Hierarchical algorithms of functional modelling for solution of optimal operation problems in electrical power systems

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Electrical Power and Energy Systems 30 (2008) 415–427 www.elsevier.com/locate/ijepes

Hierarchical algorithms of functional modelling for solution of optimal operation problems in electrical power systems V.A. Makeechev b, O.A. Soukhanov a,*, Y.V. Sharov c a Energy Systems Institute, 1 st Yamskogo Polya Street 15, 125040 Moscow, Russia Industrial Power Company, Krasnopresnenskaya Naberejnaya 12, 123610 Moscow, Russia c Moscow Power Engineering Institute, Krasnokazarmennaya Street 14, 111250 Moscow, Russia b

Accepted 18 February 2008

Abstract This paper presents foundations of the optimization method intended for solution of power systems operation problems and based on the principles of functional modeling (FM). This paper also presents several types of hierarchical FM algorithms for economic dispatch in these systems derived from this method. According to the FM method a power system is represented by hierarchical model consisting of systems of equations of lower (subsystem) levels and higher level system of connection equations (SCE), in which only boundary variables of subsystems are present. Solution of optimization problem in accordance with the FM method consists of the following operations: (1) solution of optimization problem for each subsystem (values of boundary variables for subsystems should be determined on the higher level of model); (2) calculation of functional characteristic (FC) of each subsystem, pertaining to state of subsystem on current iteration (these two steps are carried out on the lower level of the model); (3) formation and solution of the higher level system of equations (SCE), which gives values of boundary and supplementary boundary variables on current iteration. The key elements in the general structure of the FM method are FCs of subsystems, which represent them on the higher level of the model as ‘‘black boxes”. Important advantage of hierarchical FM algorithms is that results obtained with them on each iteration are identical to those of corresponding basic one level algorithms. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Optimization; Hierarchical algorithms; Functional modeling; Distributed processing; Economic dispatch; Optimal power flow

1. Introduction One of the promising trends of modern progress in methods and algorithms destined for planning, operating and controlling power generation and transmission systems is development of methods and algorithms oriented to solution of these problems on distributed computer systems. The most important advantages presented by this approach is an opportunity to obtain efficient parallelism of data processing and small amount of data communication necessary for handling these tasks. It makes this approach very attractive compared with traditional cen-

*

Corresponding author. E-mail address: [email protected] (O.A. Soukhanov).

0142-0615/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2008.02.001

tralized organization of solution based on formation and processing of single system of equations pertaining to a system as a whole. In this paper, we present theoretical foundations of hierarchical method intended for solution of optimization problems in electrical power systems and based on the principles of functional modeling. At present time there exist all technical conditions needful for practical implementation of algorithms oriented to parallel solution of several systems of equations (each of them pertaining to one of subsystems in large system) on various computers and exchange of information among them. The main problem in this field until nowadays was creation of such methods of optimization which on one hand would be able to use this organization of calculations and data communication and on the other hand would yield on each iteration same

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results which could be obtained from solution of single system of equation for system taken as a whole. The main difficulty in development of optimization methods which could meet these requirements is calculation of boundary variable values on each iteration. The problem is that values of these variables should be found from two conditions, which must be satisfied simultaneously: optimality condition for each variable and condition of compatibility for values of boundary variables when they are computed in two subsystems adjacent to each border between subsystems [4,5]. In spite of some achievements in this field the problem of formation such systems of equations, from which values of boundary variables satisfying these conditions could be found and which could be efficiently solved in distributed system, was till now far from having been solved. In this paper, we present theoretical foundations of hierarchical method intended for solution of optimization problems in electrical power systems and based on the principles of functional modelling (FM). The main objective pursued in creation of this method was solution in general form of the problem we mentioned above, i.e. determination of the values of boundary variables, satisfying conditions of optimality and compatibility. This problem can be solved by application of the general approach of the FM method. Principles of the FM method were presented previously in [1–3,9] and are stated as follows: (1) Representation of subsystems in large system by functional characteristics (FCs). FCs are input–output characteristics in which vectors of boundary variables of one kind are considered as input variables and boundary variables of another kind as output variables. These characteristics are obtained while meeting all constraints within subsystem. (2) Building of a model as a hierarchical structure, consisting of interconnected systems of equations. In this structure subsystems are represented by lower level systems of equations. A higher level system of equations represents borders between subsystems (boundary nodes). (3) On the lower level each subsystem is presented as an open system influenced by input variables on it’s borders. On this level two problems are solved: calculation of internal variables in subsystems and determination of their FCs. The problem of the higher level is calculation of the values of boundary variables. This problem should be solved through formation and solution of the system of connection equations (SCE), comprising equations obtained from expressions presenting conditions of compatibility (matching conditions) for values of boundary variables computed in adjacent subsystems. This formulation applies to solution by the FM method of power system modeling and analysis problems. It is necessary to modify and generalize this formulation if we want

to develop an optimization method based on the principles of FM. In [3] it has been shown in general features how the FM method can be applied to solution of power system optimization problems. A FM type algorithm for solution of economic dispatch problem was also presented for simple case when only characteristics of stations are taken into account. Neither active power losses nor network equations were included in the model. In this paper we present: (1) general formulation of power system optimal operation problem as it looks from the point of view of FM method; (2) derivation of main equations forming the higher level system of equations – SCE and (3) general structure of model and organization of the solution process in the FM algorithm. Optimality equations for boundary variables forming the SCE are derived in this paper in general form from Lagrange function, representing a system as a set of subsystems and including equations of matching conditions for boundary variables when they are determined in adjacent subsystems. General structure and convergence properties of the FM algorithms presented here are based on the principle of optimality for hierarchical models formulated in this paper. Due to application of this principle calculation results obtained by any FM algorithm on each iteration and final results are identical to the results of basic sequential algorithm from which it is derived. It is important to note that the problems considered in the first part of this paper are used mainly as material for derivation of the FM algorithms and for demonstration of the general logic of the FM method. These problems in themselves are of small interest from theoretical and practical points of view. The purpose of this paper is the presentation of the FM method as a method creating new opportunities to overcome serious difficulties and problems which arise in application of the concept of central control to large interconnected power systems. These interconnected systems often include large power systems of independent countries and it causes great organizational and political problems for implementation of centralized optimization and control in these systems. Important advantage of the general structure of the FM method and the FM algorithms is the presence of the upper level model (SCE) in which only boundary variables of subsystems are represented. In economical sense it means that in this model the problem of markets can be solved when the optimal values of power flows between subsystems are calculated. No information about internal parameters and variables of subsystems is needed for forming and solution of this problem. From the point of view of computational efficiency the FM algorithms perform as a distributed Gaussian elimination when the FCs of subsystems are calculated and as a distributed back substitution when the downward moves in subsystems are executed. Convergence properties of these algorithms are the same as in basic sequential algorithms.

