A Quadratic Robust Optimization Model for Automatic ... - IEEE Xplore

A Quadratic Robust Optimization Model for Automatic Voltage Control on Wind Farm Side Tao Ding, Qinglai Guo, Hongbin Sun*, Bin Wang and Fengda Xu Department of Electrical Engineering, State Key Laboratory of Power Systems Tsinghua University Beijing 100084, China [email protected], [email protected], [email protected] Abstract-Connecting

high

penetration

of

wind

power

into

power grid usually makes voltage fluctuate, due to the volatile nature of wind power injection. This paper therefore proposes a quadratic robust optimization model to guarantee the voltage of each wind unit within the security region, no matter how the wind power varies. Based on the wind power prediction, the prediction error is regarded as uncertainties, and the robust solution

can

be

found

by

regulating

the

reactive

power

equipment and each wind unit using the duality filter method. In the proposed model, linearized derivation instead of original non-linear expressions in the objective function and constraints has been utilized. Furthermore, inner loop iterative method is introduced to reduce the linearized error, which divides the optimal voltage control into multiple steps with piecewise values and updates the sensitivity at each step, according to different wind farm size and condition. A test system with 36 wind units has

been

simulated,

and

simulation. Comparison

the

with

result

using

traditional

Monte

method

Carlo

shows

the

effectiveness of proposed method.

Index Terms-- robust optimization, automatic voltage control, duality filter, inner loop iteration, wind power

I.

INTRODUCTION

Three-hierarchical structure of automatic voltage control (AVC) has been widely used in traditional power grid, which can be divided into tertiary voltage control (TVC) to obtain the global optimal power flow, secondary voltage control (SVC) to provide the area of network a coordinated control strategy for voltage support, and primary voltage control (PVC) to guarantee the local voltage in the assigned security region. As well, the operational time frame has been cooperated with each other [1-3]. With the increasing penetration of renewable resources into power grid, especially the wind power, the voltage security and stability of power system however has more and more issues, leading to great challenges for AVe [4-8]. Therefore, it is extremely urgent to develop AVe for wind farm, which can be similar to traditional Ave for thermal generators. However, due to the volatile wind speed, the voltage also fluctuates, which makes the Ave for wind farm difficult. Reference [9] utilized the short circuit ratio to allocate each wind unit's reactive power within a wind farm.

This work was supported in part by National Key Basic Research Program of China (973 Program) (2013CB22820 I), National Science Foundation of China (51277105) and National Science Fund for Distinguished Young Scholars of China (51025725).

978-1-4799-1303-9/13/$31.00 ©2013 IEEE

Sensitivity analysis in a wind farm was applied in [10-11] to coordinate each wind unit's reactive power. [12] proposed an optimal tracking secondary voltage control method by determining regulation margin and intelligent selection of voltage violation condition. Furthermore, several methods have been studied using forecasting technology [13]. For instance, [14-15] studied an area Ave divided into three hierarchies using the ultra-short-term wind power forecasting. Hierarchical model predictive control (MPC) has been widely used recently to provide an advanced control strategy combining the predictive data. However, the predictive error is inevitable, which might cause the optimal voltage value calculated by traditional quadratic optimization method to violate the security constraint in some conditions. There are few studies dealing with predictive error of wind power output. This paper therefore proposes a quadratic robust optimization method, which takes prediction error as uncertainties to search a robust solution that is always guaranteed within the feasible region no matter how the uncertainties vary in the uncertain region. The rest of the paper is organized as follows: Section II sets up the quadratic robust optimization model (QROM) and proposes the multiple-step method and duality filter to solve this model. In Section III, some simulations and comparisons are tested with 36 wind units to illustrate the effectiveness of the proposed method. Finally, conclusions are drawn in Section V. II.

ROBUST OPTIM IZATION MODEL AND SOLVER

Optimal voltage control for wind farm is similar to traditional secondary voltage control (SVC) in a quasi-steady­ state (QSS) setting. Its objective is to tracing the optimal solution by tertiary voltage control (TVC). The optimal control for wind farm regulates the reactive power of each wind turbines and reactive compensator equipment such as sve/SVG. Different from thermal generator, the wind power is however stochastic and intermittent, which brings in greater challenges. Therefore, it is promising to find a robust and effective control strategy so as to eliminate the impact of these uncertainties.

