OPTIMIZATION TOOLS ADRESSING FUZZY UNCERTAINTY

AT POWER FLOWS

Preparation of Camera-Ready Contributions to SciTePress Proceedings

Eduardo M. Gouveia

and Paulo Moisés Costa

1,2

School of Technology and Management, Polytechnic Institute of Viseu, Campus Poliécnico de Repeses, Viseu, Portugal

INESC Porto, Porto, Portugal

Keywords: Power Flow, Fuzzy, Symmetric, Optimization.

Abstract: Power flow studies use computational tools for the planning and operation of electrical power systems

purposes. The deterministic model is the most commonly used load flow approach. In this model, the input

data and the results are crisp values. Therefore, to account for uncertainties, the most common approach

used is the definition of scenarios, which are characterized by crisp values. This is an impractical way to

solve the problem of the uncertainty in the data. A more practical way to lead with the uncertainties is the

use of probabilistic power flows. On such approach, the uncertainties are modelled through the use of

probability density functions (pdf). However, that approach may be inappropriate, namely when there is no

available historical data in order to construct the pdf. On such cases, the fuzzy power flows (FPF) is an

interesting alternative. In this paper, a methodology named Symmetric Fuzzy Power Flow is used. That

methodology uses optimization models to solve power flow problems considering the uncertainty treated as

fuzzy numbers. A comparison between the proposed methodology and the classic ones is also provided.

1 INTRODUCTION

Power flows is one of the most used tools to support

the planning and operation activities of the

transmission network. There are two main classes of

power flows considering uncertainty, the

probabilistic power flow (PPF) (Borkowska, 1974)

and the fuzzy power flow (FPF) (Miranda and

Matos, 1989; Saraiva et al., 1991). In the PPF the

variables (generation and load) are considered as

random variables with probabilistic distributions

(pdf). The results of PPF are also in the form of pdf,

namely the voltages and power flows. This model

presupposes the existence of historical data for the

input variables that can guarantee the construction of

a statistical distribution. However, the historical data

may not be available, namely when we are leading

with emergent concepts as the generation at LV

networks and the electric mobility. Therefore, new

tools are needed in order to solve the problem

resulting from the inexistence of historical data. The

FPF is an interesting alternative on such context,

once this approach treats the uncertainty without

requiring the existence of statistical distributions for

the input data. The FPF, allows describe

mathematically qualitative statements or vague

information by using fuzzy models. The classic

formulations of FPF present some limitations that

may distort the results, namely: i) existence of a

slack bus (which aggregates all uncertainty that

comes from the all other buses of the transmission;

ii) linearization models used at the computation.

Saraiva el al, 2004 presented a model that includes

data correlation. An inclusion of a corrective

procedure for the slack bus is purposed by (Saraiva

et al, 1991), which consists in defining limits for

generation. In this situation, the slack bus still being

different from the others buses since continues to

receive the uncertainty from them.

In this paper a symmetrical model (SFPF)

purposed by Matos and Gouveia (2008) is used in

order to overcome some of the limitations of the

preceding models. In fact, the slack bus is treated as

all other buses and non linearization procedures are

adopted. The proposed model uses optimization

problems to obtain symmetrical solutions for the

power flow problem (regarding linearized and the

complete model). The main purpose of the paper is

to analyse the results of the SFPF and compare those

results with the ones of classic versions of the FPF.

219

M. Gouveia E. and Moisés Costa P..

OPTIMIZATION TOOLS ADRESSING FUZZY UNCERTAINTY AT POWER FLOWS.

DOI: 10.5220/0003760502190222

In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 219-222

ISBN: 978-989-8425-97-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

The paper is organized in the following way:

Section 2 - concepts of FPF and SFPF are reviewed;

section 3 - three case studies are presented (the IEEE

test grids with 14, 24 and 118 buses are used);

section 4 - some conclusions are extracted.

2 FUZZY POWER FLOW

There are three types of fuzzy numbers generally

used in FPF to describe input data at fuzzy power

flow models: rectangular, triangular and trapezoidal.

Triangular and rectangular fuzzy numbers are

particular cases of trapezoidal fuzzy numbers.

Triangular and rectangular fuzzy numbers are

particular cases of trapezoidal fuzzy numbers. For

instance, a trapezoidal fuzzy number (Figure 1) can

be described by the set of equations (1).

⎪

⎩

⎪

⎨

⎧

≥

≤≤

−

≤≤

−

≤

)(

axif

axaif

axif

(1)

Figure 1: Trapezoidal Fuzzy Number.

2.1 Classical Fuzzy Power Flow

The first formulation of Fuzzy Power Flow was

based on a DC model for the power system. In this

model if

is the fuzzy vector of injected power in

the nodes. an approximation of the arguments,

and

branches’ flows

, is obtained using:

PAP

(2)

1−

(3)

At these model, crisp matrices A and B corresponds

to the sensitivity and admittance matrices of the DC

power flow.