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In the final part of the paper hierarchical algorithms based on FM method for solution of economic dispatch and optimal power flow (OPF) problems (taking into account network losses and network constraints) are presented. Hierarchical algorithms of this type are especially efficient for distributed realizations. They can be applied in power systems working both in regulated and market conditions. Fig. 1. System presentation in the FM method.

2. General theory of FM optimization method In preceding paper [3] application of FM principles to solution of power system optimization problems has been considered and hierarchical FM algorithms have been proposed for particular cases of gradient basic algorithms. Here we present fundamentals of the FM optimization method based on the general approach to solution of constrained optimization problems involving construction of Lagrange function and determination of Lagrange multipliers. General structure of the FM model, proposed in [2,3] is applied in this paper. In this structure a power system is represented as a set of subsystems, adjoining each other, see Fig. 1. Subsystems are connected by boundary nodes There are only two types of variables in this model: boundary and internal variables. Boundary variables (denoted by index b in this paper) are the variables pertaining to boundary nodes, e.g. powers and currents crossing the boundary nodes, voltages at the boundary nodes. All other variables are internal variables of subsystems (denoted in this paper by indices i or in). 2.1. Derivation of the direct FM algorithm If this general approach is applied to hierarchical model, representing a system on tare the lower level as a set of subsystems adjoining each other, and if boundary variables in this model are considered as independent variables, Lagrange function for this model can be constructed as a sum of Lagrange functions of subsystems. If power flows through boundary nodes are taken as boundary variables this function will look as follows: XX LS ¼ F ðP iI Þ I

þ

iI

X I

" kI P LI 

X iI

P iI þ

X

!# þðÞP bI

ð1Þ

bI

where P iI is the power of the station i in the subsystem I, P bI is the power flow through the boundary node b adjacent to the subsystem I, P LI is the power consumed in the subsystem I. Note that P bi enters this function with sign + for one subsystem and sign  for adjacent subsystem. The necessary conditions for an extreme value of the objective function for this system can be obtained taking

the first derivatives of this function with respect to each of the boundary variables and setting these derivatives equal to zero. It results in the following set of equations: kI b  kJ b ¼ 0;

b ¼ 1; nb

ð2Þ

where index Ib denotes for one of the subsystems, adjacent to the boundary node b, and Jb denotes for another. Each of these equations applies to one of the boundary nodes between subsystems in this model and total number of these equations is equal to the number of these nodes. Now let us consider the systems of internal equations for all subsystems, i.e. the models of the lower level. If network losses are not taken into account these systems look as follows: dF iI  kI ¼ 0; iI ¼ 1; nI dP iI X X P LI P iI þ þðÞP bI ¼ 0 iI

ð3Þ ð4Þ

bI

Eq. (3) present the minimum cost operating conditions for subsystem I. According to it the first derivatives of the function (1) with respect to internal variables Pi should be equal to zero. Eq. (4) is the constraint equation for subsystem I. Finding expressions for the first derivatives of the cost functions of all stations and substituting them into (3) we get aiI P iI þ biI  kI ¼ 0

ð5Þ

In this case we assume for simplicity that the cost function has the form of a quadratic function. Solution of the economic dispatch problem presented above by the FM method consists according to [3] of the following steps. (a) Determination of the FCs of subsystems: Applying the Gaussian elimination of internal variables (i.e. Pi) to the system (4) and (5) we obtain for each subsystem the FC, having the following form: X þðÞP bI þ cI ð6Þ kI ¼ aI b

Expressions in the right-hand side of (6) found in the local computers should be communicated to the central computer.

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(b) Formation and solution of the SCE: Substituting for kI and kJ from right-hand side of Eq. (6) for FCs of the subsystems I and J into (2) yields one of the equations, forming the SCE. Each of these equations pertains to one of the boundary nodes in the hierarchical model of the system. The SCE obtained in this way is shown below [3] AP b ¼ b

(c) Determination of the optimum economic operating points in subsystems: Back-substituting the values of power flows on the borders of each subsystems into (6) we first find the value of kI and then back-substituting into Eq. (5) the values of Pi. These values are the same that can be obtained by basic algorithm from solution of the set of equations for this system as a whole. 2.2. Derivation of the iterative FM algorithms In derivation of the algorithm presented above we do not take into account that there are inequality constraints that must be satisfied for each of the stations within subsystems.. Also there are inequality constraints that restrict the amount of power transferred through the boundary nodes. Hence a solution got by this algorithm is acceptable only if no inequality constraint is violated. These conditions can be taken into account if we pass on to the more general form of the FCs. In this case the FC should represent for a subsystem the ratio between increments of boundary variables (i.e. power flows through borders of subsystem in our problem) and increment of Lagrange multiplier of this subsystem. This FC should be obtained while meeting all optimality equations and constraints within the subsystem. It means that the FC represents subsystem when it stays in optimum operating points when power flows on it’s borders are changing. Such FC can be obtained if the system (4) and (5) is rewritten as a system of equations for increments of all variables which are present in (4) and (5). In general form the resulting system is shown below aiI DP iI  DkI ¼ 0; iI ¼ 1; nI X X DP iI þ þðÞDP bI ¼ 0

