International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 38

ISSN 2229-5518

An Optimization Model for Reactive Power and Voltage Controls of Super High Voltage Grid Systems Using Pseudo-Inverse Method

.

Najim Abood Hamudi Al-Shaikhli

Abstract: - This paper develops an optimization model to control the excessive MVAR generation by super high voltage grid systems for maintaining the nodal voltages within the required acceptable margin. The model enhances an approximate solution on the condition of balanced real power. The Pseudo-Inverse Method of optimization is implemented as an optimal means of solving non square systems of equations based on Lagrange's Theory of optimization. It recommends the locations and ratings of the minimum required reactors from the preinstalled ones that to be in service for optimum nodal reactive power and voltage controls The model had been tested on the Iraqi Super High Voltage Grid System (400kV), and it proved to be efficient in the obtained results and reliable in convergence.

Index Terms:- Optimization Models, Pseudo-Inverse Method, High Voltage, Reactive Power Control, Voltage Control.

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1 Introduction

Present power flow optimization methods can be
classified into, exact and approximate methods. Exact methods (1,2,3) take into account both real and reactive flows in obtaining the solution. While approximate methods (4,5) achieve simplified representations and possibly computational efficiencies by ignoring either the real or reactive equations. Approximate models are normally tailored for particular applications and do not have the generality inherent in the exact models.
The reduced gradient method of Dommel and Tinny
(2), the Fletcher Powell method as developed by Sasson for
power flow applications (3), and Carpentier's method based
on satisfying the Khun-Tucker conditions (1), are all accurate and widely applicable.
While some have been developed to become
computationally very efficient, a number of approximate
models dealing exclusively with either the real or reactive power equations have also been developed. Hano, et Al (4) and Kumai, et Al (5), used primarily the reactive equations to develop a method for real time control of voltages and reactive powers. The need to adjust voltage magnitudes and reactive powers at times so that the overall solution is operationally implementable, led to the consideration of reactive equations.

2 Mathematical Formulation

Figure (1) is simulating a super high voltage grid
system by representing the nodes into four categories from the point of view of their voltages [either they are within (m) or out (o) of the acceptable margin] and VAR absorption installations [either there are (r) or none (w)].

Figure (1) Schematic Single Line Diagram of the Optimization Mode

Considering the system in steady-state condition and the real power mismatch ΔP is zero, then the following linearized equations describe the effect of Q on V for the four categories of the system:-

System (1) above can be rewritten as below:-

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International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 39

ISSN 2229-5518


Where E and F are unit diagonal matrices. Reducing

System (2) using Newton-Gauss technique, it will be:-

Or D ΔQR = C......................... (4)


The ranks of the main and sub matrices are:-

The objective function of this model is:-
To approach ΔQR optimum, the following algorithms will be followed during the iterative computational process:-

Algorithm I: - if No = Nr then:-

Algorithm II: - if No < Nr then:-


Algorithm III: - if No > Nr then:-

D1 +and D2 + are the Pseudo-Inverses of the rectangular matrix D(6,7), and their mathematical derivations are presented in Appendix-I.

3 Model's Flow Chart


The flow chart of the optimization model is shown in Figure (2). The dotted block represents the part where ΔQR is computed.

Figure -2- Optimization Model Flow Chart

4 Model's Performance


This optimization model had been tested successfully on the Iraqi Super High Voltage Grid System shown in Figure (3) at Minimum Load Condition of the year 2013.

Figure-3- Iraqi Super High Voltage Grid System

To recognize the significance of the optimization model, three modes for MVAR control were applied on the system at the same load condition as follows:-

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International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 40

ISSN 2229-5518

Mode I: - In this mode, No MVAR absorption assistance was made to the system (ΔQR = 0).

Mode II:-All MVAR absorption resources were linked to the system (ΔQR =Maximum).

Mode III: - In this, the optimization model was tested on the system to achieve the optimum performance of the MVAR absorption resources by recommending the locations and ratings of the minimum required reactors from the preinstalled ones.