The AC model of FPF was proposed by Miranda

et al (1990). This model uses a first-order Taylor

series expansion of the power flow equations. To

obtain the fuzzy voltages and angles a deterministic

AC power flow for the central values of the fuzzy

data is solved, using the Newton-Raphson algorithm.

Then, the variations around this operating point are

calculated by using the Jacobian of the last iteration

and fuzzy arithmetic’s. For other variables like the

power flows, a similar strategy of linearization is

used. More details about this model can be found in

Miranda el al (1990).

2.2 Symmetric Fuzzy Power Flow

The SFPF consists of solve for each α level of the

possibility distribution (Figure 1), optimization

problems in order to obtain the maximum and

minimum value that fuzzy variable may take, for all

the possible values (with degree of membership

greater than or equal to α). Note that no slack bus is

defined since fuzzy injections for input data are

considered for all the buses of the transmission

network. For the DC model of SFPF (Matos and

Gouveia, 2008) we must solve linear programming

problems (for each α level) to obtain the maximum

value of the power flow in branch k. At this model

(4), “Ref” means the reference bus and

()

is the

α-level interval of the nodal active injected power. If

is desired the injected power at a specific bus, the

objective function will be

()

(

)

()

∑

∈

≠

Refi

ikik

ibusesallPPst

PAP

max

(4)

Gouveia and Matos (2008) also extend the SFPF to

the AC case. Now the standard equations of the AC

power flow problem are included as constraints,

along with the α-limits for each fuzzy variable. The

maximum α-level value for each fuzzy variable

will be the result of following optimization

problem (5). The analysis for the minimum is

analogous. Variable Z may be any of the voltages in

PQ buses, any of the voltage angles, any of the

power flows

, Q

or S

or the power losses in a

branch or in the entire system. In (5),

and B

are,

respectively, the real and imaginary components of

the admittance matrix elements,

()

is the α-level

interval of the active injected power

and

(

)

the α-level interval of the reactive injected power

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

220

()

)cossin(

)sincos(:

max

Ref

∈

−=

∑

θθ

RefandbusesPVVV

ibusesallQQ

ibusesallPP

ibusesallBGVVQ

ibusesallBGVVPst

ikikikik

kii

ikikikik

kii

(5)

3 CASE STUDY

In this section we will perform the exercise of

comparing the results obtained from SFPF with the

ones available at the literature obtained by the

classical FPF.

3.1 Linear Models of FPF

Three different models were used to solve the power

flow problem (DC model) for the 24 Bus, 38

branches IEEE test system: the first model of FPF

created by Miranda and Matos (1989); the model

with correction procedures which consider

generations limits to the slack bus (Saraiva et al,

1991); and the SFPF (Gouveia and Matos, 2008).

Those models are indentified in the following

paragraphs and figures as “First”, “Cons” and

“Sym”. We will consider the uncertainty modelled

as trapezoidal fuzzy numbers (triangular or

rectangular fuzzy numbers also could be used). The

values indicated in Table I will be considered the

central values of the trapezoidal fuzzy numbers. For

these data the characteristic points are assumed to be

as shown at Table 2. For instance, applying this

information to node 9, we’ll get a fuzzy load of

(Figure 2). A base power of 500 MW is used and

bus and branch data are available at (Saraiva et al,

1991); The bus 1 is the reference bus.

0.2

0.4

0.6

0.8

-340 -330 -320 -310 -300 -290

Possib.

Figure 2: Fuzzy load.

The voltages for PV buses 1, 2 and 7 will be 1.075

pu and for the others 1.025 pu. At the reference bus,

depending on the model we will have different

situations: “First”- no fuzzy description for

generation is considered; “Cons” – were considered

generation limits of [0-800 MW]; “Sym” is defined

a fuzzy generation. In this fuzzy trapezoidal number

the extreme points (a

, a

) will be (0, 500, 600,

800) MW. After solving the optimization problem

(4) twice (max and min) for α cuts between 0 and 1

for all branches with large branch limitations we

obtain the ranges for the P

i-k

power flows. For

instance for branch 1-2 (Table 2, Figure 3) the main

points of interest of the possibility distribution are

referred (α=0

, α=0.7

, α=1

, α=0

). Figure 4

show the same kind of results for branch 5-10. As

can be verified formulations “First” and “Const”

provides results with larger ranges of uncertainty for

power flows since slack bus concentrates all

uncertainty which comes from all other buses.

Table 1: Characteristic points of fuzzy data.

Bus characteristic points for generation

1-13, 14-24 0.95 0.98 1.02 1.05

13 1.00 1.00 1.00 1.00

characteristic points for loads

1-5, 7-19,

21-24

0.95 0.98 1.02 1.05

6, 20 1.00 1.00 1.00 1.00

Table 2: Power flow fuzzy distributions, branch 1-2

(MW).

Branch

0 0.7 1 1 0

1-2 (First)

-68.0 11.2 45.1 196.1 309.2

1-2

(Const)

-68.0 11.2 45.1 196.1 245.3

1-2 (Sym)

-68.0 18.9 85.2 147.2 245.3

0.2

0.4

0.6

0.8

-100 -20 60 140 220 300 380 460

Poss.