ð8Þ ð9Þ

bI

In fact this system is the system of equations, representing subsystem in the Newton method, with the only difference that when it is used for determination of a FC the number of variables in this system is greater than the number of equations:

a a

  DP i      Ai b   k  ¼0   Abb   DP b 

ð10Þ

Applying the Gaussian elimination of internal variables to the upper equations in the system (8) and (9), we obtain from the lower equation: X DkI ¼ aI þðÞDP bI ð11Þ

ð7Þ

This system of linear equations has the order of the total number of boundary variables. Solving this system optimal values of power flows between subsystems can be found.

iI

  Aii  A bi

bI

If we use the FCs of this type we can develop iterative FM algorithms for solution of this and other optimization problems. The general concept for development of these algorithms is the following. We can set any values for the boundary variables on the borders of subsystems in a hierarchical model. Then optimization problems for all subsystems should be solved and values of Lagrange multipliers calculated. If we substitute these numerical values into optimality equations of higher level (2) we shall find that numerical values of the expressions in the left-hand side of. these equations are not equal to zero. We denote the vector of these numerical values as DS. Now we formulate the following problem. We have to find the correction of the vector Pb (i.e. DP b ) that would drive the vector DS to zero. To solve this problem we observe for each equation in (2) f ðkÞ þ Df ðkÞ ¼ 0

ð12Þ

where f ðkÞ ¼ DS i ; Df ðkÞ ¼ DkI b  DkJ b . Therefore DkI b  DkJ b ¼ DS b

ð13Þ

Substituting for Dk from (11) we obtain X X þðÞDP bI  aJ þðÞDP bI ¼ DS b aI bI

ð14Þ

bJ

So to achieve our goal we have to solve the linear set of equations: ADP b ¼ DS b

ð15Þ

It means that in iterative FM algorithms the problem which should be solved on the higher level of the model on each iteration is determination of the new values of boundary variables. Succession of these steps forms the outer loop iterative process, while within each of these steps the inner loop iterations are carried out on the lower level of the model. Hence at each vector of boundary variables (vector of power flows) update, each system of equations for subsystem is solved separately. This organization of solution in the hierarchical model enables easily to take into account all inequality constraints within subsystems as well as on the borders between subsystems. Since injections of power on the borders of each subsystem are fixed on one outer loop iteration any basic (conventional) iterative algorithm can be applied for solution of the lower level (subsystem) problem. These

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algorithms are suitable when internal inequality constraints in subsystems are considered. The limitations on power flows between subsystems should be considered on the upper level of the model. For this purpose special algorithm, see [9], has been developed in case if the SCE, solved on each iteration is a linear system (15), formed as was shown above from FCs of subsystems and based on optimality equation (2). 2.3. The principle of optimality For each intermediate vector of boundary variables iterative process in each subsystem goes on until converges. The whole iterative process in the FM hierarchical model should stop when stopping criterion is satisfied on the upper level (in the outer loop process). This criterion in our case is DS b 6 e

ð16Þ

This criterion is based on the principle of optimality for hierarchical optimization models formulated below. If a SCE is formed, such that two conditions are observed: (1) equations forming the SCE represent conditions of optimality for boundary variables obtained from the Lagrange function for the hierarchical model of the system; (2) conditions of optimality for internal variables in subsystems are observed in the SCE for any values of the boundary variables; then the values of the boundary variables found from the solution of this SCE are the same optimal values of boundary variables which can be obtained from solution of the optimization problem for the system as a whole. It is easy to show that these conditions are fulfilled if criterion (16) is satisfied on the higher level of the model. By definition the FCs of subsystems are input–output characteristics obtained while meeting all equations (in our case optimality equations) within the subsystems. These characteristics are used in formation of the SCE where they represent subsystems. Due to it the whole optimization problem is reduced to the problem of finding the optimal values of boundary variables, in which the second condition formulated above is observed. On the other hand the basic equations in the SCE, i.e. (2), have been obtained in accordance with the first condition formulated above. So the necessary and sufficient conditions for optimal solution are met in this case.

for solution of optimization problems in which boundary variables can be found from conditions of optimality for a system consisting of several subsystems. Two types of algorithms have been developed based on different types of systems of equations, forming a model of a power system. These systems of equations are obtained from Lagrange functions, in which equality constraints take two possible forms. In one of them only one equality constraint is present, representing condition of balance of powers in a system. In another form equality constraints represent complete set of power flow equations for all nodes of a power system. (1) The Lagrange function in the first form can be constructed for economic dispatch problem, in which network losses are taken into account. In accordance with the principles stated above Lagrange function for a hierarchical model should be written in this case as follows: LS ¼

XX I

þ

3.1. Solution of the problems with independent boundary variables It is possible to develop hierarchical FM algorithms for solution of optimal operation problems based on different sequential algorithms and different models pertaining to a system as a whole. First we present the algorithms destined

F ðP iI Þ

iI

X

" kI P LI 

I

X iI

P iI þ pI þ

X

!# þðÞP bI

ð17Þ

bI

where pI denotes for network losses in subsystem I. It is assumed in (17) that a scheme of connections between subsystems in this model has radial structure and there is not more than one boundary node between each pair of adjoining subsystems. Taking the first derivatives of this function with respect to the boundary variables, i.e. Pb, we obtain the following set of optimality equations for the power flows Pb.     op op kI þ kI  kJ þ kJ ¼0 ð18Þ oP b I oP b J where I and J denote for the subsystems adjacent to the node b. The SCE in this problem should be derived from this set of equations. The lower level models (the systems of equations for subsystems) consist in this case of the following equations:    dF iI op  kI 1  ð19Þ ¼ 0 iI ¼ 1; nI oP iI I dP iI X X P LI þ pI  P iI þ þðÞP bI ¼ 0 ð20Þ iI

3. FM algorithms for power system optimal operation problems

419

bI

Eqs. (19) and (20) together look like the well known coordination equations for the economic dispatch with the network losses considered. The only difference is that if this system is used for determination of the FCs the number of variables in this system is greater than the number of equations. On the first step of the FM iterative algorithm it is necessary to pick starting values of the power flows Pb and to solve the optimization problems (19) and (20) for all subsystems. For each subsystem numerical values of k and for all borders should be found. The next step is deter-

