7)

1 (8)
Bii' = ∑ k =1
n

xik

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 Secured Economic Dispatch Algorithm using GSDF Matrix
Slimane SOUAG, Farid BENHAMIDA

The terms of the matrix B’ are described above by Eq. (7) and Eq. (8). The dc power
flow [6] is used only to calculate the real power flow (MW) of transmission lines and
transformers. It gives no indication of the voltages or on the reactive power flow (Mvar) and
apparent power (MVA).
The power flow on each line using the dc power flow can be described by the
following equation:

1
Pik = (δ i − δ k ) (9)
xik
and
n
Pi = ∑
k =nodes
Pki (10)
conected i

Before moving on it is important to point out that one of the most obvious differences
between the two – the lack of losses in the DC solution – can be reasonably compensated for
by increasing the total DC load by the amount of the AC losses. Hence, in the DC approach
the estimated transmission system losses could be allocated to the bus loads. This requirement
to first estimate the losses is usually not burdensome since the specified total control area
“load” is actually the true load plus the losses.

Generation Shift Distribution Factor (GSDF)

The affects of single and multiple generator MW change can be linearly approximately
by calculating the state-independent GSDF [7]. Using the DC load flow model, the GSD
Factor is expressed as:

∂pm ∂ ⎛ δ j −δk ⎞ 1 ⎛ ∂δ j ∂δk ⎞

A(m, i) = = ⎜ ⎟= ⎜ − ⎟ (11)
∂pgi ∂pgi ⎝ xm ⎠ xm ⎜⎝ ∂pgi ∂pgi ⎟⎠
with m = 1, 2,…, NL and NL is the number of lines.
where Pm is the real power flow on line m from sending bus j to receiving bus k; xm is the
reactance of line m, is δj angle of bus j and Pgi is real power generated by the generator i.
From eq. (9) and (10), it is concluded that ∂δj/∂pgi = xji and ∂δk/∂pgi = xki thus,

x ji − xki
A(m, i) = (12)
xm
where xji and xki are the elements j-i and k-i of reactance matrix X of the lines, respectively

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 Leonardo Journal of Sciences Issue 24, January-June 2014
ISSN 1583-0233 p. 1-14

where X = [0 x12 x13 … x1n; x21 0 x23 … x2n; … ; xn1 xn2 … 0 ];

The GSDF matrix contains the GSDFs factor for all monitored lines [8], which
represent a good sensitivity factor to generator MW change [9].

Solution Algorithm
To apply the secured economic dispatch, the line flows should be recomputed in each
of the iteration due to the shifts in bus generation. In such kinds of applications, the total
system demands are assumed remain unchanged but the losses are variable during iterations.
If the loading levels change from the base point, a set of new line flow base should be
established. In this process the GSDF matrix will be used to penalize the generator cost, the
line that is overloaded they penalize the generators according to his sensitivity factor, this
process will be repeated until the line problem is resolved [10].

Start

Calculate power flow

Calculate B-coefficients

Economic dispatch

DC power flow

No
Line overloaded?

Yes

Calculate GSDF matrix Finish

Figure 1. Secured economic dispatch algorithm flow chart

Algorithm Steps
Step 1: To start the algorithm all data of bus and lines must be known. It is also important to
cheek if the total power demand is supported in the total addition of lines limits of the
network.
Step 2: Calculate the power flow in the power system to get the flow in each line of the

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 Secured Economic Dispatch Algorithm using GSDF Matrix
Slimane SOUAG, Farid BENHAMIDA

network and the output of the slack bus.

Step 3: Calculate the variable losses due to the change in generation output caused by the
economic dispatch affectation.
Step 4: Execute the economic dispatch iteration to have the power allocation in each
generation bus economically.
Step 5: Execution of the DC power flow must be performed to obtain the new power flow in
the line due to the change in power output of generators.
Step 6: Check the lines limits [9], if there are no violations the algorithm will stop and the
results are printed, if violations were observed the next step is executed.
Step 7: Calculate the generation shift distribution factor to have a sensitivity matrix of all
lines flow to the change in each generators output.
Step 8: The GSDF matrix is used to penalize the cost of the generators who make the
overload in the detected line and return to step 4 using the new generators cost, we repeat the
loop up to have no overloaded line .
Step 9: If there are no overloaded line detected after step 5, the loop is stopped and the final
results are printed.

