Vectorial Dynamic Optimal Power Flow Calculation Including Wind Farms Based on Step-controlled

Primal-dual Interior Point Method Zhijun Qin, Yude Yang and Jiekang Wu

College of Electrical Engineering, Guangxi University, Nanning, 530004, China

Abstract- A vectorial implementation of dynamic optimal power flow (DOPF) including wind farms was presented. The vectorization of DOPF was established by arranging the control variables and state variables according to the variable types and time intervals. The asynchronous generators in wind farms were modeled in Q-V formulation. A step-controlled primal-dual interior point framework (SCIPM) with upper and lower inequality constrains was adopted to solve this DOPF model. The gradient and Hessian matrices of each time interval had relative non-zeros position with the admittance matrix, which was constant during iterations. Hence a sparse data structure and memory allocation strategy was utilized to accelerate the construction of KKT system. The effect of ramping rates and generation contract constrains on solving KKT system was analyzed in detail. Through computation statistics, it is confirmed that approximate minimum degree (AMD) reordering algorithm is most efficient with only ramping rate constrains, and column approximate minimum degree (COLAMD) reordering algorithm is most efficient with both ramping rate and generation contract constrains. Numerical simulations on test systems ranging in size from 14 to 1040 buses over 12~96 time intervals validate the correctness and efficiency of the proposed method. Vectorization technique with step-controlled primal-dual interior point method improves the calculation speed and convergence performance of DOPF.

Index Terms-- dynamic optimal power flow; step-controlled primal-dual interior point method; sparse technology; reordering algorithm; vectorization; wind power generation

I. NOMENCLATURE .×: element-wise multiplication between vectors or

between matrices. [v] : converts vector v into a diagonal matrix. v(i) : permutes vector v with integer vector i. diag(A) : retrieves the diagonals of A into a new vector. A(i,j) : permutes matrix A with integer vectors i and j.

),,,,sparse( nmsji : uses the rows of [i,j,s] to generate an m-by-n sparse matrix. Any elements of s which have duplicate values of i and j are added together.

II. INTRODUCTION

Dynamic optimal power flow (DOPF) is a typical dynamic optimization problem with initial parameters assigned and control variables constrained [1]. It takes the time-related and energy-related constrains into consideration to optimize the

objective function (such as fuel cost, etc) for the whole research horizon. DOPF gives the optimal control strategy for the power system over a long period, which is significant to such problems as economic dispatch, customer demand management [2] and so on.

In recent years, much research has been carrying on to improve both model precision and calculation speed of DOPF problems. The interior point method for hydrothermal DOPF with the time-related water system constrains is proposed in [3]. The DOPF problem of wind power integrated system and the reduced gradient method on it are described in [4]. Reference [5] accounts for the dynamic customer’s respond to the pricing in hydrothermal power system.

The calculation burden of DOPF increases sharply as the number of time intervals and buses increases. Further more, the time-related constrains will deteriorate the numerical property in the solution of KKT system. Hence it is a challenging problem to improve the calculation speed for large-scale DOPF to online application. Reference [6] converts DOPF into a dynamic optimal control problem, the variation model and optimality condition being also derived. An algorithm based on Radau collocation method for solving DOPF is proposed in [6], which has high efficiency and accuracy. But this method remains less efficiency if the research horizon is long with more discrete time intervals. Reference [7] solves DOPF with primal-dual interior point method (PDIPM) and rearranges the static and dynamic variables so that the KKT system can be decomposed into a series of sub matrices. Reference [8] takes similar strategy as in [7] and uses predictor-corrector interior point method on DOPF. However, the decomposition method in [7] and [8] has to inverse a series of sub matrices whose sizes are equivalent to the coefficient of KKT system of an independent OPF problem. Hence the calculation will not be applicable for online dispatch when the size of power system increases.