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mination of the FCs of all subsystems. For this purpose it is necessary to transform the system (19) and (20) into the system of equations, corresponding to the Newton secondorder method. We present it as follows:    2    oL o2 L o2 L  DP i   o2 P 2 oP i ok oP i oP b   Dk   i ð21Þ ¼0   o2 L 2 2  o L o L   okoP okoP b  DP b  ok2 i Gaussian elimination of increments of internal variables DP i from this system produces for each subsystem the FC, presented below ð22Þ

Dk ¼ CDP b

where C is a row matrix, DP b is a vector of the power flow increments. Using simple manipulations with (21) and (22) we can also obtain parameters of the following expressions:   op D ð23Þ ¼ kDP b oP b This collection, Eq. (22) plus Eq. (23), constitutes the FC of each subsystem. In order to form the SCE in this algorithm we have to calculate the mismatches for each of the optimality equation (18) substituting the numerical values k and (found previously in the subsystems) into these equations. So the vector DS will be found. After that it is necessary to obtain from the system (18) a system of the same type as (21), in which the variables will be Dk and Dðop=oP b ). Substituting for these variables the expressions from the FCs (22) and (23) we obtain the SCE of the same type as (15) ADP b ¼ DS

ð24Þ

Then the new vector of the power flows between subsystems should be found P kþ1 b

P kb

¼

þ

DP kb

ð25Þ

and subvectors of this vector sent to the proper subsystems for the next step on the lower level. The stopping criterion for this algorithm is presented above, see (16). (2) In case if Lagrange function is constructed with complete set of power flow equations included, it takes for a hierarchical model the following form: XX X t LS ¼ F iI ðP iI Þ þ kiI giI ðxI ; xbI ; uI ; pI Þ I

þ

iI

X

I

ktbI gbI ðxI ; xbI ; uI ; pI ; P bI Þ

In explicit form the Lagrange function for one of subsystems in this system can be written as X F iI ðP iI Þ þ ktiI giI ðxI ; xbI ; uI ; pI Þ LI ¼ iI

þ ktbI gbI ðxI ; xbI ; uI ; pI ; P bI Þ

ð27Þ

The optimality conditions for boundary variables can be obtained for the function (26) in the same way as it was done above for the function (17). Taking the derivatives of this function with respect to the boundary variable we come to the set of equations kbI  kbJ ¼ 0

ð28Þ

Essential difference between these equations and equations in (2) is that Lagrange multipliers in (28) pertain to constraints for boundary nodes of subsystem and not to constraints for subsystem as a whole as in (2). Eq. (28) are the only equations, connecting variables in different subsystems. They are key equations for further formation of the upper level system of equations – SCE. Equations for one subsystem based on the Lagrange function (26) can be obtained if we set the gradient of the Lagrange function of this subsystem to zero. It brings about the following equations for the parts of the gradient vector of this subsystem rLx ¼ oL ¼0 ox rLu ¼ oL ¼0 ou oL rLk ¼ ok ¼ 0

ð29Þ

oL rLxb ¼ ox ¼0 b oL rLkb ¼ ok ¼0 b

In this system the equations pertaining to the boundary variables are marked out and placed into lower part of this system. In explicit form this system should be presented as follows:  T  T oF þ og k þ ogoxb kb ¼ 0 ox ox  T oF þ og k¼0 ou ou gðx; u; pÞ ¼ 0 h iT h iT ogb og oF þ k þ kb ¼ 0 oxb oxb oxb

ð30Þ

gb ðx; xb ; u; P b Þ ¼ 0 ð26Þ

I

where xI, ui and pi are correspondingly: vector of internal state variables, vector of control variables and vector of fixed parameters in subsystem I; ktiI – vector of Lagrange multipliers for internal constraints in this subsystem, giI – set of internal equality constraints in this subsystem, ktbI – vector of Lagrange multipliers for boundary constraints in this subsystem and gbI – set of boundary constraints in this subsystem. Lagrange function (26) consists of Lagrange functions of subsystems within a power system.

To minimize the cost function of a power system comprising this set of subsystems it is necessary to observe simultaneously Eq. (30) for all subsystems and Eq. (28) for all ties between subsystems. It can be achieved by the following FM algorithm, based on the second-order Newton method. This hierarchical algorithm consists on each iteration of the following steps. (1) Determination of the FCs of subsystems It is necessary to form for each subsystem a system of equations, representing this subsystem according to the

V.A. Makeechev et al. / Electrical Power and Energy Systems 30 (2008) 415–427

second-order Newton looks as follows:    Dx    rLx    Du      rLu   2    o L  Dk   rLk  ¼  2  oV  Dxb     rLxb    Dkb    rLk  b  DP

b

method. This system of equations              

ð31Þ

For this purpose we first assume a starting set of boundary variables of subsystems and control variables within subsystems. Then it is necessary to find numerical values of all components of the gradient vectors in the left-hand part of this system and to calculate all second derivative matrices in the right-hand side of the system. Determination of the FCs of each subsystem from the system (31) is standard for the FM method. Applying the Gaussian elimination to the upper equations of this system it should be reduced to the following form:    DX     0  rL   Aii Aib       ð32Þ  a  ¼  0 A  Dkb  bb   DP b The FC of subsystem will be obtained after this transformation from the lower part of the transformed system (32). It looks as shown below Dkb ¼ ADP b  a0

ð33Þ

All operations on this step are executed on separate systems of equations, corresponding to different subsystems and can be performed on different computers. (2) Formation and solution of the SCE. This higher level system of equations should be formed by substituting for kIb and kJb to each of Eq. (28) their expressions from right-hand side of FCs (33). In this way a SCE will be formed ADP b ¼ b

ð34Þ

After solution of this linear system new values Pb of power flows through the borders of subsystems should be found ¼ P kb þ DP kb P kþ1 b

ð35Þ

Subvectors of P kþ1 and DP kb , pertaining to each subsysb tem, should be sent to the computers, carrying out operations in subsystems and used on the next step. (3) Determination of the increments of internal variables in subsystems. In each subsystem values of increments to internal variables on current iteration are calculated using back substi-