Test System Data

The IEEE 14 bus system has different bus type coded as 1 for slack bus, 2 for PV bus
and 0 for PQ bus (Table 1 & 2). The lines are coded by 0 and transformers by theirs tap
changing. The cost data for this experiment are presented in Table 3.
Table 1. Bus data of IEEE 14 bus system
N° Load Load Gen Gen Gen Gen
Type Voltage Angle Cond. Suscep.
bus MW Mvar MW Mvar Qmin Qmax
1 1 1.06 0 0 0 0 0 0 0 0 0
2 2 1.045 0 21.7 12.7 40 0 -40 50 0 0
3 2 1.01 0 94.2 19 0 0 0 40 0 0
4 0 1 0 47.8 -3.9 0 0 0 0 0 0
5 0 1 0 7.6 1.6 0 0 0 0 0 0
6 2 1.07 0 11.2 7.5 0 0 -6 24 0 0
7 0 1 0 0 0 0 0 0 0 0 0
8 2 1.09 0 0 0 0 0 -6 24 0 0
9 0 1 0 29.5 16.6 0 0 0 0 0 0.19
10 0 1 0 9 5.8 0 0 0 0 0 0
11 0 1 0 3.5 1.8 0 0 0 0 0 0
12 0 1 0 6.1 1.6 0 0 0 0 0 0
13 0 1 0 13.5 5.8 0 0 0 0 0 0
14 0 1 0 14.9 5 0 0 0 0 0 0

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 Leonardo Journal of Sciences Issue 24, January-June 2014
ISSN 1583-0233 p. 1-14

Table 2. Line data of IEEE 14 bus system

N° From To R X 1/2 B line line
line bus bus (pu) (pu) (pu) code Limits
1 1 2 0.01938 0.05917 0.0264 0 200
2 1 5 0.05403 0.22304 0.0246 0 100
3 2 3 0.04699 0.19797 0.0219 0 100
4 2 4 0.05811 0.17632 0.017 0 100
5 2 5 0.05695 0.17388 0.0173 0 100
6 3 4 0.06701 0.17103 0.0064 0 50
7 4 5 0.01335 0.04211 0 0 100
8 4 7 0 0.20912 0 0.978 50
9 4 9 0 0.55618 0 0.969 50
10 5 6 0 0.25202 0 0.932 100
11 6 11 0.09498 0.1989 0 0 50
12 6 12 0.12291 0.25581 0 0 20
13 6 13 0.06615 0.13027 0 0 50
14 7 8 0 0.17615 0 1 50
15 7 9 0 0.11001 0 1 50
16 9 10 0.03181 0.0845 0 0 20
17 9 14 0.12711 0.27038 0 0 20
18 10 11 0.08205 0.19207 0 0 20
19 12 13 0.22092 0.19988 0 0 20
20 13 14 0.17093 0.34802 0 0 20

Table 3. Generator cost data of IEEE 14 bus system

Unit N° Pimin Pimax ai bi ci
1 50 500 0.007 7 240
2 20 200 0.0095 10 200
3 20 300 0.009 8.5 220
4 20 150 0.009 11 200
5 20 200 0.008 10.5 220

The IEEE 14 Bus Test Case represents a portion of the American Electric Power
System which is located in the Midwestern US as of February, 1962 [17]. Basically this 14
bus system has 14 buses, 5 generators and 11 loads presented in table 1 and 20 transmission
lines and transformers presented in Table 2.
A better version is provided by Rich Christie at the University of Washington in
August 1993. The 14 bus test case does NOT have line limits. And to achieve this work we
propose the lines and transformers limits presented in Table 2.
This section shows the most important case of the program, and the line power flow
after the economic dispatch show the good state of all the lines (Figure 2-6), in this state the
secured process is finished with zero security iteration it execute only the economic dispatch
with total cost equal to 3415.99$/h shown in Figure 7.

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 Secured Economic Dispatch Algorithm using GSDF Matrix
Slimane SOUAG, Farid BENHAMIDA

Figure 2. Bus and line data of IEEE 14 bus system in LabVIEW program

Figure 3. Power flow of IEEE 14 bus system in LabVIEW program

Figure 4. Cost data of IEEE 14 bus system in LabVIEW program

Figure 5. Economic dispatch of IEEE 14 bus system in LabVIEW

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 Leonardo Journal of Sciences Issue 24, January-June 2014
ISSN 1583-0233 p. 1-14

Figure 6. Line power flow of IEEE 14 bus system in LabVIEW program

To show the efficiency of the secured economic dispatch to resolve the problem of line
overloaded, we increase the total MW load in the system to cause en overload in one or
multiple lines, by make some adjustments in bus 4 and 5, and add to each bus 100 MW of
load and see what it gives (Figure 7).