Vectorization is an important area of high performance scientific calculation, which plays a key role in improving the calculation speed and is also applied in power system simulation [9,10]. The direct solver and reordering algorithm for speeding up the solution of large scale KKT system are also investigated in [11]. Approximate minimum degree (AMD) algorithm is applied on the reactive optimization with interior point method in [12]. A step-controlled primal-dual interior point method (SCIPM) framework with upper inequality constrains is proposed in [13], which improves the convergence performance

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while keeping the high speed of PDIPM. A vectorial DOPF implementation including wind farms is

presented in this paper. The SCIPM framework with upper and lower inequality constrains is adopted to solve this model. The construction of KKT system is accelerated with vectorization technique. Though numerical simulation and comparison, AMD [14] and column approximate minimum degree (COLAMD) [15]

reordering algorithm with LDLT decomposition [16] are selected to solve the KKT system. Extensive simulations data on test systems demonstrate the correctness and efficiency of the proposed algorithm.

III. LIST OF PRINCIPAL SYMBOLS The following symbols definitions will be used to describe

the model and formulas of DOPF. F : objective function q : quadratic generation cost curve nt : number of time intervals for research horizon nw : number of wind farms xt : optimization variables at time interval t et , ft : real and imaginary part of voltages at time

interval t Pgt : active power output of controllable generators at

time interval t Qgt : reactive power output of controllable generators

or compensators at time interval t Pgs, Qgs : number of controllable real power generators and

reactive power generators or compensators Plt, Qlt, : active and reactive power load at time interval t

Qct : reactive power output of capacitor banks in wind farms at time interval t

Qwt : reactive power output of asynchronous generators in wind farms at time interval t

G, B : real and imaginary part of admittance matrix Pw.i.t, Qw.i.t : active and reactive power output of wind farm i

at time interval t ei.t, fi. t : real and imaginary part of bus voltage of wind

farm i at time interval t xm : excitation reactance of asynchronous wind

generator x : sum of stator and rotor reactance of asynchronous

wind generator i, j : start and end buses of tie lines. Pijt : active power transmission in tie lines at time

interval t Rup, Rdown : upper and lower limits of ramping rates Pc : active power generation contract δ : allowable contract error

IV. VECTORIAL MODEL OF DOPF The objective function to be minimized is the total fuel cost

of traditional units over the periods. The optimization variables of each time intervals are arranged in blocks as T T T T T T T

g g c w[ , , , , , ]t t t t t t t=x e f P Q Q Q ( 1,2,..., tt n= ). The overall

variables are arranged and denoted as T T T T1 2[ , ,... ]

tn=x x x x .

The DOPF model can be expressed as the following form:

g1

min ( ) ( )tn

tt

F q=

=∑x P (1)

subject to: 1) Equality constrains include the power flow equations and

Q-V formulation [4] of the output of wind farms. The load and the active power output of wind farms can be regarded as determinate quantities in each time intervals. The latter is calculated as the expectation of active power output by forecasting the wind velocity.

g w l

g c w l

. ( ) . ( ). ( ) . ( )

t t t t t t t t t

t t t t t t t t t t

+ − − × − − × +⎡ ⎤=⎢ ⎥+ + − − × − + × +⎣ ⎦

P P P e Ge Bf f Gf BeQ Q Q Q f Ge Bf e Gf Be

0

1,2,..., tt n= (2) 2 2 2 2 2 22 2. . . . w. .. .

w. .( ) ( ) 4

2i t i t i t i t i ti t i t

i tm

e f e f P xe fQ

x x− + + + −+

= − +

1,2,..., tt n= , w1,2,...,i n= (3) 2) Inequality constrains include (a) time-separated

constrains as upper and lower constrains of control variables and state variables; (b) time-related constrains, i.e. ramping rate constrains and generation contract constrains of all traditional generation units.

gg g

g gg

cc c( )

. .. . .

t

t

tt

t t t t

ijtij ij

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥≤ = ≤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥× + ×⎢ ⎥× ⎢ ⎥ ×⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦ ⎣ ⎦

P PPQ QQ

QQ g x Qe e f fV V V V

PP P

1,2,..., tt n= (4) ( ( ). ( ) ( ). ( )