421

tution of the values DP b into upper part of the transformed system (32) and successive back substitution in the upper part equations in order to obtain the values of the increments of all internal variables. Then the new values of the internal variables in each subsystem on this iteration should be found xkþ1 ¼ xk þ Dxk ;

ukþ1 ¼ uk þ Duk ;

kkþ1 ¼ kk þ Dkk ;

xkþ1 ¼ xkb þ Dxkb b

ð36Þ

There are special features in calculation of new values of h in the vector of state variables in this algorithm. In the subsystem, in which the reference bus is placed (we define it as subsystem I R ), new values of h should be determined according to (36). In all subsystems, adjacent to the subsystem IR, the new value of hb in boundary node, connecting such subsystem with the subsystem I R (which was determined in the subsystem I R Þ should be assumed as a new value of hb in this adjacent subsystem. New values of the internal variables in each subsystem as well as new values of the boundary variables on their borders will be used on next iteration in order to calculate the gradient of LI and matrices of the second derivatives in the system (31). 3.2. Solution of the problems with dependent boundary variables General concept of the FM method also can be applied to the solution of the optimization problems in power systems, which cannot be presented as the problems with independent boundary variables. In these problems apart from conditions of optimality for boundary variables it is necessary to consider conditions which should be imposed on the values of the second boundary variables (voltage angles and magnitudes on the boundary nodes) if we want to obtain a valid solution for the real and reactive power flows at the borders of subsystems. They can be presented as the following equations: hbI  hbJ ¼ 0 U bI  U bJ ¼ 0

ð37Þ

showing that voltage angles and magnitudes in the boundary node b found in adjacent subsystems I and J should be equal. The SCE should include in this case supplementary equations derived from these equalities. The whole SCE for determination of optimal power flows can be obtained in this problem from the expanded Lagrange function, having the form, presented in [3]. In this function supplementary terms and multipliers are added to the original Lagrange function (1) in order to take into account the boundary matching constraints (37). The number of these additional terms and multipliers in a problem should be equal to the number of these constraints. Here we present this function in general form for a system, consisting of N subsystems with nb boundary nodes between subsystems. The matching conditions, which are

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taken into account in this function include both equalities of the voltage angles and voltage magnitudes in the boundary nodes, when they are determined in adjacent subsystems. The expanded Lagrange function looks in this case as follows: XX X t LS ¼ F iI ðP iI Þ þ kiI giI ðxI ; xbI ; uI ; pI Þ I

þ

iI

X I

I

ktbI gbI ðxI ; xbI ; uI ; pI ; P bI Þ 

J

þ rt gb xIin ; xIb ; xJin ; xb

ð38Þ

where r is the vector of additional Lagrange multipliers and gb is the set of boundary equality constraints. The total number of these constraints and their composition can be determined from the following considerations. For each intersection between subsystems the number of constraints for voltage angles in boundary nodes should be equal to the number of boundary nodes minus one. It means that in each intersection only differences of voltage angles of one boundary node and of all other boundary nodes should be taken into account. The number of constraints for voltage magnitudes is equal to the total number of boundary nodes in the model. The set of added terms in (38) does not change the problem because at any solution these terms are equal to zero. The hierarchical FM algorithm which we present now is based on the one hand on the general idea of the Newton second-order method and on the other hand on the general organization of the FM algorithms. In accordance with the principle of optimality, presented above, the first step in formation of the upper level system of equations, i.e. SCE, in this case is to find expressions for the derivatives of the function (38) with respect to the upper level variables, P b , Qb and r and equate them to zero. It brings about the following system of equations: r oPogb

rLS b ¼ kbI  kbJ þ ¼0  I I J J rLr ¼ gb xin ; xb ; xin ; xb ¼ 0

ð39Þ

Upper equation in this system consists of a vector of derivatives of the Lagrange function with respect to the boundary variables, i.e. P b and Qb . Lower equation is the gradient of the Lagrange function with respect to the additional Lagrange multipliers, forming the vector r. Actually this equation consists of the compatibility equation (37) themselves. The second step in formation of the SCE consists in substitution for k and k of their expressions from the FCs of subsystems. It should be done also for the voltage angles hb and voltage magnitudes U b in boundary nodes. General organization of this FM algorithm is the same as that of the FM algorithm, intended for solution of the problems with independent boundary variables, Eqs. (26)–(36). It consists of the following steps. (1) Determination of the FCs of subsystems

The system of equations for one subsystem has in this algorithm a slightly different form than in (30) because the added terms in (38) should be taken into account. So it looks as shown here  T  T þ og k þ ogoxb kb ¼ 0 rLx ¼ oF ox ox  T  T þ og k þ ogoub r ¼ 0 rLu ¼ oF ou ou rLk ¼ gðx; u; pÞ ¼ 0 h iT h iT ogb og oF rLxb ¼ ox þ k þ kb ¼ 0 ox oxb b b

ð40Þ

rLkb ¼ gb ðx; xb ; u; S b Þ ¼ 0 rLr ¼ g0b ðu; S b Þ From this system a system of equations should be obtained, representing a subsystem in the second-order Newton method. In this case such system of equations is    Dx       rLx      Du    rL    u      2  Dk    rLk   o L  ¼  Dxb  ð41Þ    2   oV   rLxb   Dk    b  rLkb       Dr   rL    r  DP b  It is necessary to make the same operations in order to get the system (41) as those which were necessary in order to get the system (31). Elimination of internal variables from this system brings it to the same form as (32). The FCs of subsystem will be obtained from the lower part of the transformed systems, obtained from (41). These FCs are Dkb ¼ ADS b  a0

ð42Þ

Dg0b ¼ B1 DS b  b0

ð43Þ

(2) Formation and solution of the SCE. Substitution of the FCs (42) and (43) of subsystems into optimality equations of upper level (39) (written in incremental form) results in formation of the following SCE    DS  ¼b ð44Þ jAj Dr  The increments of variables in the vectors DP b , DQb and Dr, which should be found from this system, are the increments of active and reactive power flows crossing the borders of subsystems and the increments of the additional Lagrange multipliers. (3) Determination of the increments of internal variables in subsystems. Subvectors of the vectors DP b , DQb and Dr, found in the previous step, are to be substituted into the systems of equations, obtained after the Gaussian elimination of

