FUZZY SET BASED MULTIOBJECTIVE ALLOCATION OF RESOURCES AND ITS POWER ENGINEERING APPLICATIONS R.C. Berredo Department of Engineering and Coordination of Distribution Expansion Planning Minas Gerais State Energy Company Av. Barbacena, 1200, 30190-131, Belo Horizonte, MG, Brazil
[email protected] R.L.J. Carnevalli Department of Engineering and Coordination of Distribution Operation Minas Gerais State Energy Company Av. Barbacena, 1200, 30190-131, Belo Horizonte, MG, Brazil
[email protected] P.Ya. Ekel J.G. Pereira Jr. Graduate Program in Electrical Engineering Pontifical Catholic University of Minas Gerais Ave. Dom José Gaspar, 500, 30535-610 - Belo Horizonte - MG, Brasil
[email protected] [email protected] Abstract – This paper presents results of research into the use of the Bellman-Zadeh approach to decision making in a fuzzy environment for solving multiobjective optimization problems. Its application conforms to the principle of guaranteed result and provides constructive lines in obtaining harmonious solutions on the basis of analyzing associated maxmin problems. The use of the BellmanZadeh approach has served as a basis for solving problems of multiobjective allocation of resources (or their shortages) and developing a corresponding adaptive interactive decision making system (AIDMS1). Its calculating kernel permits one to solve maxmin problems using procedures of a nonlocal search (modification of the Gelfand's and Tsetlin's "long valley" method). The principles of their construction provide solutions that are, indeed, Pareto. The AIDMS1 includes procedures for constructing linguistic variables to take into account conditions that are difficult to formalize as well as procedures for forming and correcting vectors of importance factors for goals. The use of these procedures permits one to realize an adaptive approach to processing information of a decision maker to provide successive improving the solution quality. The application of the Bellman-Zadeh approach is illustrated by considering problems of load management in power systems and tuning models related to fuzzy control of system voltage and reactive power (the procedures of tuning are based on a deep connection between production rules used in fuzzy control technology and multiobjective optimization models). Keywords – Multiobjective optimization, Bellman-Zadeh approach, Allocation of resources 1. Introduction Investigations of recent years show the possibility and advisability of applying fuzzy set theory [1,2] for considering and overcoming diverse kinds of uncertainty. Its use in problems of optimization character offers advantages of both fundamental nature (the possibility of validly obtaining more effective, less "cautious solutions") and computational character [3,4]. The uncertainty of goals is an important kind of uncertainty related to a multiobjective character of many optimization problems. When analyzing multiobjective models, a vector of objective functions
F ( X ) = {F1 ( X ),..., Fq ( X )} is considered, and the problem consists in simultaneous optimizing all objective functions, i.e.,
F p ( X ) → extr, X ∈L
p = 1,..., q ,
(1)
where L is a feasible region in R n . The first step in solving the problem (1) is associated with determining a set of Pareto solutions Ω ⊆ L [5]. This step is useful, however it does not permit one to obtain unique solutions. It is necessary to choice a particular Pareto solution on the basis of information of a decision maker (DM). There are three approaches to using this information [6]: a priori, a posteriori, and adaptive. The most preferable approach is the adaptive one. When using this approach, the procedure of successive improving the solution quality is realized as a transition from X α0 ∈ Ω ⊂ L to X α0 +1 ∈ Ω ⊂ L with considering information
I α of DM. The solution search may be presented in the following form:
Iα I1 α −1 ω −1 X 10 , F ( X 10 ) → X α0 , F ( X α0 ) → → X ω0 , F ( X ω0 ) . ... I→ ... I
(2)