( ). ( ) ( ). ( )). diag( ( , ))( ( ). ( ) ( ). ( )). diag( ( , ))

ijt t t t t

t t t t

t t t t

= × + ×

− × − × ×+ × − × ×

P e i e i f i f i

e i e j f i f j G i je i f j e j f i B i j (5)

down upg g( 1) , 1, 2,...,t t tt n−≤ − ≤ =R P P R (6)

c g c1

tn

tt =

− ≤ ≤ +∑P P Pδ δ (7)

V. FRAMEWORK OF SCIPM

A generic optimization problem with both upper and lower inequality constrains is as following:

min ( )subject to ( ) ( )

c=

≤ ≤

xH x

g G x g0 (8)

where c(x) is the objective function, H(x) is equality constrains , G(x) is inequality constrains, ,g g are lower and upper limits of G(x) , respectively. The primal-dual interior point method framework applied on model (8) can be described as :

(1) slack variable vectors ( , ) 0≥l u are introduced hence

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inequality constrains are converted into: ( )

( )

− − =⎧⎪⎨

− + =⎪⎩

G x g l

G x g u

0

0

(2) Lagrangian function is defined as: T T

T

( , , , , , ) ( ) ( ) [ ( )

] [ ( ) ] ( ln ln )i i

L c

u lμ≡ − − −

− − + − − +∑ ∑x l u y z w x y H x z G x

l g w G x u g , where (y,z,w) are Lagrangian multipliers. (3) KKT system with the Lagrangian function is:

( ) ( ) ( )( )( )

( ) ( )

x

y

z

w

c= ∇ − ∇ − ∇ + =⎧⎪ = =⎪⎨ = − − =⎪⎪ = + − =⎩

L x H x y G x z wL H xL G x l gL G x u g

00

00

[ ][ ]

[ ][ ] l

u

μ

μ

μμ

⎧ = − =⎪⎨

= + =⎪⎩

L l z e e

L u w e e

0

0,

where lμL , u

μL are perturbed complementary conditions, μ is perturbed factor, 0≤w ； 0≠y ； 0≥l ； 0≥u ； 0≥z . (4) By applying Newton’s method, the correction equations of the reduced system can be written as follows:

T

( )0⎡ ⎤ Δ⎡ ⎤ ⎡ ⎤

= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥Δ⎣ ⎦ ⎣ ⎦⎣ ⎦

xM Jy H xJ

ψ (9)

2 2 2

1 1 T

T

10

1 1 10

( ) ( )( ) ( )

( )([ ] [ ] [ ] [ ]) ( )

( )

( ) ( ) ( )([ ] [ ]

[ ] [ ] ([ ] [ ] ) )w

z

c

c

μ

− −

−

− − −

⎧ ≡ ∇ + ∇ + − ∇⎪

≡ + ∇ − ∇⎪⎪ ≡ ∇⎨⎪ ≡ ∇ −∇ + ∇⎪⎪ − − −⎩

F H x y G x z w x

M F G x u w l z G x

J H x

ψ H x y x G x u w L

l z L u l e

(10)

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−+Δ∇=Δ

+−Δ∇−=Δ

+Δ∇−=Δ

+Δ∇=Δ

−−

−−

)]([][)G(][][

)]([][)G(][][

))G((

)G(

0011

0011

0

0

μ

μ

uwT

lzT

wT

zT

LLwuxxwuu

LLzlxxzlz

Lxxu

Lxxl

(11)

where ),,,,,( 000000μμulwzyx LLLLLL denote the residuals of the

perturbed KKT systems. (5) The variables are updated according to:

⎭⎬⎫

⎩⎨⎧

<ΔΔ−<Δ

Δ−= 1 ;0: ;0:min9995.0 i

i

ii

i

iP u

uul

llα

⎭⎬⎫

⎩⎨⎧

>ΔΔ−<Δ

Δ−= 1 ;0: ;0:min9995.0 i

i

ii

i

iD w

wwz

zzα

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔuΔlΔx

ulx

ulx

Pα , ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

wD

ΔΔzΔy

wzy

wzy

α (12)