V.A. Makeechev et al. / Electrical Power and Energy Systems 30 (2008) 415–427

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internal variables from the systems (41). It should be done for all subsystems in order to calculate the increments of internal variables using successive back substitutions in the upper part equations of the transformed systems, obtained from (41). It is important to note that the numerical values of the increments Dk, Du, and Dx in subsystems are influenced in this algorithm not only by the values DP b and DQb but also by the values Dr. 4. General structure of the FM algorithms All hierarchical algorithms, based on the FM method, have important common features, determinative for their properties and efficiency. These features are (1) Systems of equations, representing each subsystem in these algorithms, are formed in such way that the number of variables in these systems is greater the number of equations. These systems are formed in the same way as usual systems of equations in Newton method for a system as a whole in every respect with the exception that supplementary variables are introduced which are considered as unknown. These supplementary variables in this class of problems are control boundary variables (P b and Qb ). (2) System of equations for each subsystem should be written in such way that equations containing control boundary are placed in the lower equations of these systems. Gaussian elimination of internal variables in these linear systems brings each of them to such form that in the lower equations of each system only control boundary variables and variables, representing subsystem as a whole in the equations of optimality of upper level. These variables are k, kb , hb , U b . (3) From the lower equations of these transformed systems FCs of subsystems are obtained representing functional dependence of these variables upon the control boundary variables of subsystems. (4) Conditions of optimality of upper level should be obtained from expressions for derivatives of the Lagrange functions (representing a power system as a set of subsystems) with respect to control boundary variables. (5) Substitution of FCs of subsystems into optimality equations of upper level gives the SCE, from which increments of control boundary variables should be found. (6) Increments of internal variables in subsystems are obtained by back substitution of the values of boundary variables in each subsystem. (7) Within one of these iterative cycles several iterations can be introduced within one or more subsystems using any basic algorithm. All optimization algorithms presented in this paper perform in the same way when implemented on parallel and

Fig. 2. General structure of the hierarchical FM algorithms. (1) Calculation of the FCs of subsystems. (2) Formation and solution of the SCE. (3) Calculation of the internal variables of subsystems.

distributed computer systems. These algorithms consist of three main steps presented on the Fig. 2. Steps 1 and 3 in these algorithm are performed in parallel on computers of lower level, pertaining to subsystems. Formation and solution of the upper level problem (SCE) is executed on the computer of upper level (server). This determines the general organization of processing and data transfer in the implementation of these algorithms on computer systems. 5. Conclusions In this paper, the foundations of the FM optimization method have been presented. Unlike the decomposition methods [4–8] the FM method is based on the solution of the upper level problem in the hierarchical model: determination of the boundary variables. The main point in the solution of this upper level problem is the reduction of the whole system of equations to the system of equations, having the order of the number of ties between subsystems. This upper level model is built on the condition that all internal constraints within subsystems are observed. Owing to this fact the results of the solution in these hierarchical models are the same as in the conventional one level models. Therefore the convergence properties and final results of the FM algorithms are identical to those of the basic one level optimization algorithms. At the same time they have all advantages of the parallel and distributed processing. Application of the FM theory resulted in creation of the efficient algorithms for solution of the power system problems presented in this paper. Appendix A Below we present an example of FM method application for the case when the boundary variables can be considered as independent variables.

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The algorithm of Eqs. (17)–(25) is applied to the example of a four station system. The model of this system is shown in Fig. 3. It includes two interconnected subsystems having stations parameters presented below, see Eq. (5). a1 ¼ 1:25  102 ;

b1 ¼ 5

a2 ¼ 2  102 ;

b2 ¼ 2

a3 ¼ 2  102 ;

b3 ¼ 2

a4 ¼ 1:25  102 ;

b4 ¼ 5

Loads in subsystems areP LI ¼ 350 MW P LJ ¼ 125 MW: A simplified expression for losses in the line 1–0 is p ¼ 0:001P 21 . Solution of the economic dispatch problem by the FM method looks as follows in this case (1) On the first step we find solution of this problem without taking into account the network losses. System of equations for subsystem I, formed in accordance with (3) and (4) may be written as 2:5  102 P 1 þ 5  kI ¼ 0 4  102 P 2 þ 2  kI ¼ 0 350  P 1  P 2  P b ¼ 0

ðA1Þ

By elimination of internal variables from the upper equations of this system the following FC is obtained kI ¼ ð600  P b Þ=65

ðA2Þ

2P b ¼ 225 From this equation P b ¼ 112:5 MW.Back substitution of this value of P b into (A1) and (A3) gives kI ¼ 7:5;

P 1 ¼ 100;

kJ ¼ 7:5;

P 3 ¼ 137:5;

P 2 ¼ 137:5 P 4 ¼ 100

(2) On the second step optimal dispatch problem is solved for subsystems I and J with network losses taken into account when it is assumed that the power flow between subsystems is equal to its value found on the previous step.System of equations which should be solved in this case in subsystem I looks as follows: 2:5  102 P 1 þ 5  kð1  0:002P 1 Þ ¼ 0 4  102 P 2 þ 2  k ¼ 0; 350 þ 0:001P 21  P 1  P 2  P b ¼ 0

ðA5Þ

Substituting the values of variables found on the previous step as starting values and using Newton method for solution of this problem in the subsystem I, we obtain the following system of equations on the first iteration. ð2:5  102 þ 0:002kI ÞDP 1  ð1  0:002P 1 ÞDkI ¼ 1:5; 4  102 DP 2  Dk ¼ 0; DP 1 þ DP 2  0:002P 1 DP 1 ¼ 10

ðA6Þ

System of equations for subsystem J can be presented as

Solution of this linear system gives the following values

4  102 P 3 þ 2  kJ ¼ 0

DP 1 ¼ 18;