The process (2) serves for two types of adaptation: computer to preferences of DM and DM to the problem. The first type of adaptation is based on information received from DM. The second type of Iα adaptation is realized as a result of carrying out several steps X α0 , F ( X α0 ) → X α0+1 , F ( X α0+1 ) , which permit DM to understand the correlation between own needs and possibilities of their satisfaction by the model (1). When analyzing multiobjective problems, it is necessary to solve some questions related to normalizing criteria, selecting principles of optimality, and considering priorities of criteria. Their solution and, therefore, development of multiobjective methods is carried out in the following directions [7-9]: scalarization techniques, imposing constraints on criteria, utility function method, goal programming, and using the principle of guaranteed result. Without discussion of these directions, it is necessary to point out that an important question in multiobjective optimization is the quality of obtained solutions. The quality is considered as high if levels of satisfying criteria are equal or close to each other (harmonious solutions) if we do not differentiate the importance of objective functions [10]. From this point of view, it should be recorded the validity and advisability of the direction related to the principle of guaranteed result. The lack of clarity in the concept of "optimal solution" is the basic methodological complexity in solving multiobjective problems. When applying the Bellman-Zadeh approach to decision making in a fuzzy environment [1,2,11], this concept is defined with reasonable validity: the maximum degree of implementing goals serves as a criterion of optimality. This conforms to the principle of guaranteed result and provides constructive lines in obtaining harmonious solutions. The Bellman-Zadeh approach permits one to realize an effective (from the computational standpoint) as well as rigorous (from the standpoint of obtaining solutions X 0 ∈ Ω ⊆ L ) method of analyzing multiobjective models [3,12]. Finally, its use allows one to preserve a natural measure of uncertainty in decision making and to take into account indices, criteria, and constraints of qualitative character. The present paper is dedicated to applying the Bellman-Zadeh approach to solving a problem of multiobjective allocation of resources (or their shortages) and its applications to power engineering problems. In particular, the following problems are considered: multiobjective power and energy shortage allocation as applied to load management in power systems and tuning fuzzy models related to fuzzy control of system voltage and reactive power.
2. Multiobjective resource allocation When using the Bellman-Zadeh approach, each objective function F p ( X ) is replaced by a fuzzy objective function or a fuzzy set
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A p = { X , µ A ( X )}, X ∈ L,
p = 1,..., q ,
p
(3)
where µ Ap ( X ) is a membership function of A p [1,2]. A fuzzy solution D with setting up the fuzzy sets (3) is turned out as a result of the intersection q
D = I A p with a membership function p =1
µ D ( X ) = min µ A ( X ),
X ∈L.
p
p =1,..,q
(4)
Its use permits one to obtain the solution proving the maximum degree max µ D ( X ) = max min µ A ( X )
(5)
p
X ∈L p =1,..,q
of belonging to the fuzzy solution D and reduced the problem (1) to X 0 = arg max min µ A ( X ) .
(6)
p
X ∈L p =1,...,q
To obtain (6), it is necessary to build membership functions µ A ( X ), p = 1,..., q reflecting a degree of p
achieving "own" optima by the corresponding Fp ( X ), X ∈ L , p = 1,..., q . This condition is satisfied by the use of membership functions
F p ( X ) − min F p ( X ) X ∈L µ A (X ) = p max Fp ( X ) − min F p ( X ) X ∈L X ∈L for maximized objective functions or by the use of membership functions
λp
max F ( X ) − F ( X ) p p µ A (X ) = p max F p ( X ) − min F p ( X ) X ∈L X ∈L
(7)
λp (8)
for minimized objective functions. In (7) and (8), λ p , p = 1,..., q are importance factors for the corresponding objective functions. The construction of (7) or (8) demands to solve the following problems: F p ( X ) → min ,
(9)
F p ( X ) → max ,
(10)
X ∈L
X ∈L
providing X 0p = arg min F p ( X ) and X p00 = arg max F p ( X ), respectively. X ∈L
X ∈L
Thus, the solution of the problem (1) on the basis of the Bellman-Zadeh approach demands analysis of 2q + 1 monocriteria problems (9), (10), and (5), respectively. Since the solution X 0 is to belong to Ω ⊆ L , it is necessary to build q
µ D ( X ) = ∧ µ A ( X ) ∧ µ π ( X ) = min { min µ A ( X ), µ π ( X )} , p =1
p
p =1,..,q
p
(11)
where µ π ( X ) = 1 if X ∈ Ω and µ π ( X ) = 0 if X ∉ Ω . The procedures for solving the problem (5), discussed in Section 3, provide the line in obtaining X 0 ∈ Ω ⊆ L in accordance with (11). Thus, it can be said about equivalence of µ D ( X ) and µ D ( X ) .
