A model for the Optimal Allocation of Voltage Regulators ...Introduction 10/11/2017 3 •Utilities are constantly working to provide better voltage and reactive power levels •Voltage

A model for the Optimal Allocation of

Voltage Regulators and Capacitors

based on MILP Applied to Distribution

Systems in Steady State

Code: 19.021

M. S. Medrano, S. M. C. Tome, L. L. Martins, P. L. Cavalcante,

H. Iwamoto, M. R. R. Malveira and T. M de Moraes

CPqD and Energisa

10/11/2017 1

Objectives

10/11/2017 2

Development of a model for the simultaneous allocation of Cs and VRs in the electrical network

Minimization of power losses, equipmentinvestments and O&M costs

Tests with different linearizations of model’snonlinear constraints

Introduction

10/11/2017 3

• Utilities are constantly working to provide better voltageand reactive power levels

• Voltage levels – Voltage Regulator

• Reactive power – Capacitor

• Equipment allocation is very important and it is not a trivialtask

• Optimization models can help to improve the power qualitytogether with the return on investment

Introduction

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Methodology

10/11/2017 5

As

su

mp

tio

ns

Steady state distribution system – radial operation

Balanced distribution system load-flowbalanced monophasic equivalent

Different load level - Constant active (P) and reactive(Q) power

Fixed or switched capacitor, with different capacities (kVAr)

Voltage regulators with regulator range (r%) and fixednumber of steps (𝒏𝒕)

Acquisition (investment) and O&M costs according to the equipment and its capacity

Objective function

Min { p1 + p2 + p3 }

Constraints

• Equations for the linearized power flow

• Related to the capacitor and voltage regulator allocation

• Voltage and current limit

𝑉𝑚𝑖𝑛2 ≤ 𝑉𝑖,𝑑

𝑞𝑑𝑟≤ 𝑉𝑚𝑎𝑥

2 and 𝐼𝑖𝑗,𝑑𝑞𝑑𝑟

≤ 𝐼𝑚𝑎𝑥2

Methodology

10/11/2017 6

Power

Losses costs

Investments

(new equipment)

costs

O&M costs

• Optmized model solved using MILP

Methodology

10/11/2017 7

MIL

P

Objective function to minimize, composed by a linear equation

Feasible area defined by linear constraints

Integer and real variables

Some of the previous constraints are nonlinear

Power flow equations:

𝑗𝑖∈𝐴

𝑃𝑗𝑖,𝑑 −

𝑖𝑗∈𝐴

𝑃𝑖𝑗,𝑑 + 𝑅𝑖𝑗 + 𝑃𝑖,𝑑𝑆 = 𝑃𝑖,𝑑

𝐷 ∀𝑖 ∈ 𝐴, ∀𝑑 ∈ 𝐷

𝑗𝑖∈𝐴

𝑄𝑗𝑖,𝑑 −

𝑖𝑗∈𝐴

𝑄𝑖𝑗,𝑑 + 𝑋𝑖𝑗 + 𝑄𝑖,𝑑𝑆 + 𝑄𝑖,𝑑

𝐵𝐶 = 𝑄𝑖,𝑑𝐷 ∀𝑖 ∈ 𝐴, ∀𝑑 ∈ 𝐷

− 2 𝑅𝑖𝑗 . 𝑃𝑖𝑗,𝑑 + 𝑋𝑖𝑗 . 𝑄𝑖𝑗,𝑑 − 𝑍𝑖𝑗2 . − = 0 ∀𝑖𝑗 ∈ 𝑉, ∀𝑑 ∈ 𝐷

. = 𝑃𝑖𝑗,𝑑2 + 𝑄𝑖𝑗,𝑑

2 ∀𝑖𝑗 ∈ 𝑉, ∀𝑑 ∈ 𝐷

Methodology

8

𝐼𝑖𝑗,𝑑2

𝐼𝑖𝑗,𝑑2

𝑉𝑖,𝑑2 𝑉𝑖,𝑑

2𝐼𝑖𝑗,𝑑2

𝑉𝑖,𝑑2 𝐼𝑖𝑗,𝑑

2


𝑗𝑖∈𝐴


𝑖𝑗∈𝐴


𝐷 ∀𝑖 ∈ 𝐴, ∀𝑑 ∈ 𝐷

𝑗𝑖∈𝐴


𝑖𝑗∈𝐴





2 ∀𝑖𝑗 ∈ 𝑉, ∀𝑑 ∈ 𝐷

Methodology

9

𝐼𝑖𝑗,𝑑2

𝐼𝑖𝑗,𝑑2


2𝐼𝑖𝑗,𝑑2


2

Constant loads for

each load level d


𝑗𝑖∈𝐴


𝑖𝑗∈𝐴


𝐷 ∀𝑖 ∈ 𝐴, ∀𝑑 ∈ 𝐷

𝑗𝑖∈𝐴


𝑖𝑗∈𝐴





2 ∀𝑖𝑗 ∈ 𝑉, ∀𝑑 ∈ 𝐷

Methodology

10

𝐼𝑖𝑗,𝑑2

𝐼𝑖𝑗,𝑑2


2𝐼𝑖𝑗,𝑑2


2

Reactive power due to the installation of fixed and switched capacitors


𝑗𝑖∈𝐴


𝑖𝑗∈𝐴


𝐷 ∀𝑖 ∈ 𝐴, ∀𝑑 ∈ 𝐷

𝑗𝑖∈𝐴


𝑖𝑗∈𝐴





2 ∀𝑖𝑗 ∈ 𝑉, ∀𝑑 ∈ 𝐷

Methodology

11

𝐼𝑖𝑗,𝑑2

𝐼𝑖𝑗,𝑑2


2𝐼𝑖𝑗,𝑑2


2

Nonlinearities


𝑗𝑖∈𝐴


𝑖𝑗∈𝐴


𝐷 ∀𝑖 ∈ 𝐴, ∀𝑑 ∈ 𝐷

𝑗𝑖∈𝐴


𝑖𝑗∈𝐴





2 ∀𝑖𝑗 ∈ 𝑉, ∀𝑑 ∈ 𝐷

Methodology

12


2

𝑉𝑖,𝑑2

Nonlinearities

𝐼𝑖𝑗,𝑑𝑞𝑑𝑟





𝑗𝑖∈𝐴


𝑖𝑗∈𝐴


𝐷 ∀𝑖 ∈ 𝐴, ∀𝑑 ∈ 𝐷

𝑗𝑖∈𝐴


𝑖𝑗∈𝐴





2 ∀𝑖𝑗 ∈ 𝑉, ∀𝑑 ∈ 𝐷

Methodology

13

𝑉𝑖,𝑑2

Nonlinearities





𝑉𝑖,𝑑𝑞𝑑𝑟



𝑗𝑖∈𝐴


𝑖𝑗∈𝐴


𝐷 ∀𝑖 ∈ 𝐴, ∀𝑑 ∈ 𝐷

𝑗𝑖∈𝐴


𝑖𝑗∈𝐴





2 ∀𝑖𝑗 ∈ 𝑉, ∀𝑑 ∈ 𝐷

Methodology

14

Nonlinearities








Voltage regulator model:

• Voltage regulators are modeled using the expression:

𝑽𝒋,𝒅 = 𝟏 + 𝒓%𝒏𝒕𝒊𝒋,𝒅

𝒏𝒕. 𝑽𝒋,𝒅,

where 𝑛𝑡𝑖𝑗,𝑑 is an integer variable with −𝑛𝑡 ≤ 𝑛𝑡𝑖𝑗,𝑑≤ 𝑛𝑡

Methodology

10/11/2017 15

Re

gula

tor

ran

ge 𝑉𝑗,𝑑 𝑉𝑗,𝑑

1 + 𝑟% . 𝑉𝑗,𝑑

1 − 𝑟% . 𝑉𝑗,𝑑

𝑉𝑗,𝑑

𝑛𝑡 𝑡𝑎𝑝𝑠

−𝑛𝑡 𝑡𝑎𝑝𝑠

• In addition, for each theoretical system, the model was solved

considering the following approximations for the term 𝑉𝑖,𝑑𝑞𝑑𝑟

. 𝐼𝑖𝑗,𝑑𝑞𝑑𝑟

:

The results were compared with those obtained using a piecewise linear

function with 5 parts:

• Maximum relative error;

• Average relative error;

• Standard deviation of relative errors.