Four condition parameters are defined in SCIPM framework as:

0cond

max(max( ( ) ), max( ( )), )

1 max( , , )y

f ∞

∞ ∞ ∞

− −=

+

G x g g G x L

y z w,

0cond 1 max( , , )

xg ∞

∞ ∞ ∞

=+

Ly z w

,

Gapcond 1

Cc

∞

=+ x

,

1cond

( ) ( )1 ( )

k k

k

c co

c−−

=+

x xx

,

which denote the feasibility, optimality, complementary conditions and objective function value, respectively. To monitoring the conditions, sudden change of derivative will be detected and the Newton step will be shortened to the quadratic approximation of the Lagrangian function. Hence the convergence performance is improved.

By defining: T T

01( ) ( ) ( ) ( )2xθ Δ ≡ Δ − Δ × × Δx L x x F x

( , , , , , ) ( , , , , , )( )( )

L Lρθ

+ Δ −Δ ≡Δ

x x l u y z w x l u y z wxx

,

the pseudo code of SCIPM is showed as below. Step 0 0=k , 1.0=σ , 0,0 >> ul , 0,0,0 =<> ywz ,

510−=ε , 01.0,5.0 == ηκ , scipm=false.

Step 1 determines fcond, gcond, ccond, ocond. WHILE( 50<k AND ε≥∈∀ ),,,( condcondcondcond ocgf ）

Step 3 calculates zwulyx ΔΔΔΔΔΔ ,,,,, according to (9)~(11). Step 4 IF scipm=true THEN

4.1 calculates )( xΔρ . 4.2 WHILE( ηρ −<Δ 1)( x OR ηρ +>Δ 1)( x )

zzww

uullyyxxΔ=ΔΔ=Δ

Δ=ΔΔ=ΔΔ=ΔΔ=Δκκ

κκκκ,

,,,,

END DO Step 5 updates variables according to (12). Step 6 determines fcond, gcond, ccond, ocond. Step 7 IF

1−≥

kk condcond ff AND 1−

≥kk condcond gg THEN

scipm=true. Step 8 1+= kk .

END DO

VI. VECTORIAL IMPLEMENTATION OF DOPF

A. C++ Vectorization Calculation Library The principle of vectorization is to utilize the computer storage hierarchy and make computation on consecutive data in cache. Thus the data pipeline and instruction pipeline can be guaranteed and the overall efficiency is improved considerably.

In the high-performance scientific calculation field, the typical application of vectorization is the highly optimized Basic Linear Algebra Subprograms (BLAS) [17]. To develop a portable, high-performance, vectorization-based scientific

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calculation package for power system simulation, we develop a C++ vectorization calculation library, which implements the basic linear algebra in the equivalent performance of C-BLAS used by Matlab v7.0. It is integrated with AMD, COLAMD and LDLT. It optimizes the memory allocation strategy for huge matrices to avoid unconscious virtual memory page exchange by operating system, which keeps the instruction pipeline consecutive. DOPF code is implemented based upon this library.

B. Characteristics of Matrices in Vectorial Model Reference [9] and [18] present the vectorization formulas for gradient matrix and the Hessian matrices of power flow equations. The formulas reconfigure the execution sequence of the floating point operation instructions, the efficiency of the program being improved. Meanwhile, matrices are partitioned matrices, the sub matrices of which have the same sparse pattern with the admittance matrix. This noteworthy property can be used to pre-allocate memory for matrices in conventional compressed-column-storage (CCS) structure, which is efficient in both storage and non-zero access. The matrices relating to the tie-line active power transmission constrains also have relative sparse pattern with the admittance matrix. For instance, a power grid of N buses and L controlled tie lines, ∂P ijt/∂e can be formulated as:

1 sparse( , 1: , diag( ( , )), , ) [2 ( ) ( )] sparse( , 1: , diag( ( , )), , ) [ ( )]

t t

t

L N LL N L

= × − −×

T i G i j e i e jj G i j e i

2 sparse( , 1: , diag( ( , )), , ) [ ( )] sparse( , 1: , diag( ( , )), , ) [ ( )]

t

t

L N LL N L

= × −×

T i B i j f jj B i j f i

1 2ijt∂

= +∂P

T Te

Non-zero positions in all matrices can be determined after the admittance matrix is figured out. Before iterations, matrices are initiated and stored in CCS format. For an nt time intervals DOPF problem, the static memory demand to hold matrices is (nt +1) times as the OPF memory needed. However, the dynamic peak memory needed is related to the constrain property, reordering algorithm and symbolic factorization, which is discussed in section VII.

VII. TEST RESULTS AND DISCUSSION

A. Introduction to Test Systems The proposed method has been coded and compiled in

Microsoft Visual Studio 6.0 with compile option /O2 (maximize speed) on Dell personal computer (Intel-Pentinum4 2.8GHz, 512MB RAM). IEEE-14, IEEE-118, IEEE-300, P-700, P-1040 are used as test systems to evaluate the performance of the DOPF code. All test systems are modified to hold wind farms. For simplicity, just a small proportion of load buses of original test systems are set to be wind farm buses. The size of test systems are listed in Table I. The size of correction equation of OPF is in reduced form, which is also listed in Table I.

The size of correction equation of DOPF is in proportion to

the number of time intervals, which is easily figure out with parameters in Table I and nt. However, the time-related constrains will affect the non-zeros in the coefficient of correction equation. To improve efficiency of memory allocation and impose vectorization, the sparse pattern of matrices should be determined before iterations. Non-zeros in coefficient of KKT system over different time intervals in test systems are listed in Table II. The non-zeros’ positions are redundantly sufficient and fixed during iterations.

TABLE I INTRODUCTION TO TEST SYSTEMS

Name of systems

Number of buses/lines

Controllable source

(Pgs,Qgs,nw)

Inequality constrains

(Vi2,Pij)

Size of correction equation

IEEE-14 14/20 9(3,5,1) (14,4) 67 IEEE-118 118/179 73(16,54,3) (118,22) 551 IEEE-300 300/409 95(21,69,5) (300,35) 1305

P-700 700/800 164(54,100,10) (700,83) 3984 P-1040 1040/1183 235(70,150,15) (1040,112) 4425

TABLE II NON-ZEROS IN COEFFICIENT OF KKT SYSTEM

OVER DIFFERENT TIME INTERVALS Test system nt=12 nt=24 nt=48 nt=96

IEEE-14 8 514 17 034 34 074 68 154

IEEE-118 74 392 148 816 297 664 595 360 IEEE-300 172 320 344 830 689 620 1 379 286

P-700 356 218 712 546 1 425 202 2 850 514

P-1040 528 128 1 056 392 2 112 920 4 225 976

B. Calculation Time of DOPF Program First, we use AMD with LDLT decomposition to solve the

correction equation. The calculation time and iterations needed for model (1) ~ (6) of test systems are listed in Table III. Since the number of iterations is affected by load curve of the research horizon with the influence of ramping rate, the statistics in Table III represents typical cases. It is showed from Table III that the optimization for P-1040 system over 96 time intervals with ramping rate constrains is accomplished within 156 seconds, which is promising for online applications.