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It permits one to give up the necessity of implementing a cumbersome procedure for building the set Ω⊆L. Finally, the existence of additional conditions (indices, criteria, and/or constraints) of qualitative character, defined by linguistic variables [1,2], reduces (6) to X 0 = arg max min µ A ( X ) , X ∈L p =1,..., q + s
p
(12)
where µ A ( X ), X ∈ L, p = q + 1,..., s are membership functions of fuzzy values [1,2] of linguistic p
variables which reflect additional conditions (criteria, and/or constraints of qualitative character).
3. Multiobjective power and energy shortage allocation as applied to load management Different conceptions of load management (for example, considered in [13,14]) are united by the following: elaboration and realization of control actions are performed on the two-stage bases. On the level of energy control centers, optimization of allocating power and energy shortages (natural or associated with the economic feasibility of load management) is carried out for different levels of territorial, temporal, and situational hierarchy of planning and operation. This allows one to draw up tasks for consumers. On their level, control actions are realized in accordance with these tasks. Thus, the questions of power and energy shortage allocation are of a fundamental importance in a family of load management problems. These questions should be considered not only from the economical and technological points of view, but from the social and ecological points of view as well. Besides, it is necessary to account for considerations of creating incentive influences for consumers. Considering this, it should be pointed out that methods of power and energy shortage allocation, based on fundamental principles of allocating resources [15], have drawbacks [14]. Their overcoming is possible on the basis of formulating and solving the problems within the framework of multiobjective optimization models. This permits one to consider and to minimize diverse consequences of power and energy shortage allocation and to create incentive influences for consumers. Substantial analysis of the problems of power and energy shortage allocation, systems of economics management as well as real, readily available reported and planned information has permitted the construction of a general set of goals. The list includes 17 types of goals. Without listing all of them, it is possible to indicate the following goals: 1. Primary limitation of consumer with more low cost of produced production and/or given services on consumed 1 kWh of energy (achievement of a minimal drop in total produced production and/or given services); 4. Primary limitation of consumers with a more low level of payment in the state budget and/or a more low level of lease payment for basic production resources (funds) on consumed 1 kWh of energy; 5. Primary limitation of consumers with a more low level of payment in the state budget for working resources on consumed 1 kWh of energy. 15. Primary limitation of consumers with a more low value of the demand coefficient (primary limitation of consumers with greater possibilities of production out the peak time); 16. Primary limitation of consumers with a more low duration of using maximum load in twentyfour hours (primary limitation of consumers with greater possibilities in transferring maximum load in the twenty-four hours interval); 17. Primary limitation of consumers with a more low duration of using maximum load in month (quarter, year) (primary limitation of consumers with greater possibilities in transferring maximum load in the month (quarter, year) interval). The general set of goals is sufficiently complete because is directed to decreasing diverse negative consequences for consumers and creating incentive influences for them. This set is universal because
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can serve as the basis for building models at different levels of load management hierarchy by aggregation of information and posterior decomposition of the problems in accordance with different indices. The concrete list of goals can be defined at every case by DM, who can be individual or group (for example, it may be leading organizations of the country or state, a council of directors of enterprises, etc., whose decision regarding the concrete list of goals can be considered as the legislative one for the corresponding level). From the formal standpoint, an achievement of the goals of power and energy shortage allocation is associated with optimizing linear objective functions n
F p ( X ) = ∑ c pi x i .
