Electrical Energy Supply Control Support via Meters

Available online at www.sciencedirect.com

ScienceDirect Procedia Computer Science 103 (2017) 287 – 294

XIIth International Symposium «Intelligent Systems», INTELS’16, 5-7 October 2016, Moscow, Russia

Electrical energy supply control support via meters data ellipsoidal approximation in smart grids G. Chrysostomoua, V. Shikhinb*, A. Shikhinaс a b

Frederick University of Technology, Nicosia, Cyprus Moscow Power Engineering Institute, Moscow, Russia c RT-Soft LLC, Moscow, Russia

Abstract Paper discusses the approach that supports the decision-making under need to provide total multi-line electrical energy supply under appointed maximum level. The data available from power meters allows to estimate the boarders of acceptable level for the external power supply and to predict its overshoot. It leads to rapid decision-making and allows to avoid the penalties on behalf of the power distribution utilities. Data approximation in the form of ellipsoid gives clear graphics and concise analytical description of the desired boarder. Introduced in the paper algorithm involves logarithmic functional transformation of the target criterion that allows to convert the meter data approximation via maximum volume ellipsoid problem to the convex nonlinear mathematical programming. © 2017 2017The TheAuthors. Authors. Published Elsevier © Published by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the XIIth International Symposium «Intelligent Systems». Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems” Keywords: electrical energy; ellipsoidal approximation; nonlinear programming; power distribution; smart grid

1. Introduction The need to optimize end-to-end electrical grid efficiency closely related with the mathematical tools applied for measurement data mining, presentation and treatment. The demand from utility dispatchers and operators for clear and ready for direct use forms of measurement data representation is great. Collected data are utilized not only for

* Corresponding author. E-mail address: [email protected]

1877-0509 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems” doi:10.1016/j.procs.2017.01.109

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G. Chrysostomou et al. / Procedia Computer Science 103 (2017) 287 – 294

online power consumption control but also can be used for the working capacity indices estimation and analysis that matches the Smart Grid concept1,2 and supports the decision-making under need to provide total multi-line electrical energy supply under appointed maximum level3,4,5. The method for such data approximation in the form of ellipsoid is represented in this paper and called the Balancing Ellipsoid Method. Method follows the well-known mathematical approaches to approximate the areas via second order geometrical figures6,7,8,9,10. 2. Formulation Let the Area of Required Level (ARL) is considered to be the area S in :x, where target criterion J(X) satisfies: S:

^ x : x | J ( X ) d c` ,

(1)

while outside the area S: (2)

J(X)>c,

where c is any pre-assigned positive constant; X=[x1,x2,...,xp]T is the vector of measured variables with components from the space :x:={xRm} on the variety of real numbers Rm. Accepted restrictions: a). Criterion J(X) to be numerically valued and continuous, bounded and positivelydefined; b). ARL supposed to be convex and constrained; c). Let the absence of sub-regions with non-measurable variables. Let exists nonempty set S* which contains elements-points of number N*+No=N: S*

{ x* R m | J ( x* ) d c}, i i

i

1, N *

(3)

and also exists nonempty set: So

{xio R m | J ( xio ) ! c}, i

1, N o .

(4)

ARL construction can be formulated as the construction of ellipsoid Е* which includes the most elements from the set S* and strictly no one from the set Sо. This requirement induces new formulation for ellipsoid search6. 3. Main result Suppose ellipsoid with center x o appropriates to the symmetric positively-defined matrix W,[mum] with elements wi , j , i, j 1, m : E

{x : ( x xo )T W ( x xo ) d 1}

For clarity put ellipsoid Е center into origin xo VS

VE

(5) О . Well-known formula for ellipsoid volume:

(6)

det W

where the volume of m-dimensional hyper-sphere VS of radius R is calculated through gamma-function Г(m/2+1): VS

S

m

2

*  (

m 2

1) R m

(7)

289


From (6), (7) is clear, that VE varies while varies the determinant of W, and hence the formulation will be: wij

ij VE o max

det W o min, w w ,

w

ij T

i, j

(8)

1, m,

ji

x Wx d1, g j (W ) ! 0,

o

x S o

where gj(W) denotes the main co-factors of the matrix W calculation. The restriction on matrix symmetry (wij=wji ) adopts the requirement for Oi to be real because it defines the half-axle Ui of the ellipsoid. The problem (8) can be converted to mathematical programming through transformation the restrictions on matrix symmetry and positive definition to the form of functional limitations. The restriction for matrix W to be positively defined can utilize the orthogonal transformation about matrix W to the diagonal form:

FWF diag (O , O

W

T

1

2

,..., O m )

O 12 O 2 2 ... O m 2 ,

here F- orthogonal transformation matrix, FTF=FFT=Im; O i , i 1, m - matrix eigenvalues; W m

- matrix norm:

m

¦¦w

2 ij

W

(9)

trW T W

(10)

i 1 j 1

Since it is known that for positive definite matrix W all eigenvalues O i , i 1, m should be positive to form the spectrum of the matrix, hence always can be chosen sufficiently large number /, such that / t O i , i 1, m . Thus: W d/ m

(11)

Note that the value / defines the length of semi-minor axis Umin. For the sufficiently small number O: 0 O d O i, i

1, m ,

(12)

Value Umax defines the length of semi-major axis. Formula (12) corresponds to the limited range of the matrix W spectrum: 0 O d O min ,..., O i ,..., O max d / f .

(13)

Thus, problem (8) is converted to nonlinear programming problem formulation: ij VE o max

w

wij

trW o min, i, j w w , ij

(14)

1, m

ji

W d/ m o

x S o

Problem (8) can be transferred to the convex nonlinear mathematical programming that makes much easier to find the optimum. The following functional transformation is proposed: f (W )

2 ln

VE VS

ln(det W )

(15)

290


where f(W) is supposed to be obviously convex. In this case the task (14) converts to the next formulation: ij VE o max

w

f (W )

wij

ln(det W ) o max, w w ,

i, j

1, m

(16)

ln(det W ) o max, i, j w w ,

1, m

(17)

ij

ji

g j (W ) ! 0,

o

x S o

and after applying result (11): ij VE o max

w

f (W )

wij

ij

ji

W d/ m , o

x S o

Result (17) contains restriction on the length of minimal half-axis and does not contain the restriction on the maximum half-axis. In this regard, it is proposed to change the limitations in the formulation (17) with considering the regularization parameter by Tikhonov: W d / m G ; G O d O i, i

(18)

1, m

Regularization parameter G guarantees the convergence and the degeneracy of the numerical procedure. Thus, the final formulation of the problem to construct maximum volume will be in the following form: ij VE o max

w

f (W )

wij

ln(det W ) o max, i, j w w , ij

1, m

ji

(19)

W d / m G

o

x S o

Here W is a positively-defined symmetric matrix with nonzero determinant, hence the gradient for f(W) is always defined. The next algorithm will be based on the result (19). Separating rules should be developed for narrowing or extension of the space be approximated by ellipsoid. It is proposed to start from the premise stated above, that it is permissible to exclude from consideration acceptable experimental point x i* , but strictly not allowed entering the unsatisfactory points xio . If at the k-th iteration the next condition from (19) is not satisfied: W d (/ m G ) ,

(20)

then the cut-off vector should be associated with the semi-major axis Umax of the ellipsoid. Regularization parameter

G in (20) is recommended to be a function of the specified for particular study accuracy threshold Vо: V o d VˆE * (Wˆ ) VE * (W )

(21)

* of where VE*(W) accepted as the volume the true (but unknown) ellipsoid, thus G=G(Vо). Because of VE is unknown * then it can be used its estimate VE (W ) . For applications is recommended to choose Vо as the minimum value distance between the center of the ellipsoid x o and any point x i* :

Vo

min xi* xo ,

i

1, N * ,

or through the semi-minor axis of the ellipsoid:

(22)

291


Vo

1

min

Oi

i

,

1, m ,

(23)

or through radius rS for the initial sphere of the minimum volume:

Vo

min xi* xo ,

rS

i

1, N * ,

(24)

or through acceptable accuracy in the preparation of the original array of ( yi , xi ), i 1, N . The coordinates of the center x o* is proposed to determine over weighed acceptable measurements-points xo* , i 1, N * : N*

x

¦x

* o

* i

i 1

N*

.