TABLE III CPU TIME AND ITERATIONS OF DOPF WITH

RAMPING RATE CONSTRAINTS Test system nt=12 nt=24 nt=36 nt=48 nt=96

IEEE-14 0.098 s/15 0.226 s/15 0.302 s/15 0.516 s/15 1.180 s/15

IEEE-118 1.297 s /16 2.516 s/17 3.804 s/18 5.820 s/18 13.88 s/19

IEEE-300 3.160 s/20 6.914 s/20 10.12 s/21 16.70 s/22 35.22 s/23

P-700 6.537 s/20 15.26 s/21 23.95 s/21 39.76 s/22 78.90 s/23

P-1040 11.04 s /23 26.26 s/23 45.68 s/23 77.28 s/24 155.2 s/25

C. CPU Time of Different Modules The memory allocation and symbolic factorization for

KKT system is time-consuming for manipulating large mount of memory while the floating point operation capability of CPU is not utilized sufficiently. Since the reordering and symbolic factorization for KKT system just use the sparse pattern of the coefficient of KKT system, these two tasks can be accomplished

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before iteration. Matrices are represented in CCS format during iterations, which keep the vectorization applicable for calculation. The time for constructing KKT system is also reduced considerably. Since no memory allocation is performed during iteration, the program makes extreme use of the floating point operation capability of CPU, the overall efficiency being improved sharply.

With the algorithm and memory allocation strategy proposed above, CPU time for iterations is almost constant, other than varying randomly due to virtual memory mechanism.

CPU time of calculation modules in test systems over 48 time intervals is listed in Table IV. It is showed that time for memory allocation and symbolic factorization increases rapidly as the sizes of test systems increase. If these two modules are performed in all iterations, the calculation efficiency of DOPF will drop as a result of reduplicated operation and failure of vectorization.

TABLE IV CPU TIME OF CALCULATION MODULES IN TEST SYSTEMS

OVER 48 TIME INTERVALS WITH RAMPING RATE CONSTRAINTS

Test system

Memory allocation

Reordering & symbolic

Each iteration

GradientKKT

residual Hessian

Correction (Solving KKT)

IEEE-118 0.31s 0.05s 0.026s 0.023s 0.017s 0.23s(0.052s)

IEEE-300 0.76s 0.17s 0.057s 0.055s 0.036s 0.51s(0.132s)

P-700 1.73s 0.36s 0.15s 0.14s 0.092s 1.24s(0.275s)

P-1040 2.68s 0.58s 0.23s 0.22s 0.170s 1.88s(0.420s)

D. Impact of Time-related Constrains on Solution of KKT The impact of time-related constrains on solution of KKT

is reviewed by comparison of calculation burden and peak virtual memory demand between including time-related constrains and without time-related constrains. The calculation statistics for solving KKT system of P-1040 system over 48 time intervals by AMD with LDLT is showed in Table V, from which we can see that the time-related constrains improve the fill-in elements and memory demand in the factorization and solution of KKT system. The impact of generation contract constrains is more tremendous than ramping rate constrains.

TABLE V CALCULATION STATISTICS FOR KKT SYSTEM OF P-1040

SYSTEM OVER 48 TIME INTERVALS WITH TIME-RELATED CONSTRAINTS

Inequality constrains

Time for solution

Floating point operation executed

Fill-in element in factorization

Peak virtual memory demand

No time-related constrains 0.856s 13 934 640 15 457 728 2 17.8MB

With only ramping rate

1.118s 44 048 318 131 291 797 1 278.1MB

With only generation contract

32.56s 5.1805×109 401 851 207 3 763.3MB

With ramping rate & generation contract

36.02s 5.4196×109 445 118 274 4 192.8MB

If the decoupling-inversion strategy proposed in [7] is applied, the time for inversion of the coefficient of KKT system for IEEE-300 OPF problem is 2.03s, which is much longer than time for solution of KKT system of P-1040 DOPF problem with

only ramping rate constrains. Thus the algorithm proposed in this paper preponderates in terms of calculation speed.

E. Influence of reordering algorithm on Solution of KKT The efficiency of symbolic and numerical factorization of KKT system is affected greatly by the reordering strategy, which determines the fill-in elements during decomposition. The rearranging strategies proposed in [3, 7] are essentially reordering algorithm aiming to improve speed. Various reordering algorithm are tested on the solution of KKT system of DOPF. The coefficient of KKT system of P-1040 system over 48 time interval is shown in Fig. 1 and Fig.2. As arranged in order mentioned in section IV, the optimization variables make the coefficient as banded block structure. After permuted by AMD, the coefficient is transformed as border-blocked, which is suitable for efficient sparse direct solvers.