P1-2 Fir st

P1-2 Cons .

P1-2 Sy m.

Figure 3: Fuzzy load flow in branch 1-2.

0.2

0.4

0.6

0.8

-80 -30 20 70 120 170 220

Poss.

P5-10 First

P5-10 Cons.

P5-10 Sy m.

Figure 4: Fuzzy load flow in branch 5-10.

OPTIMIZATION TOOLS ADRESSING FUZZY UNCERTAINTY AT POWER FLOWS

221

3.2 Complete Models of FPF

Now we will perform the exercise of comparing AC

SFPF with classical AC FPF models. Considering

the model based on FPF, the boundary load flow

(BLF) (Dimitrovski and Tomsovic, 2004) we will

use the IEEE 14 and 118 test systems. Data of these

networks can be seen at (http://www.ee.washington

.edu/research/pstca/. The BLF uses an interactive

procedure in order to enhance the results accuracy in

cases of considerable non linearity due to large

uncertainty in input data. The characteristic points

for IEEE 14 bus test system at α=0 are assumed to

be 0.0 and 2.0 of the central values (rectangular

fuzzy numbers). Performing the comparison with the

exact values calculated with the SFPF shows that

some “artificially uncertainty” is still present in the

BLF results. Table 3 shows results for voltage

magnitudes. The same kinds of results (not shown)

were also obtained for active power flows. Using the

118 bus test system also Gouveia and Matos (2008)

found differences between the FPF, BLF (that falls

under same philosophy of FPF) and SFPF. Those

differences are exposed for some of the variables of

this network (Table 4). Now the characteristic points

for α=0 are assumed to be 0.6 and 1.4 of the central

values.

Table 3: Voltage values (pu) IEEE 14 bus test system.

α=0, [0.0-2.0]

Bus BLF SFPF AC

4 0.978 1.045 0.992 1.045

5 0.981 1.042 0.996 1.042

7 1.024 1.089 1.051 1.088

9 1.001 1.101 1.028 1.101

10 0.997 1.096 1.020 1.096

11 1.025 1.084 1.038 1.083

12 1.036 1.072 1.039 1.072

13 1.023 1.074 1.028 1.074

14 0.972 1.089 0.992 1.088

Table 4: Voltage and Power Flow Values (p.u.) for some

variables of IEEE 118 test system.

Var. FPF BLF Sym. FPF

V44 0.94 1.03 0.87 1.01 0.96 1.01

P68-69 -17.44 14.92 -21.10 13.27 -3.74 1.39

Q68-69 0.39 1.86 0.84 9.25 1.07 1.33

4 CONCLUSIONS

The use of SFPF doesn’t consider a slack bus or

linearization procedures as happens at the traditional

formulation of FPF. At FPF the use of these

simplifications have as a consequence a distortion of

the results mainly due the influence of the slack bus.

While in the deterministic power flow formulations

the slack bus only compensate the uncertainty in loss

estimation, at classical FPF compensates the

uncertainty that comes from all other buses. Adding

to this the linearization procedures used, results

shown “excessive” uncertainty as shown in this

paper at the several IEEE networks tested. Finally,

the SFPF models have the drawback of requiring

solving a great number of optimization problems but

are completely symmetric regarding the buses. This

is not a real problem since SFPF are addicted to

long-term planning studies. However justify future

work in order to improve simulation times.

REFERENCES

Borkowska, B., 1974. Probabilistic load flow. IEEE

Transactions on PAS 23 (3), 752– 759.

Dimitrovski A, Tomsovic K, 2004. Boundary Load Flow

Solutions, IEEE Trans. Power Systems, vol. 19, no.1,

February 2004, pp 348-355.

Gouveia E M, Matos M A, 2008 ”Symmetric AC Fuzzy

Power Flow Model”, European Journal of Operational

Research, DOI: 10.1016/j.ejor.2007.12.044”

Matos M A, Gouveia E M, 2008. Fuzzy Power Flow

Revisited, Power Systems, IEEE Transactions on,

Volume 23, Issue 1, Page(s):213 – 218, February 2008

Miranda, V., Matos, M. A., 1989. Distribution system

planning with fuzzy models and techniques. In:

Proceedings of the CIRED 89, Brighton, pp. 472–476

Miranda, V., Matos, M. A., Saraiva, J. T., 1990. Fuzzy

load flow - new algorithms incorporating uncertain

generation and load representation. In: Proceedings of

the 10th PSCC, London, pp. 621–627

Saraiva, J T, Fonseca N, Matos M A, 2004. Fuzzy Power

Flow – An AC Model Addressing Correlated Data, in:

Proc PMAPS’04, Iowa, September 2004, pp 519-524.

Saraiva, J. T., Miranda, V., Matos, M. A., 1991.

Generation and load uncertainties incorporated in load

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424 Ljubljana, pp. 1339-1342

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