2

2:5  10 P 4 þ 5  kJ ¼ 0 125  P 3  P 4 þ P b ¼ 0

Dk ¼ 0:975

Then we have after the first iteration ðA3Þ

The FC of subsystem J found in the same way as (A2), is kJ ¼ ð375 þ P b Þ=65

DP 2 ¼ 24; 375;

ðA4Þ

Substituting for kI and kJ from (A2) and (A4) gives the following SCE

P 1 ¼ 82;

P 2 ¼ 161; 875;

k ¼ 8:475

Substituting these values into left-hand side of Eq. (A5) we find that the mismatches in these equations are small enough to accept these values as optimal.Optimal values of variables in the subsystem J are the same as on the previous step. (3) On this step the FCs of subsystems I and J should be found, which will be used in formation of the SCE. For this purpose systems it is necessary to form equations for subsystems I and J, representing them in Newton method when the boundary variable P b changes, but optimality equations within subsystems remain satisfied. Such system for subsystem I can be written as ð2:5  102 þ 0:002kI ÞDP 1  ð1  0:002P 1 ÞDkI ¼ 0; 4  102 DP 2  Dk ¼ 0; DP 1 þ DP 2  0:002P 1 DP 1 þ DP b ¼ 0

ðA7Þ

Elimination of internal variables P 1 and P 2 gives in lower equation the following FC Fig. 3. Power system model.

DkJ ¼ DP b =41:653

ðA8Þ

V.A. Makeechev et al. / Electrical Power and Energy Systems 30 (2008) 415–427

The FC for subsystem J can be obtained in the same way and is DkJ ¼ DP b =65

ðA9Þ

kIb ¼ 0:02P b System of equations obtained for subsystem J from (31) consists of the following equations:

We have also to find mismatch in optimality equation (18), corresponding to the values of variables found on the previous step. Since the derivatives opI /oP b and opJ / oP b in this system are equal to zero, this equation becomes

oL=ok ¼ 4:34ðhL Þ  P 2 ¼ 0

kI  kJ ¼ 0

oL=ohb ¼ 8:68kL þ 8:68kb ¼ 0

ðA10Þ

Substituting to this equation the values kI ¼ 8:475 and kJ ¼ 7:5 gives Ds ¼ 0:975. Then the following SCE can be formed in accordance with (13) ð1=41:653  1=65ÞDP b ¼ 0:975

ðA11Þ

Solving this equation and finding the new value of P b we obtain DP b ¼ 24:745;

P b ¼ 112:5 þ 24:745 ¼ 137:245

oL=oP 1 ¼ 0:02P 1  k1 ¼ 0 oL=ohL ¼ 4:34k1 þ 4:34kL þ 8:68kL  8:68kb ¼ 0 oL=okb ¼ 8:68hb  8:68hL  P b ¼ 0

ðB2Þ

(This system was obtained from (31) using linearization of characteristics P ¼ f ðhÞ for lines with values h and k instead of Dh and Dk in the equations. It is supposed that h are measured in grad in these equations). Applying the Gaussian elimination of internal variables to this system we obtain in the last equation 173:6  50kb  P b ¼ 0

kJb ¼ 0:02P b þ 3:472

The example we present below illustrates how the algorithm (26)–(36) can be applied for solution of economic dispatch problem in the system, consisting of two subsystems, shown in Fig. 4. Generator 1 in the subsystem I is considered as a generator at the reference bus. Parameters of network and stations in this system are as follows: xbL ¼ 20;

F 2 ¼ 0:01p22 ;

oL=okL ¼ 4:34hL þ 8:68ðhL  hb Þ  ð173:6Þ ¼ 0

It follows from this equation that the FC of this subsystem is

Appendix B

x1b ¼ 20;

425

xL2 ¼ 40; F 1 ¼ 0:01P 21 ;

P L ¼ 173:6 MW

We consider in this example the case when all lines in the network are presented with their reactances. System of linear equations, obtained from (31) for subsystem I should be written in this case as 2

oL=ohb ¼ 0:02  ð8:68Þ hb þ 8:68kb ¼ 0; oL=okb ¼ 8:68hb þ P b ¼ 0

ðB1Þ

where hb is the voltage angle in the boundary node. Eliminating hb in the lower equation we obtain the following FC of the subsystem I.

Substituting the expressions from the FC of subsystems into (28) we obtain a SCE 0:04P b ¼ 3:472 which gives P b ¼ 86:8 Values of internal values of subsystems can be obtained by back substitution of this value of P b into the systems (B1) and (B2). It is necessary to note that any node in subsystem J can be chosen as basic in subsystem J in this formulation of optimization problem. In usual applications of this algorithm when characteristics of lines are not linearized, only increments of variables should be calculated on each step and iterative process should go on until convergence conditions are satisfied. If not only the values of reactances x, but also the values of resistances r in the lines are taken into account in the nodal equations, it is possible to solve by this algorithm the OPF problems with network losses considered. If simplified loss expressions in subsystems I and J are applied in this example, calculation of optimal power flow P b can be organized in the same way as in the previous example. Appendix C

Fig. 4. Power system model.

The examples presented below illustrate application of the algorithms of Eqs. (37)–(44), intended for solution in the power systems in which there are more than one boundary node between subsystems (see Fig. 5).

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phase angles in the boundary nodes b1 and b2 should be the same when it is determined in the subsystem I and subsystem J, see Eq. (37). This equation can be easily formed from the FCs of subsystems, representing this angle as a function of boundary variables P b1 and P b2 . These FCs should be obtained together with the FCs (43) in the process of elimination of internal variables. In this case these FCs for two subsystems are h12I ¼ 0:071P b1  0:213P b2 þ 28:4

ðC4Þ

h12J ¼ 0:213P b1 þ 0:071P b2  28:4

ðC5Þ

The supplementary equation obtained from these FCs together with the optimality equation form a system Fig. 5. Power system model with two boundary nodes.