Experimental results

10/11/2017 16

𝑉𝑚𝑎𝑥𝑞𝑑𝑟

= 𝑉𝑛𝑜𝑚𝑞𝑑𝑟

𝑉𝑚𝑖𝑛𝑞𝑑𝑟

𝑉𝑚𝑒𝑑𝑞𝑑𝑟

= 𝑉𝑚𝑖𝑛𝑞𝑑𝑟

+ 𝑉𝑚𝑎𝑥𝑞𝑑𝑟


10/11/2017 17

Numerical results were obtained considering three theoreticaldistribution systems, composed by:

70, 202 and 400 nodes

Different capacities were considered:

Fixed capacitor – 300, 600 and 900 kVAr

Switched capacitor– 300, 600 and 900 kVAr

Voltage regulators that enables ±10% regulator range and 33 steps

Minimum distance between the nodes was set in 100m andbetween capacitors was set in 500m

• Distribution system with 70 nodes


10/11/2017 18

0,08

3,69

0,04

0,44

0,04

1,5

0,010,28

0,12

1,57

0,04

0,79

1 2 3 4

Average relative errors (%)



= 𝑉𝑚𝑖𝑛2 𝐼𝑖𝑗,𝑑

𝑞𝑑𝑟



= 𝑉𝑚𝑒𝑑2 𝐼𝑖𝑗,𝑑

𝑞𝑑𝑟



= 𝑉𝑚𝑎𝑥2 𝐼𝑖𝑗,𝑑

𝑞𝑑𝑟

𝑉𝑖,𝑑𝑞𝑑𝑟 𝐼𝑖𝑗,𝑑

𝑞𝑑𝑟𝑃𝑖𝑗,𝑑 𝑄𝑖𝑗,𝑑

0,31

1,41

0,06

1,28

0,16

0,77

0,02

0,95

0,34

1,32

0,07

2,84

1 2 3 4

Standard deviations – averageerrors

Série1 Série2 Série3𝑉𝑖,𝑑𝑞𝑑𝑟



𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟



10/11/2017 19

1,33

5,07

0,29

14,72

0,672,1

0,08

8,98

1,35

4,57

0,24

30,87

1 2 3 4

Maximum relative errors (%)




𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟



𝑃𝑖𝑗,𝑑 𝑄𝑖𝑗,𝑑

Only in onebranch



10/11/2017 20




𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟

0,01

1,66

0,05

0,3

0,01

0,85

0,01 0,060,01

1,5

0,04

0,23

1 2 3 4





𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟

0,01

3,97

0,03 0,150,01

1,33

0,01 0,040,01

3,41

0,03 0,13

1 2 3 4





𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟


0,02

7,41

0,3

3,43

0,01

3,05

0,070,39

0,02

6,35

0,21

2,96

1 2 3 4



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10/11/2017 22




𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟

0,27

3,65

0,05

0,770,51

1,47

0,010,180,25

3,71

0,05

0,57

1 2 3 4


0,33

1,88

0,1

2,43

0,64

1,02

0,02

0,560,31

1,69

0,07

1,67

1 2 3 4





𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟


0,71

7,45

0,61

19,27

1,343,09

0,16

3,91

0,66

6,32

0,24

14,33

1 2 3 4



10/11/2017 23




𝑞𝑑𝑟




𝑞𝑑𝑟




𝑞𝑑𝑟




Conclusions

10/11/2017 24

The model ensures that the set of equipment is the mostappropiate for the voltage and reactive control

The approximation for the product 𝑉𝑖,𝑑𝑞𝑑𝑟


was analyzed

using two different approaches and the relative errors(maximum and average) and the standard deviation werecompared.

The use of 𝑉𝑚𝑒𝑑𝑞𝑑𝑟

to approximate 𝑉𝑖,𝑑𝑞𝑑𝑟

in the product



presented lower errors, with indices under 10%

10/11/2017 25

Luana Laquanette [email protected]

+55 (19) 3705-4398

Thanks for your Attention

mailto:[email protected]

Documents

A model for the Optimal Allocation of Voltage Regulators ...Introduction 10/11/2017 3 •Utilities are constantly working to provide better voltage and reactive power levels •Voltage