0.0

0.4

0.8

1.2

1.6

2.0

0 1 2 Colum index(105)

(a) Non-zeros positions before permuted

Row

inde

x(10

5 )

0.0

0.4

0.8

1.2

1.6

2.0

0 1 2

Column index(105) (b) Non-zeros positions after permuted

Row

inde

x(10

5 )

Fig.1. Non-zeros of coefficient of KKT system with ramping rate constrains before

and after permuted by AMD TABLE VI

CPU TIME IN SOLVING KKT SYSTEM OF P-1040 OVER 48 TIME INTERVALS DOPF MODEL WITH DIFFERENT REORDERING ALGORITHM

Reordering algorithm Time for reordering Time for solution COLAMD 1.485s 15.84s

AMD 0.688s 31.78s SYMMD 12.56s 31.96s

COLMMD 22.67s 39.95s Other reordering algorithm as COLAMD, SYMMD [19]

(symmetric minimum degree), COLMMD [20] (column minimum degree) are adopted to solve model (1)~(7) of P-1040 system, with the statistics result in Table VI. Conclusion can be

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drawn that COLAMD is most efficient to solve DOPF with both ramping rate and generation contract constrains.

0.0

0.4

0.8

1.2

1.6

2.0

0 1 2 Column index (105)

(a) Non-zeros positions before permuted

Row

inde

x(10

5 )

0.0

0.4

0.8

1.2

1.6

2.0

0 1 2 Column index(105)

(b) Non-zeros positions after permuted

Row

inde

x(10

5 )

Fig.2. Non-zeros of coefficient of KKT system with ramping rate and generation

contract constrains before and after permuted by AMD

VIII. CONCLUSION The vectorial model and implementation of DOPF is

presented. Numerical studies showed that 1) Vectorization technique improves the efficiency of the construction of KKT system. It plays an important role in large-scale power system optimization such as DOPF. 2) AMD and COLAMD reordering algorithm are more efficient than other reordering algorithm such as SYMMD, COLMMD, 4ⅹ4 block structure. 3) An appropriate reordering algorithm with LDLT decomposition is more efficient on solution of KKT system than the strategy of decoupling-inversion.

REFERENCES

[1] Yuan Guichuan, Wang Jianquan, "Optimal power flow considering dynamic and stability constrains", Proceedings of the EPSA, vol.15, no. 3, pp. 1-9, 2003.

[2] A.S. Costa, W. Uturbey, S. Nakanishi, A. de Marco, S.S. Pacheco, "An OPF based tool for demand relief programs", in Proc. IEEE Transmission and Distribution Conference and Exposition, Latin America, 2004, pp.820-826.

[3] Hua Wei, H. Sasaki, J. Kubokawa, R. Yokoyama, "An implementation of interior point algorithm for large-scale hydro-thermal optimal power flow problems" IEEE Transactions on Power System, vol.15, no.1, pp. 396-403, 2000.

[4] Haiyan Chen, Jinfu Chen, Xianzhong Duan, "Multi-stage Dynamic Optimal Power Flow in Wind Power Integrated System, " in Proc.

Transmission and Distribution Conference and Exhibition : Asia and Pacific, Dalian, China, 2005, pp.1-5.

[5] W. Uturbey, A.S. Costa, "Dynamic optimal power flow approach to account for consumer response in short term hydrothermal coordination studies", IET Generation, Transmission &Distribution, vol.1, no.3, pp.414-421, 2007.

[6] Sun Yingyun, He Guangyu, Mei Shengwei, Wang Wei, Zhang Wangjunl, "A novel algorithm for dynamic optimal power flow problems based on variation model", Automation of Electric Power Systems, vol.31, no.17, pp.16-20, 2007.