0:284P b1  0:284P b2 þ 56:4 ¼ 0;

Parameters of network and stations in this system are as follows: x1 ¼ 25; x5 ¼ 12:5;

x2 ¼ 12:5;

x3 ¼ 12:5;

x6 ¼ 12:5;

F 2 ðP 2 Þ ¼ 0:01P 22 ;

x4 ¼ 25;

F 1 ðP 1 Þ ¼ 0:01P 21 ;

P LI ¼ 200 MW;

P LJ ¼ 200 MW

System (40) for subsystem I looks in this case as shown below. oL=ohL ¼ 0:98hL þ 21kL þ 1:96hb1  14kb2 oL=okL ¼ 21hL  14hb2 þ 200 oL=ohb1 ¼ 1:96hL þ 3:92hb1 þ 14kb1 oL=ohb2 ¼ 14kL þ 14kb2 oL=okb1 ¼ 14hb1  P b1 oL=okb2 ¼ 14hL þ 14hb2  P b2

ðC1Þ

kb1 ¼ kb2  50kb1 þ 200  P b1  P b2 ¼ 0 Then the FC of this subsystem is ðC2Þ

The FC of the subsystem J calculated in the same way is kb ¼ 4 þ 0:02P b1 þ 0:02P b2

ðC6Þ

Solution of this system gives P b1 ¼ 100P b2 ¼ 100 Here we present also solution of optimization problem for this system illustrating general approach, based on application of the expanded Lagrange function (38). It is assumed in this example that losses are taken into account and that voltage magnitudes are constant. So only the matching conditions for voltage angles should be applied and they can be reduced in this example to only one equality (for the difference between voltage angles in the two boundary nodes). The expanded Lagrange function looks in this case as follows: LS ¼ ½F 1 ðP 1 Þ þ kI ðP LI  P 1 þ pI  P b1  P b2 Þ

Applying the Gaussian elimination to (C1) we obtain in two lower equations

kb ¼ 4  0:02P b1  0:02P b2

P b1 þ P b2 ¼ 0

ðC3Þ

Substituting these FCs to optimality equation (28) gives P b1 þ P b2 ¼ 0 It means that using the system of optimality equation (28) it is possible in this case to determine the total optimal power flow between subsystems, but not the optimal power flow in each line. Yet using the SCE, based on the system (39) it is possible to solve this problem also in this particular case. In this system one more equation is introduced in this case, found from the condition that the difference between

þ ½F 2 ðP 2 Þ þ kJ ðP LJ  P 2 þ pJ þ P b1 þ P b2 Þ þ rðhI  hJ Þ

ðC7Þ

where r is the additional Lagrange multiplier, hI and hJ are the difference angles, found in the subsystem I and J. Parameters of network and stations in this example are the following: x1 ¼ x2 ¼ x3 ¼ x4 ¼ x5 ¼ x6 ¼ 25r5 ¼ r6 ¼ 5; F 1 ¼ 0:025P 21 ;

F 2 ¼ 0:01P 22 ;

U 1 ¼ U 2 ¼ U 3 ¼ U 4 ¼ 100 kV The loads in subsystems are P 3 ¼ 200 MW;

P 4 ¼ 150 MW

On the first step as in previous examples optimal values of boundary and internal variables were found without taking losses into account. These values of variables are P b1 ¼ 8:4; P b2 ¼ 108:4; P 2 ¼ 250 kI ¼ 5kJ ¼ 5

P 1 ¼ 100;

V.A. Makeechev et al. / Electrical Power and Energy Systems 30 (2008) 415–427

On the next step optimal values of internal variables within subsystems were found with losses in subsystems considered and fixed values of boundary variables found on the previous step. It gave P 1 ¼ 105:88;

P 2 ¼ 255:88;

kI ¼ 5:294;

kJ ¼ 5:1176

On the last stage the mismatches of optimality equation (39) were calculated and a system for determination of increments of boundary variables (43) for compensation of these mismatches formed. It looks on the first iteration as shown below 0:07DP b1 þ 0:0667DP b2 þ 0:0075Dr ¼ 0:1764 0:0667DP b1 þ 0:0747DP b2  0:0065Dr ¼ 0:955 0:0075DP b1  0:0065DP b2 ¼ 0:028

ðC8Þ

Solution of this system gives DP b1 ¼ 0:8216

DP b2 ¼ 5:2577

Dr ¼ 77:9226

New values of boundary variables according to (35) were found after it and new values of internal variables in subsystems. P b1 ¼ 9:22;

P b2 ¼ 103:145;

r ¼ 77:9226

After two iterations of this kind the mismatches in optimality equations became negligible and solution was found.

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In all examples presented above solution found by the FM algorithm was the same as solution found by basic algorithm for a system as a whole. References [1] Venikov VA, Soukhanov OA. Cybernetic models of electrical systems. Moscow: Energoizdat; 1982. p. 328 [in Russian]. [2] Soukhanov OA, Shil SC, Kovalev VD, Kovalev SV. Functional modeling algorithms for fast solution of electrical power systems steady-state and dynamic problems. In: Proceedings of the second international conference on digital power systems simulators, Montreal, Quebec, Canada; 28–30 May, 1997. p. 163–7. [3] Soukhanov OA, Shil SC. Application of functional modeling to the solution of electrical power systems optimization problems. Int J Electric Power Energy Syst 2000(2). [4] Cohen G. Auxiliary problem principle and decomposition of optimization problems. J Optim Theory Appl 1980;32(3):277–305. [5] Kim BH, Baldick R. Coarse-grained distributed optimal power-flow. IEEE Trans Power Syst 1997;12(2):932–9. [6] Kim BH, Baldick Ross. A comparison of distributed optimal power flow algorithms. IEEE Trans Power Syst 2000;15(2). [7] Shahidehpour Mohammad, Wang Yaoyu. Communication and control in electric power systems. Application of parallel and distributed processing. John Wiley & Sons Inc.; 2003. [8] Pandelis N, Biskas A, Bakirtzis G, Macheras Nikos I, Pasialis Nikolaos K. A decentralized implementation of DC optimal power flow on a network of computers. IEEE Trans Power Syst 2005;20(1). [9] Soukhanov OA, Sharov YV. Hierarchical models in power system analysis and control. Moscow: MPI Publishing House; 2007. 310pp [in Russian].