(13)
i =1
However, our experience shows that the use of these functions can lead to very "strict" solutions. Considering this, it is possible to use objective functions of the following type: n c d pi pi (14) Fp ( X ) = ∑ d i =1 pi − x i to obtain more flexible solutions. The use of damage characteristics for consumers as the objective functions is associated with minimizing F p ( X ) = a pi x i2 + b pi xi . (15) Finally, the consideration of integrated damage for a group of consumers demands to minimize n
F p ( X ) = ∑ ( a pi xi2 + b pi xi ) .
(16)
i=1
Taking into account the possibility to apply the objective functions of the types (13)-(16), the problem (1) is to be solved with the box-triangular structure of a feasible set: n
∑ xi = A ,
(17)
i =1
0 ≤ xi ≤ Ai ,
i = 1,..., n ,
(18)
where X = ( x1 ,..., x n ) is the sought for a vector of limitations; Ai is the permissible value of limitation for the ith consumer; A is a total value of limitations for all consumers participating in load management. To describe a general scheme of solving the problems formalized within the framework of the model (1), (17), and (18), it is advisable to introduce into consideration a linguistic variable Q - Limitation for Consumer to provide DM with the possibility to consider conditions that are difficult to formalize. Thus, the general scheme assumes the availability of a procedure for building a term-set T (Q ) of the linguistic variable and membership functions for its fuzzy values. In addition, if the solution X α0 with the values F p ( X α0 ) or µ Ap ( X α0 ) , p =1,..., q is not satisfactory, DM has to have the possibility to correct it, passing to X α0+1 with changing the importance of one or more objective functions. Thus, the general scheme also assumes the availability of the procedure for forming and correcting the vector Λ = (λ1 ,..., λ q ) of importance factors. The general scheme of solving the problem (1), (17), and (18), which has served for implementing an adaptive interactive decision making system (AIDMS1), is associated with the following sequence of blocks: 1. Solution of the problems (9) and (10) to obtain X 0p , p = 1,..., q and X 00 p , p = 1,..., q , respectively. 2. Construction of the membership functions defined by (7) or (8).
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3. Construction of an initial vector λ = (λ1 ,..., λ q ) of the importance factors. 4. Analysis of the availability of initial conditions defined by the linguistic variables. If these conditions are lacking, then go to Block 8; otherwise go to Block 5. 5. Verification of compatibility of the conditions and, if necessary, their interactive correction. 6. Solution of the problem (5) with the goal to obtain X α0 defined by (12). 7. Analysis of the current solution X α0 . If DM is satisfied by the solution, then go to Block 10; otherwise go to Block 8, taking α := α + 1 . 8. Correction of the vector λ = (λ1 ,..., λ q ) of the importance factors. 9. Insertion of additional conditions defined by the linguistic variables; then go to Block 5. 0 10. Calculations are completed because the solution X is obtained. The main functions of a calculating kernel of the AIDMS1 are associated with obtaining X 0p , 0 p = 1,..., q and X 00 p , p = 1,..., q defined by solving the problems (9) and (10) and with obtaining X
defined by (6). The solution of the problems (9) and (10) creates no difficulties. The maximization of (11) is based on a non-local search that is a modification of the Gelfand's and Tsetlin's "long valley" method [16]. Experimental calculations show that variables of (4) can be divided into two groups: inessential and essential. The change of inessential variables leads to essential variations of (4). The change of essential variables leads to inessential variations of (4). Thus, a structure of (4) may be considered as a multidimensional "long valley". This circumstance, if we use direct search methods [9], requires the ascent from different initial points X 0p (Pareto points), if we must minimize F p ( X ) , or X 00 p (Pareto points), if we must maximize F p ( X ) , to find the most convincing solution X 0 . This explains the use of a non-local search, which can be presented as follows: 1. The sequence { X (l ) }, l = 1,..., q is built from points X 0p , if we must minimize F p ( X ) , or X 00 p , if we must maximize F p ( X ) , obtained as a result of execution of Block 1 of the general scheme. This sequence has the following property: min µ Ap ( X (l ) ) ≥ min µ Ap ( X (l +1) ), l = 1,..., q - 1 . 1≤ p ≤ q
2. The local search of X point X
(1) 0
0
1≤ p ≤ q
is carried out from X
with corresponding µ Ap ( X
(1) 0
(1)
(l = 1) . As a result of this search, we obtain a
), p = 1,..., q .