[7] K. Xie, Y.H. Song, "Optimal power flow with time-related constrains by a nonlinear interior point method", in Proc. IEEE Power Engineering Society Winter Meeting, Singapore, 2000, pp. 1751-1759.

[8] Liu Fang, Yan Wei, Xu Guoyu, "Dynamic optimal power flow with decomposed predictor-corrector interior point method", Automation of Electric Power Systems, vol.31, no.14, pp.38-42, 2007.

[9] Yude Yang, Zhijun Qin, Hua Wei, "Power Flow Calculation Based on Non-linear Programming Model and Vectorization Mode, " in Proc. 2007 IEEE International Conference on Mechatronics and Automation, Harbin, China, 2007, pp.1729-1733.

[10] R.D Zimmerman, C. E. Murillo-Sánchez, D. Gan. MATPOWER 3.0.0 User’s Manual. [Online].Available: http://www.pserc.cornell.edu/matpower.

[11] O. Schenk, A. Wachter, M. Hagemann. (2006, Apr.). Solution of KKT Systems in Large-Scale Interior-Point Optimization. Department of Computer Science, University of Basel, Switzerland. [Online]. Available: http://iacs.epfl.ch/colloqnum06/posters/59_Schenk.pdf.

[12] Yu Juan, Yan Wei, Xu Guoyu, Du Peng, Liu Fang, "A new model of reactive optimization based on predictor corrector primal dual interior point method", Proceedings of the CSEE, vol.25, no.11, pp.146-151, 2005.

[13] H.Wang, C. E. Murillo-Sánchez, R.D Zimmerman, R.J Thomas, "On computation issue of market-based optimal power flow, " IEEE Transactions on Power System, vol.22, no.3, pp. 1185-1193, 2007.

[14] P. Amestoy, T.A. Davis, I.S. Duff, "Algorithm 837: AMD, An approximate minimum degree ordering algorithm," ACM Transactions on Mathematical Software, vol. 30, no. 3, pp. 381-388, 2004.

[15] T.A. Davis, J.R. Gilbert, S. Larimore, E.G. Ng, "A column approximate minimum degree ordering algorithm," ACM Transactions on Mathematical Software, vol. 30, no. 3, pp. 353-376, 2004.

[16] T.A. Davis, "Algorithm 849: A concise sparse Cholesky factorization package," ACM Transactions on Mathematical Software, vol. 31, no. 4, pp. 587-591, 2005.

[17] L.S. Blackford, J. Demmel, J.Dongarra, I. Du, S. Hammarling, G. Henry, et al, "An Updated Set of Basic Linear Algebra Subprograms (BLAS),” ACM Trans. Math. Soft., vol. 28, no. 2, pp. 135-151, 2002.

[18] G.L Torres, M.A de Carvalho Jr, "On efficient implementation of interior-point based optimal power flow in rectangular coordinates, " in Proc. 2006 IEEE Power Systems Conference and Exposition, Atlanta, USA, 2006, pp.1747-1752.

[19] J. R. Gilbert, C. Moler, R. Schreiber, "Sparse Matrices in MATLAB: Design and Implementation," SIAM Journal on Matrix Analysis and Application, vol.13, no.1, pp. 333-356, 1992.

[20] G. Alan, Liu Joseph, "The Evolution of the Minimum Degree Ordering Algorithm," SIAM Review, vol.31, no.1, pp. 1-19, 1989.

BIOGRAPHIES

Zhijun Qin was born in Nanning in Guangxi province, P.R. China on September 27, 1977. He received BS and MS degree in automation of power system from Huazhong University of Science and Technology, P.R. China in 2000 and 2003, respectively. Now he is a PH.D candidate with Guangxi University. His research interest includes optimization of power system and high performance scientific calculation. Email: hust_qzj@hotmail.com.

Yude Yang was born in LiuZhou in Guangxi province, P.R. China on April 23, 1971. He graduated and received master degree from Electrical Engineering College of Guangxi University, P.R. China. Now he is PHD candidate of Guangxi University. His research interest is optimal power flow of power system.

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