As for its physical significance, the robust solution may not be the optimal solution, since the optimal solution is usually on the boundary of constraints just like the Fig.I, whereas the optimal robust solution is to guarantee the solution always in the feasible region no matter how the uncertainties vary in the uncertain region.

(QROM):

njr�

fO:' llu= t/J Upcc �+:tdU tJ.Q+UfO:'-U:0�+1JfO:' (4) k=1 ag '�I ag 2

l

l

Subject to

au aU au u; ::; I' � + I' !1G + I' !1Q +U/+lZ::;U; - k=1 a� .=1 ag 1=1 dQ n

m

n

(5) (6)

!1Qk ::; !1Qk ::; !1Qk !1Q, ::; !1Q, ::; !1Q, i=l,

Uncertian Variable XI

Due to enough reactive power reserve and fast response of reactive power equipment (e.g. wind turbines, SVC/SVG), the wind farm operational time frame for SVC and TVC is usually 1O�20 seconds and 5�10 mimutes respectively. However, wind power has lower accuracy of prediction and volatilely fluctuates, which needs a more robust control method to reduce these uncertainties as little as possible. Furthermore, this robust method must be fast computed to satisfy the requirement of real-time control. Therefore, a QROM has been proposed in his work, using the linearized derivation instead of primary non-linear functions in the objective and constraints where the objective uses the Q-U matrix in (3) and the constraints use the Jaccobi matrix in (2). 1/pcc and 1/; are the error caused by linearization, which can be dealt with inner loop iterative method, which has been widely used in the practical engineering nowadays. That is because the inner loop iterative method has to give the piecewise linear optimal value, and update the sensitivity at each iteration where the error of linearized objective can be reduced with small step. Certainly, this method would be more time consumed, although the QROM is similar to quadratic programming (QP) in the form, which can be fast calculated. Therefore, a few iterations will be needed to approach the referred voltage magnitude value of PCe. However, in order to satisfy the operational time requirement of wind farm for SVC, proper steps should be chosen according to different wind farm size and condition.

][ ]

JPO JpU !18 !1Q JQB JQU !1U !1Q = [JQU -JQBJ;�JpU J!1U =

k=l,

. . .

,n,

l=l,

. . .

,m

[JQU -JQBJ;�JpU JI

aupcc aupcc and belong to Q-U matrix S; aQk aQk aU au --' and ' belong to the inverse of power flow matrix; a� aQk U�cc is the voltage magnitude of PCC at time t; U;�� is the

respectively,

Fig. l The difference between optimal and robust solution

S

,n,

(8)

Where llI\ and llQk denote the active and reactive power deviation of wind unit k, llQI denotes the reactive power deviation of SVC/SVG device I, nand m is the total number of wind units and reactive power compensation devices

Optimal Solution

[M ] [

. . .

(7)

--

referred voltage of PCC at time t.

Receive the TVC referred voltage value

"rej Upcc

If

rej,I>UI Upcc pcc y

?

N

i=i+l Updata sensitivity coeffients and

u;��;)

=

U;��i-l) + /'"u;�

(1) (2) (3)

Fig.2 Flow chart of proposed method

The algorithm flow chart is presented in Fig.2, where multiple-step is adopted with each given iterative step Mr� and the initial value is assigned by TVC which is assumed as

Uref(O)

pee

=

�uef pee ' then � U >0, else

'

The general fonn of ROM with uncertain variables wand control variable x addressed in the work is as follows. Obviously, QROM model is the special robust optimization model, whose objective is quadratic and constraints are linear, we will mainly talk about the general form of ROM as follows. mmmax x

S.t.

w

J(x,w)

g(x,w)::sO,

(9)

VWE W

(10)

Where the operator"::s " includes inequality and element­ wise constraints according to QROM. To solve the robust optimization model, lots of methods have been proposed, such as duality, explicit maximization, relaxation and so on, to eliminate the uncertainties. However, different method will have different problem size, and how to choose the proper method is largely depended on the fonn of uncertainties, as well as the structure of optimization model. The reference [16] concluded that "The duality filter is applicable to element-wise constraints with coefficients linearly parameterized in the uncertain variable and the uncertainty constrained to an intersection of linear", so the duality properties are strongly recommended in this work. Such that mm x,z

S.t.