(25)

The semi-major axis Umax and semi-minor axis Umin at the initial step of the approximation: U max

max xi* xo , U min i 1, N *

min xi* xo

(26)

i 1, N *

To activate the procedure the needed Vо can be determined, for example, from (24). The regularization parameter G can be calculated from the next equation: max xi* xo

/ m G ,

(27)

i 1, N *

G

max xi* xo / m

(28)

i 1, N *

Constraints in the formulation (19) under (26), (27), (28):

/=1/U2min , O=1/U2max , wij

w ji , i , j

1, m ,

W d ( / m G ), xio S o

(29)

In the case of indicating the violations it causes iterative calculations by means of cuts off using cut-off radius U:

U

[1 (O G )] max xi* xo*

(30)

This operation leads to the loss of some x i* suitable for criterion (3) and induces the narrowing of ARL and can be interpret as a "pay" for getting simple geometric surface like ellipsoid. Using the calculated radius of truncation U the formation of the truncated set Sˆ * is evaluated. Further the search procedure is repeated, accompanied by compressing the semi-major axis and a change in the orientation of the ellipsoid. A reasonable choice of parameters / and O is an important stage in the developed computational procedure. Accordingly, the following Lemma is proposed. Lemma. The values for / and O can be defined as: / d rS2 ,

(31)

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G. Chrysostomou et al. / Procedia Computer Science 103 (2017) 287 – 294 2

§ rSm 1 · , m ¸ © RS ¹

(32)

min xi* xo

(33)

max xi* xo

(34)

O t¨

where: rS

RS

i 1, N *

i 1, N *

Proof of Lemma is not given here because of restricted space. Lemma allows to modify algorithm and accept the initial approximating surface not in the form of the maximum volume ellipsoid but in the form of maximum volume sphere of radius RS. Therefore, to calculate O instead of (29) should be used (32) and the data area must be represented in the form of a cube that requires a proper normalization. 4. Numerical Example Load monitoring follows the need to control the total consumption of the oil-pump station that should be under appointed maximum value. In the case of exceeding the appointed level the penalties are very high. Under discussion is 3-lines power supply with total power 64 MW for 3 parallel pipelines and 9 pumps from 2.5 to 12 MW each. Fig.1 represents the sampled data of volume N=240 from power-meters with 30 minutes discreet. "Nulls" and "Stars" appropriate to the sets S*, So in accordance with (3) and (4) in three-dimensional space of the loads through 3 power-supply lines P1, P2, P3. Following the results of Lemma procedure starts from the center x o* calculation over N*=163 "*"-points: N*

xo*

¦x i 1

N*

* i

(0.0094, 0.0811, 0.9398) T

Fig. 1. Classified and normalized measurement data

Scaling values P1o102,P2o101,P3o100 are applied to get a cubic form. From (33) and (31):

(35)

293


rS /

min xio xo* 1 rS 2

48.22,

§ 0.00798 10 2 · § 0.0094 10 2 · ¨ 0.0805 101 ¸ ¨ 0.0811 101 ¸ ¨ ¸ ¨ 0.9398 ¸ 0.914 © ¹ © ¹ 4 3 S rS 0.0125 . VS min 3

0.144,

(36)

From (32) and (34) the radius RS will be:

RS

O

§ 0.00133 10 2 · § 0.0094 10 2 · ¨ 0.115 101 ¸ ¨ 0.0811 101 ¸ 0.857, ¨ ¸ ¨ 0.9398 ¸ 1.73 © ¹ © ¹ 2 § rS2 · 4 S RS3 2.6351 . 0.00108, VS max ¨ 3¸ 3 © RS ¹

max xi* xo* § rSm 1 · ¨ m ¸ © RS ¹

2

(37)

In this particular case the regularization parameter G is related with rS . Because the volume of desired ellipsoid VE cannot be smaller than VSmin: G

(1%)VS min

4 (1%) S rS3 | 0.0001 3

(38)

At the first phase the ellipsoid includes forbidden "o"-points and so the check for the constraints (18) xio E , W d (/ m G ), wij

w ji , g j (W ) ! 0 ,

(39)

[1 (O G )] RS 0.856 gives a negative result. Using cut-off radius (30) U [1 (O G )] max xi* xi* truncation is evaluated and leads to some loss of the "*"-points. Consequently, this shifts the suspended center x o* and decreases the semi-major axis Umax.. For matrix W from (5):

W

§ 4.4240 1.4040 0.2286 · ¨ 1.4040 3.6635 0.3959 ¸ , ¨ 0.2286 0.3959 4.4794 ¸ © ¹ W 7.5745

det W

63.0673,

(40)

For O1,O2,O3 and normalized eigenvectors p1 , p2 , p3 : det(W O I )

0,

(W O i I ) pi

0,

pi

1 O1=2.572,

O2=4.338, O3=5.651,

(41)

From (36), (37) and (41) follows that boundedness of the matrix W spectrum of the eigenvalues is performed: 0