3. The local search of X is carried out from X (l +1) . As a result of this search, we obtain a point X (l +1) 0 with corresponding µ Ap ( X (l +1)0 ), p = 1,..., q . 0
4. Analysis is executed: a) if X (1)0 ≠ X (l +1)0 , then go to Operation 5; b) if X (1)0 = X (l +1)0 for l ≠ q − 1 , then go to Operation 3, taking l := l + 1 ; c) if X (1)0 = X (l +1)0 = X ( q )0 , then go to Operation 8, taking X 0 = X (1)0 . 5. A line between points X (t ) 0 and X (t +1) 0 is "constructed" to generate points X s(t ,t +1) , s = 1, 2, 3 (see Figure 1). Among them (if they are acceptable from the point if view of the constraints (18)), a point X (t ,t +1)0 = arg max min µ Ap ( X s(t ,t +1) ) is selected to define a direction for a future search. t
1≤ p ≤ q
6. The next local search of X 0 is carried out from X (t ,t +1)0 . As a result of this search, we obtain a point X (t +2)0 (see Figure 1). 7. Analysis is executed: if three "last" points X (t )0 , X (t +1)0 , and X (t +2)0 differ on min µ Ap ( X (t ) 0 ) , 1≤ p ≤ q
min µ Ap ( X
1≤ p ≤ q
( t +1) 0
) , and min µ Ap ( X 1≤ p ≤ q
( t + 2) 0
) less than the accuracy desired, then go to Operation 8, taking
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X 0 = arg max [ min µ Ap ( X (t +2) ), min µ Ap ( X (t + 2) ), min µ Ap ( X (t + 2) )] ; otherwise go to Operation 5, 1≤ p ≤ q
1≤ p ≤ q
( t ,t +1) 0
( t +1) 0
1≤ p ≤ q
(t + 2)0
taking X := X and X := X . 8. Calculations are completed because the solution X 0 ∈ Ω ⊆ L is obtained. (t )0
X (1)
X (l+1)
X1(t, t+1) X (1)0 =X (1)0 X2(t, t+1) X (l+1)0 =X (t+1)0 X (t+2)0
X3(t, t+1)0
Figure 1. Non-local search of X 0 The execution of Operations 2, 3, and 6 of the algorithm is possible on the basis of any search method (in particular, a modification of the univariate method [9] was implemented within the framework of ( m) ( m +1) the AIDMS1). If X is a current point, the transition to a point X is expedient if (∀p = 1,..., q ) : µ Ap ( X ( m+1) ) ≥ min µ Ap ( X ( m ) ) .
(19)
(∃p = 1,..., q) : µ Ap ( X ( m +1) ) < min µ Ap ( X ( m ) ) ,
(20)
1≤ p≤ q
In contrast, if 1≤ p ≤ q
( m +1)
is not expedient from the point of view of maximizing (4). This way of the transition to X ( m +1) evaluating the expediency of the transition to the next point X leads to the solution (5) that is Pareto, if all inexpedient transitions are rejected. The AIDMS1 includes the procedure for constructing and correcting the term-set T (Q ) and membership functions for fuzzy values of the linguistic variable Q - limitation for consumer. The initial term-set available for the DM, is T (Q ) =〈Near, Approximately, Slightly Less, Considerably Less, Slightly More, Considerably More>. The corresponding membership functions are: 2
µ ( xi ) = e − k (Ti − xi ) ;
(21)
1 − e − k (Ti − xi ) 2 , x ≤ T , i i µ ( xi ) = 0 , xi > Ti ;
(22)
1 − e - k (Ti − xi )2 , x ≥ T , i i µ ( xi ) = 0 , xi < Ti ,
(23)
where k is a coefficient defined by a given solution accuracy; Ti is a "specific value" which is related to the condition that is to be taken into account.