Figure 7. Economic dispatch results of IEEE 14 generation system in LabVIEW

Results and Discussion

In this section the same IEEE 14 bus system is texted by adjusting the load of bus 4
and 5, the total MW load will be increase to 459 MW to cause an overload in lines and
resolve it by the new algorithm.
We have implemented our algorithms in the LABVIEW power system simulation
package LABPOWER and verified them on the IEEE power system test cases [17] with the
modified load demands. All experiments are conducted on a Windows 7 with 2.66 GHz DUO

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 Secured Economic Dispatch Algorithm using GSDF Matrix
Slimane SOUAG, Farid BENHAMIDA

CPU and 4G RAM.

The new adjustment of the load caused a multiple overload in the network shown in
figure 8; before test the new algorithm we will make a simple economic dispatch in the
network to save some money and look what will happen in the line flow state [11], certainly
there will be a new power distribution.
The new values of powers generated make in figure 9 the new state of the lines power
flow after the execution of the economic dispatch, look that the problem is slightly improved
but we have always an overload in the line number 6; some problem is solved, but it is by
chance.
To solve the problem we must face set and include a verification process and
measured the sensitivity with respect to different party involved, and this is what our new
algorithm is doing.
There are many theories for finding the best compromise solutions between the
economic dispatch and secured power flow, and most of this solution includes the security as
constraint in the process of economic dispatch which will transform to a strongly constrained
nonlinear optimization, particularly when there are a large number of transmission lines.
Hence the solution is not always guaranteed, or it will be found after many iterations.
Finally we execute the new algorithm to resolve problem in the network economically
using the GSDF matrix which will be calculated just after the economic dispatch [12].
Innovation in this algorithm is to preserve the economic dispatch as it is (Figure 8),
and added a process that will economically penalize generator according to their impact on
safety on the grid. ie the generator causing more overload on the line reported it will be
considered more expensive and automatically the process of the economic dispatch will
reduce the production of the latter.
The GSDF value correspond to each line power flow sensitivity to each generator
power output [13], will penalize the cost of the latter will he have forced to produce less and
therefore decrease the power in overloaded line [14].
Figure 9 showed that after the execution of the new algorithm in the overloaded
system [15], the total cost of production is slightly increased and this is normal because it is a
system constraints but the problem is directly resolved after 2 security iteration which proves
the efficiency and speed of this algorithm (Figure 10 and 11). This algorithm was tested on
more complex and large system and it showed a very large facility resolution [16].

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 Leonardo Journal of Sciences Issue 24, January-June 2014
ISSN 1583-0233 p. 1-14

Figure 8. Line power flow of the adjusted IEEE 14 bus system before (left) and after (right)
economic dispatch

Figure 9. Line power flow of the adjusted IEEE 14 bus system after the secured economic
dispatch

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 Secured Economic Dispatch Algorithm using GSDF Matrix
Slimane SOUAG, Farid BENHAMIDA

Figure 10. Economic dispatch results of adjusted IEEE 14 bus system with just economic
dispatch in LabVIEW

Figure 11. Economic dispatch results of adjusted IEEE 14 bus system with secured economic
dispatch in LabVIEW

In this paper, we have proposed a novel algorithm to solve economic and security
problem in the network. We have also integrated the sensitivity matrix with the contingency
constrained economic dispatch problem. With such extension, the original economic dispatch
problem becomes a difficult optimization problem. We then propose an elegant way to
transform the problem as a more tractable problem by adopting the GSDF matrix element as
penalty factor of the generators who make overload in reported lines. They preserve the
network in a good state of power flow and preserve a good price of generation cost.
The results showed using this algorithm are satisfactory, which checks the validity of
this study concerning the execution time. The performance of our method is much faster.
In the future, we will further study how to take the variation of load into consideration,
the value of which also varies with time and locations.

Conclusion

Our experiments based on IEEE power system test cases have shown that the
proposed algorithm can achieve speed-up at a similar generation cost when compared to the
conventional practice.

References

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ISSN 1583-0233 p. 1-14

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Slimane SOUAG, Farid BENHAMIDA

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