Figure (4) translates nodal voltages in ( p.u.) of the system into three margins.

V(p,u.)

Figure 4- Nodal Voltages Margins for the three modes of MVAR Control at Minimum Load Condition of the Iraqi Super High voltage Grid System

Margin A: Nodal Voltages Margin at (Mode 1). Margin B: Nodal Voltages Margin at (Mode 11). Margin C: Nodal Voltages Margin at (Mode 111).


While figure (5) translates reactive power generation / absorption required by the power stations for the three modes at minimum load condition of the system.

Figure -5- Reactive Power Generation/Absorption by the power stations for the three modes of MVAR control.

5 Conclusions

The proposed optimization model for the reactive and
nodal voltage controls is based on linearized load flow reactive power equations and Lagrange’s Theory of
optimization. The Pseudo-Inverse method of solving rectangular matrices is implemented via the algorithms of computing [ΔQR ] , when this matrix is not square.
The model proved to be reliable in convergence, the
solution is obtained in few iterations and it needs less memory requirements due to that the Jacobian matrix is formulated with !4 of the linearized load flow equations (reactive power equations only) in the first iteration of the solution and it is reduced when ignoring the healthy nodes iteratively.
Three modes of MVAR control are implemented on
the Iraqi National Super High Voltage Grid System at its
worst operating condition, to recognize the significance and the efficiency of the proposed optimization model concerning the nodal voltage and MVAR controls. It has given better nodal voltage regulation at 25% of savings of total reactors ratings and has improved the performance of the generating stations from the point of view of reactive power generation/absorption, and then it improves the stability of the system.

Appendix- I

Having the system of equations [A][X]= [b], Where [A]= {aij }
, i = l. ...m and j = l.. ..n. Its method of solution is dependent on whether [A] is square or not, and it can be obtained by one the following three algorithms:-

Algorithm I :-

If [A] is a square matrix, (m = n ), then :-

[X]= [A]-1[b]..,………………... (I.1)

Algorithm II :-



If m<n, an optimal solution is achieved when:- Optimum [X], is approached by Lagrange’s theory:-
Equation (1.5) gives:-


Substituting for [X] in equation (1.4), results:-

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International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 41

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Substituting for [λ] in equation (l,6), gives: -

And [A1 ]+ is the Pseudo-Inverse of the rectangular matrix
[A], when m<n.

Algorithm III :-

If m >n , an optimal solution is achieved when :-

Minimum ɸ(x) is approached when;-

And [A2 ]’ is the Pseudo-Inverse of the rectangular matrix
[A], when m>n.

M.Sc. in Power & Machines, Member of College Staff

University of Baghdad - IRAQ

1. J. Carpentier,”Contribution a’l’ Etude du Dispatching Economique “Bulletin de la Societe Francaise des Electriciens, Ser. 8, Vol. 3, 1962

2. H.W.Dommel, W.F. Tinney,”Optimal Power Flow

Solutions” IEEE; PAS, Vol. 87, 1968, PP. 1866-1876.

3. A.M. Sasson’Tombined Use of the Powell and Fletcher- Powell Nonlinear Programming Methods for Optimal Load Flows”, IEEE, PAS, Vol. 88, 1969, PP. 1530-1537.

4. Hano, Y. Tamura, S. Narita, ”Real Time Control of System Voltage and Reactive Power”, IEEE,PAS, 88, 1969, PP. 1544-1559.

5. K. Kumari and K. Ode,”Power System Voltage Control by Using a Process Control Computer”, IEEE, PAS, 87,

1968, PP. 1985 -1990.

6. V.A. Stroev, I.S. Rokotian “The Algorithms for Alleviating Overloads Using Pseudo-Inverse Method” Paper from Moscow Power Eng. Institute,. 1991.

7. Brian D.Burctay,”Basic Optimization Methods”, PP (47-

90), Edward Arnold (publishers), 1985.

References

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