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The membership function (21) corresponds to the terms Near and Approximately, (22) - to Slightly Less and Considerably Less, and (23) - to Slightly More and Considerably More. Furthermore, the AIDMS1 includes several procedures for forming and correcting the vector Λ = (λ1 ,..., λ q ) of importance factors. These procedures are oriented to the individual DM as well as to the group DP. In particular, one of the procedures is associated with processing of the results of paired qualitative comparisons of the importance for different goals. The use of this type of information is rational because psychological experiments show that the DM is faced with difficulties in direct estimating the importance factors. In accordance with [17], the DM is to indicate which among two goals is more important and to estimate his or her perception of the distinction degree using a scale which includes the following ranks: Identical Significance, Weak Superiority, Strong Superiority, Evident Superiority, and Absolute Superiority. The comparisons allow one to construct the matrix [b pt ], p, t = 1,..., q . The components of the eigenvector corresponding to the maximum eigennumber of the matrix (normalized in a certain way, if necessary) can serve as estimates for λ p , p = 1,..., q. Experience in the use of the AIDMS1 shows the efficiency of applying the Bellman-Zadeh approach and the possibility of obtaining the most harmonious solutions on relative levels of satisfying goals. It is illustrated by computing experiments given below. Furthermore, the AIDMS is flexible and opened for extending its functions, correcting goals and their mathematical descriptions as well as for including new goals. It permits one to use the AIDMS1 for multiobjective allocation of diverse types of resources or their shortages.
4. Computing experiments Consider the solution of the following problems of multiobjective power shortage allocation on the basis of the Bellman-Zadeh approach and well-known Boldur method [18], based on the scalarization approach. It is necessary to allocate power shortages A1 =20000 kW, A 2 =40000 kW, and A 3 =60000 kW among six consumers with considering goals 1, 15, 16, and 17 indicated above. These goals are described by the linear objective functions (13) and we have: 6
F p ( X ) = ∑ c pi xi ,
p = 1, 15, 16, 17 .
(24)
i =1
Here xi , i = 1,...,6 are limitations of power supply for consumers. The coefficients c pi , p = 1, 15, 16, 17 , i = 1,...,6 are determined by specific characteristics of the consumers. Table 1 provides initial information for the problems according to (24) and (18). The results of the solution on the basis of the Bellman-Zadeh approach ( X 0 ) and the Boldur method ( X ∗ ) are presented in Table 2 and Table 3. The comparison of corresponding levels of the membership functions µ Ap ( X 0 ) and µ Ap ( X ∗ ) shows that the solutions X 0 are more harmonious than X ∗ : we have obtained a correlation min µ Ap ( X 0 ) > min µ Ap ( X ∗ ) in all cases. p
p
To show the possibility of correcting solutions as a result of changing the importance of the objective functions, we can assume, for example, that the fifteenth objective function has the level "Strong Superiority" relative to other objective functions (other objective functions have the level "Identical Significance" relative to each other). The paired comparisons permit one to obtain λ1 =0.5, λ15 =2.50,
λ16 =0.50, and λ17 =0.50. The corresponding solution for A 3 =60000 kW is: x13,0 =9179 kW, x23,0 =3604 kW, x33,0 =4000 kW, x 43,0 =5000 kW, x53,0 =23000 kW, and x 63,0 =15217 kW with µ A15 ( X 3,0 ) =0.836 and µ A1 ( X 3,0 ) =0.491, µ A16 ( X 3,0 ) =0.477, µ A17 ( X 3,0 ) =0.408.