J(x,z)

(11)

g(x,z) ::;;0

(12)

Infinite Bus

power optimization for reducing the active power losses is accomplished and the optimal referred voltage magnitude for wind farm is assigned. The method proposed in the work is solved by MATPOWER [17] and YALMIP [18], where we choose the duality filter to solve the quadratic robust optimization model. Besides, considering the active power of each wind unit randomly varies between 0.9 to 1.3 MW and the wind power prediction is assumed as 80% to reflect the randomicity. From the perspective view of wind farm, the exterior system can be equivalent using Thevenin parameters. At the normal condition, let the power factor of each wind unit is 1.0 (i.e. the reactive power is 0), and the total active power generation of wind farm is 40MW.1t can be easily obtained by power flow that the voltage magnitude of pee is 1.044 p.u., and the Thevenin equivalent parameters are estimated using the high- and low-voltage solutions of power flow proposed in [19]. If let the angle of equivalent voltage angle is zero, the equivalent voltage is Eeq=1.02LO° and the equivalent impendence is 0.084 + jOA93 p.u ..

Simulation of Inner Loop Iterative Methodfor QROM

A.

At the normal condition, the optimal voltage magnitude of pee is 1.070 p.u. by the optimal reactive power flow, . . rer -1. 070 p.u. Accordmg ' ' 4, urer IS I.e. upee to the FIg. pee Iarger than Upcc, so �uef >0. Suppose each step is 0.005 p.u., then five steps are needed to approach the optimal value. The iterative process is shown in FigA, and for convenience, one wind unit from each feeder is studied in TABLE I and TABLE II, where the minus of reactive power denotes the reactive power is from wind units to grid, and the plus one is opposite. TABLE I shows all of wind units' reactive power is negative, definitely making the voltage of each wind unit and pee rising to the optimal value. 1.075 r-----,--r---,

1---------------- -------- 220kv--1

I I I Substation I I L ____������-

I I I I

-=T--....,..-...J,-� ] _

1.W ----------------------------------1.0643>1.06 ::;1.065 Q. 1.0597>1.06 03 '0 .� 1.06 c 1.0548->1.0597 0> to

� 1.055 0>

El

�

1.0497->1.0548

1.05 1.045 •

1.04

Multiple-step method

'--__--'___---'___---'___---L___--'

o

2

Steps

3

4

5

Fig.4 Comparison between directed and multiple-step method Fig.3

A tested wind farm with 36 wind units

III.

SIMULATION AND RESULTS

The detail modeling for wind farm is shown in Fig.3 with 36 wind units, and the capacity of each wind unit is 1.5MW. The 3375-bus test system in [17] is used to simulate the exterior power grid, and the bus"10325" is regard as the point common coupling (pee) of wind farm. Furthermore, reactive

If using inner loop iterative method, compared with direct method by sensitivity instead of iteration, the control results of which can be seen in FigA, illustrating that the latter result of voltage is 1.0630 p.u., whereas the former one is 1.0693 p.u. with five iterative steps. While the optimal value is 1.070 p.u., it manifested the superiority of inner loop iterative method. TABLE I also shows the time-consumption of proposed

objective value of QROM is zeros, it means the optimal value can be reached by controlling wind units and reactive power equipment and finding a robust reactive power solution; if not, any robust solution cannot be found to satisfy the optimal value, so the nearest value to optimal value would be used instead to search a robust solution.

method using five steps is feasible for the time operational framework. B.

Comparison Between Traditional Optimization Method andQROM

Voltage of each wind unit usually fluctuates with the volatile power output, and sometimes may exceed the security region. TABLE II illustrates that the robust solution of QROM is the same as Non-QROM solution when the optimal value is 1.075 p.u .. In this condition, the voltage never violates the security region (voltage should be in 0.9�1. 1 p.u.) no matter how the active power of each wind unit varies with 20% uncertainty using Monte Carlo simulation. Furthermore, if the TABLE I Initial State

When the optimal value is however 1.080 p.u., the Non­ QROM can approach this value, but the QROM cannot because the objective is not zero. That means robust solution is null and uncertainty is sure to make the voltage violated if using this optimal value (see Fig.5-(a)), so the nearest value 1.0767 p.u. is instead of optimal value to find a robust solution which guarantees the wind units' security (see Fig.5-(b)).

Result of proposed multiple-step method for QROM

First Step

Second Step

Third Step

Fourth Step

Fifth Step

Regulation

Regulation

Regulation

Regulation

Regulation

UlQ.u.

Q/Q.u.

U/Q.u.

Q/Q.u.

UlQ.u.

Q/Q.u.