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Table 1 - Initial information
i c1,i , monetary units/kWh c15,i
1
2
3
4
5
6
1.65
3.24
1.47
2.22
1.12
2.13
0.53
0.33
0.23
0.19
0.22
0.27
c16,i , hours
18.64
19.87
21.96
14.99
17.72
22.40
c17,i , hours
5400
6800
6200
5600
4900
7000
Ai , kW
16000
5000
4000
5000
23000
16000
5 18093 20000 23000 23000 23000 23000
6 0 0 2349 0 13316 12000
Table 2 - Power shortage allocation
i
1 0 0 4339 8000 11777 16000
1,0
X X 1,∗ X 2,0 X 2,* X 3,0 X 3,*
2 0 0 1312 0 2907 0
3 0 0 4000 4000 4000 4000
4 1907 0 5000 5000 5000 5000
Table 3 - Levels of the membership functions
p
1
15
16
17
)
0.920
0.983
0.920
0.968
µ Ap ( X 1,∗ )
1.000
0.972
0.864
1.000
µ A p ( X 2, 0 )
0.807
0.801
0.801
0.801
µ A p ( X 2, ∗ )
0.906
0.650
0.880
0.911
µ A p ( X 3, 0 )
0.626
0.620
0.620
0.620
µ Ap ( X 3,∗ )
0.986
0.446
0.727
0.932
µ Ap ( X
1, 0
Let us consider the influence of the linguistic variable Q - Limitation for Consumer. For example, the introduction of the condition "Considerably Less than 5000 kW" for the fourth consumer leads to the change of the solution given in Table 2 to: x 43,0 =2755 kW and x13,0 =13530 kW, x 23,0 =3312 kW,
x33,0 =4000 kW, x53,0 =23000 kW, x 63,0 =13403. At the same time, the introduction of the condition "Approximately 15000 kW" for the fifth consumer leads to the change of the solution given in Table 2 to: x53,0 =18965 and x13,0 =11995 kW, x 23,0 =4543 kW, x33,0 =4000 kW, x 43,0 =5000 kW, x 63,0 =15497 kW.
5. Multiobjective optimization in fuzzy model tuning The results of investigations on solving an important problem of system voltage and reactive power control on the basis of fuzzy logic technology are reflected in [19]. These results have been developed [20]. In particular, as it is shown in [20], flexible general rules, which can serve for different levels of control hierarchy, have the following form:
IF AND
bus voltage violates the operational limit a controller is available for effective voltage control adjusting its output
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AND AND THEN
there is adequate margin of output adjustment to eliminate the restriction violation the controller is available for loss reduction (increase) increase (decrease) the controller output.
In a similar way the general rule may be presented, for example, IF system element load is above its power-handling capability. The application of the general rules requires the availability of several types of sensitivity indices. In particular, the evaluation of influence of a control action at bus j (with a voltage regulating or reactive power compensating device) on a voltage level at bus i is associated with the voltage sensitivity S ijV . In a system with I controlled buses and J buses with regulating and compensating devices, it is necessary to have a matrix [ S ijV ], i = 1,..., I , j = 1,..., J . Besides, it is necessary to have power sensitivities [ S kjS ], k = 1,..., K , j = 1,..., J (reflecting the power flow change on line k due to the action at bus j) and reactive power sensitivities [ S kjQ ], k = 1,..., K , j = 1,..., J (reflecting the reactive power flow change on line k due to the action at bus j allowing the estimation of loss increments [21]). The rational way of constructing sensitivity indices is based on the use of experimental design [20]. Since the general rules reflect a general control strategy, they are to be presented by a set of concrete rules defined by fuzzy values of the linguistic variables, which are Control Efficiency, Adequate Margin, and Loss Increment as the input variables and Utilization Intensity as the output variable. Determination of ranges for base variables [1,2] of the linguistic variables Utilization Intensity and Control Efficiency presents no special problems. The variable Utilization Intensity has the unit range for its base variable to reflect the degree of utilizing available capacity in accordance with upper x +j and lower x −j limits ( u j = x +j − x j in the case of increasing the output or u j = x j − x −j in the case of its decreasing) of regulating or compensating device j. The base variable range of Control Efficiency is defined by the maximum magnitudes of S ijV or S kjS , which are significant to eliminate the violation of the bus voltage level restriction or the system element power-handling capability restriction, respectively. Determination of the base variable range for Adequate Margin is based on using S ijV or S kjS . It is possible to evaluate the necessary degree d j = ∆Vi
S ijV or d j = ∆Sk
S kjS of utilizing device j to
eliminate the violation ∆Vi of the bus voltage level restriction or the violation ∆Sk of the system element power-handling capability restriction. The degree d j serves as the range for the base variable. The construction of the base variable range of Loss Increment is similar to determining the base variable range for Control Efficiency. It is based on utilizing a vector [ S ∆j P ], j = 1,..., J of loss sensitivities. Its components are specific loss increments calculated as follows:
S ∆j P =
∆L( min {u j , d j }) min {u j , d j }
.
(25)
The next step in constructing the linguistic variables is associated with the choice of their fuzzy values and corresponding membership functions. Considering this, it should be pointed out that in the majority of cases fuzzy control systems [22] manipulate with triangular or trapezoidal membership functions. In developing a control system [20], they were also utilized: trapezoidal membership functions as extremes and the triangular membership functions as intermediate. To tune fuzzy models from the viewpoint of adequacy, considering its complexity, the following approach has been used. It is not difficult to observe that "a controller is available for effective voltage control adjusting its output" of the rule given above, from the substantial point of view, is equivalent to "primary utilization
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of controllers with the more high voltage sensitivity". Thus, indicated part of the rule is in accordance with the following problem:
F1 ( X ) =
, ∑ SijV x j → max X ∈L
(26)
j∈J s
where J s is the set of variables, whose coefficients are significant. It is natural that the similar problem may be formulated in the case of violating the system element power-handling capability restriction. In like manner, it is possible to say, "there is adequate margin of output adjustment to eliminate the restriction violation" of the rule fitted with uj (27) F2 ( X ) = ∑ x j → max . X ∈L j∈J s d j Finally, "the controller is available for loss reduction (increase)", considering that it may be interpreted as "primary utilization of controllers with the more low loss sensitivity", is consistent with the following problem:
F3 ( X ) =
. ∑ S ∆j P x j → min X ∈L
(28)
j∈J s
Thus, the rule given above is equivalent to the multiobjective problem with the objective functions (26)-(28). This has made it possible, using the AIDMS1, to obtain multiobjective solutions for typical control hierarchy situations. These solutions have served as patterns for tuning the fuzzy models (to estimate the acceptable number of fuzzy values for the variables, levels of their overlapping, etc.) on the basis of the results of [22].
6. Conclusions When utilizing the Bellman-Zadeh approach to decision making in a fuzzy environment, the concept of "optimal solution" is defined with reasonable validity: the maximum degree of implementing goals serves as a criterion of optimality. This conforms to the principle of guaranteed result and provides constructive lines in obtaining harmonious solutions on the basis of analyzing associated maxmin models. The use of the Bellman-Zadeh approach has served as a basis for solving a problem of multiobjective allocation of resources and developing the corresponding adaptive interactive decision making system (AIDMS1). Its calculating kernel is based on a non-local search (modification of the Gelfand's and Tsetlin's "long valley" method. The peculiarities of its construction permit one to obtain solutions that are, indeed, Pareto. The AIDMS1 includes procedures for constructing linguistic variables (to consider conditions that are difficult to formalize) as well for forming and correcting vectors of importance factors for goals. The use of these procedures permits one to realize an adaptive approach to provide successive improving the solution quality on the basis of processing information of a decision maker. The results of the paper are illustrated by considering problems of load management and tuning models related to fuzzy control of system voltage and reactive power.
7. Acknowledgements This research is supported by the National Council for Scientific and Technological Development of Brazil – CNPq and Minas Gerais State Energy Company – CEMIG.
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