UlQ.u.

Q/Q.u.

UlQ.u.

Q/Q.u.

U/Q.u.

PCC

1.0440

0

1.0497

0

1.0548

0

1.0597

0

1.0643

0

1.0963

Q/Q.u. 0

W-IO

1.0566

-0.1176

1.0633

-0.1204

1.0709

-0.1766

1.0764

-0.1997

1.0823

-0.2219

1.0878

-0.2463

W-16

1.0586

-0.1008

1.0656

-0.1481

1.0727

-0.1612

1.0785

-0.1828

1.0848

-0.2351

1.0900

-0.2412

W-20

1.0566

-0.0817

1.0637

-0.1168

1.0713

-0.1815

1.0769

-0.1925

1.0832

-0.2186

1.0885

-0.2346

W-28

1.0571

-0.0806

1.0661

-0.1532

1.0728

-0.1624

1.0782

-0.2075

1.0855

-0.2285

1.0909

-0.2532

W-32

1.0565

-0.1187

1.0637

-0.1245

1.0701

-0.1505

1.0766

-0.1730

1.0831

-0.2056

1.0883

-0.2363

W-40

1.0544

-0.0863

1.0617

-0.1351

1.0681

-0.1507

1.0748

-0.2045

1.0804

-0.2102

1.0863

-0.2476

Objective Value CPU Time

2.7918e-018

2.7647e-018

3.5237e-018

7.1557e-018

4.3368e-018

1.398047s

1.384008s

1.433626s

1.426676s

1.413471s

Note: w- denotes the wind unit, U denotes the voltage magnitude and Q denotes the reactive power.

TABLE II Comparison between QROM and Non-QROM Urer=1.075 (Both)

Urer=1.080(Non-QROM)

Q/p.u.

U/p.u.

Q/p.u.

U/p.u.

PCC

1.0742

0

1.0799

0

1.0767

0

W-IO

1.0945

-0.2835

1.0990

-0.2971

-0.2959

W-16

1.0979

-0.2617

1.1003

-0.1488

W-20

1.0958

-0.2643

1.0999

-0.2810

1.0975

-0.2526

W-28

1.0970

-0.3050

1.0995

-0.0938

1.0973

W-32

1.0960

-0.3092

1.0995

-0.2841

-0.0798 -0.2460

W-40

1.0928

-0.2713

1.0997

1.0966 1.0951

i

1.095

:g

1.093

F

1.089

---.a.-- Lower Bound of Voltage

•

�

.e-

]

II

.� i

l\ I

F

II

:g

II

1.091

---.a.-- Lower Bound of Voltage

13

19

25

31

1.098

0

1.096 1.094

36

wind unit number

(5-a) Monte Carlo for voltage using Non-QROM method

1.088

,I

�;

1.092

�

1.09

� Upper Bound of Voltage Monte Carlo Simulation

1

-0.2783

0.0032

1.102 ,---�--�---�---,

Security boundary

1.099 1.097

.�

-0.0870

Security Boundary � Upper Bound olVoltage 1 . 1 - --------------------------.--..,.-

.e-

]

-0.2916 6.347Ie-019

9.7578e-019

1.103 ,-----.---�--�---,--__,

�

Q/p.u.

1.0963 1.0977

Objective Value

1.101

Urer=1.080(QROM)

U/p.u.

1

Monte Carlo Simulation

�

[

II 13

19

25

31

wind unit number

(5-b) Monte Carlo for voltage using QROM method

Fig.5 Comparison between QROM and Non-QROM for voltage with random disturbance using Monte Carlo

36

IV.

CONCLUSIONS

A quadratic robust optimization model (QROM) has been set up in this work to solve the automatic voltage control problem for wind farm. Compared with traditional method, the QROM method could find a robust solution that guarantees each of wind unit in the security region and not violating the constraints no matter how the active power volatilely fluctuates. Then, inner loop iterative method is introduced to reduce the error of linearized model with piecewise linear optimal value and update the sensitivity at each iteration. Furthermore, the duality filter is chosen to solve this QROM model. The computation time can meet the engineering requirement. Simulation has been tested using 36-wind-unit wind farm. Through comparison, the result shows the effectiveness of QROM method. If any robust solution cannot be found to satisfy the optimal value, the nearest one to optimal value would be used instead to search a robust solution. REFERENCES [I]

[2]

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[5]

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[9]

[10]

[II]

[12]

[13]

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