Electric Power Systems Research 68 (2004) 93 /101 www.elsevier.com/locate/epsr
Using multi-objective optimal power flow for reducing magnetic fields from power lines Lucio Ippolito *, Pierluigi Siano Department of Electrical and Electronic Engineering, University of Salerno, Via Ponte don Melillo no. 1, I-84084 Fisciano (SA), Italy Received 21 February 2003; received in revised form 8 May 2003; accepted 12 May 2003
Abstract Over the past several years, concerns have been raised over the possibility that the exposure to 50/60 Hz electromagnetic fields (EMFs) from power lines, substations, and other power sources may have detrimental health effects on living organisms. As a result of these concerns, some European States, as Belgium, Italy, Switzerland and Estonia, have set limits, which are more stringent then the Council Recommendation making reference to the precautionary principle. This stricter legislation is leading not only to an ambiguous legal situation but, above all, to controversy, delay, and costs increases in the construction of utility lines and facilities. Consequently, a number of techniques for mitigating EMFs associated with power lines have been proposed, but many of them are mainly applicable to future constructions and may not be appropriate for existing transmission or distribution lines due to high implementation cost. From these considerations, the study analyses the feasibility of using optimal power flow (OPF) for limiting EMF levels. The mitigation is obtained solving a multi-objective optimal power flow (MO-OPF) problem with a specific objective function for the EMFs. In order to validate the usefulness of the approach suggested herein, a case study using a modified IEEE 30bus power system is presented and discussed. # 2003 Elsevier B.V. All rights reserved. Keywords: Electric fields; Magnetic fields; Electromagnetic fields; Fuzzy logic; Goal programming; Optimal power flow
1. Introduction Several scientific studies have raised general public concerns about possible detrimental health effects from the electromagnetic fields (EMFs) in the 50/60 Hz extremely low frequency (ELF) band that surround power lines. The public pressure has forced the state authorities to take regulatory actions to protect citizens against the potential risks posed by EMFs. To date, many states, applying the prudent avoidance policy, have taken regulatory actions to limit the intensity of electric and magnetic fields on the edge of the transmission line right-of-way (ROW). As a part of the scientific effort, extensive evaluations of induced electric field and current density in the human body have been performed. During the last few
* Corresponding author. E-mail address:
[email protected] (L. Ippolito). 0378-7796/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0378-7796(03)00151-2
years the interest has shifted from electric to magnetic fields, which are now generally believed to be more potential hazard to humans. To date all countries use the ICNIRP guidelines [1] and the 12th July Council Recommendation [2] as the scientific basis for their recommended levels of exposure. However, Belgium, Italy, Switzerland and Estonia have implemented stricter legislation making reference to the precautionary principle. More in particular, in Italy ‘zones’ have been installed, near hospitals, schools, and places where the density of population is high, etc, where allowed EMF frequency limits are more restrictive. For such zones limits for the magnetic flux density are set to 0.2 mT, which is 500 times lower the limit suggested by the EU Council Recommendation. Moreover, in Italy there is also a regional variation for protection regimes, having some bylaws, which introduce the limit of 0.5 mT for long time of exposure. It is worth to note that for electric power transmission companies it would be a very big problem to comply
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with the present legislation because rebuilding existing power lines is extremely expensive. This difficulty to comply with law is leading to an increasing number of controversy between the electric power transmission companies and citizens, those demand for the safeguard of their health. The consequences of this process are the delay and the cost increases in the construction of power lines and electrical stations, which are essential for the growth of the national grids in the new liberalized power market. In this scenario, designers of power lines are searching for technically and economically acceptable measures of magnetic field mitigation. As well known various ‘hard measures’ can be used to avoid the higher strength ELF magnetic fields: . increasing the safe distance from power lines; . shielding the ELF field source or the person near it; . balancing the phase currents, reducing the distances between conductors; . splitting the conductors; . re-arranging the phases for multiple circuits tower structure; . installing passive or active loop conductors near phase conductors, and so on. Even though all these methods of reducing the distant external fields from conductors of overhead power lines can contribute significantly to cancel the major part of the fields, they could be extremely expensive. In order to achieve the objective of magnetic field reduction, in this paper a ‘soft measure’ for field mitigation, consisting in the redistribution of the line power flows based on the optimal power flow (OPF) solution, is presented. To be specific, the idea is to introduce a specific objective function in the OPF problem, trying to minimize the effects of magnetic fields near those overhead power lines representing more potential hazard to humans. In the subsequent sections this goal will be achieved solving a multi-objective optimal power flow (MO-OPF) problem, using a goal programming technique to pursue the satisfaction of objectives, which have different characteristics. In particular, a specific objective function for minimizing the magnetic fields is considered. By varying, hour by hour, the weighting factors inside this objective function the magnetic fields at the edge of the ROW can be modified, dynamically, in accordance to the degree of dangerousness of the power lines, which are function not only of parameters such as the line current, the conductor sag and the line edification distance from the power line, but also of some other aspects as the kind of edification, the daily hour, and the day of the week. In the following sections, after having recalled some basic elements for overhead power lines magnetic fields
calculation, the MO-OPF for reducing magnetic fields from power lines is discussed and formalized. The effectiveness of the proposed method is tested on a modified IEEE 30-node system comprising six generators, eight phase shifters, and 20 loads.
2. Theory of overhead power line magnetic fields To date the calculations of the magnetic fields of infinitely long horizontal conductors above a homogeneous earth has been widely studied [3]. With reference to the geometrical variables defined in Fig. 1, a useful approximation to the RMS fields of a conductor, which carries power system frequency current, is: (y yn ) (y yn a) [Tesla] Bxn 210In R2cn R2in
(x xn ) (x xn ) Byn 210In R2cn R2in
[Tesla]
Bzn 0
(2:1)
In the previous equation In is the RMS phasor current in the nth conductor in amperes, (xn , yn ) and (x , y ) are the locations of the n th conductor and the field point, respectively in meters, Rcn, Rin, and a have the following expressions: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rcn (xxn )2 (yyn )2 Rin
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (xxn )2 (yyn a)2
Fig. 1. Geometric variables for magnetic field model.
(2:2)
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pffiffiffi jp a 2de 4 sffiffiffiffiffi rg d k f
(2:3)
where d is the skin depth of the earth where rg is the earth resistivity in V meters and f is the power system frequency in Hertz. Eqs. (2.1) and (2.2) are valid for field points above the earth’s surface and whose distance away from the conductors (Rcn) is less than l /20 where l is the free space wavelength. The total magnetic field of N parallel conductors is simply the superposition of the fields from N conductors. Bx
N X
Bxn
n1
By
N X
Byn
n1
Bz 0
(2:4)
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which considers both the fuel cost of thermal units and the transmission loss, a magnetic flux density index for the magnetic fields is defined. In details, the following indices have been defined and included into the vector f (x) of the objective functions: f1 (x)
NG X
(ai bi PGi ci P2Gi )
n X
c(Pi Qi ) [$=h]
i1
i1
(economy index) f2 (x)
n X
(3:2)
wBij j(BLij (x)BL) ij [mT]
i;j1
(3:3)
(magnetic flux density index) where NG : total number of generator busbars of the system; n : total number of the nodes of the system; ai , bi , ci : cost coefficients; Pi : active power at the i th node; Qi : reactive power at the ith node; c : cost coefficient; x x 0 /j(x) / 0 x 50 wBij : line magnetic flux density weighting factor; BLij (x): magnetic flux density from the line between node i to j; /BL: reference value of magnetic flux density. ij /
3. MO-OPF for reducing magnetic fields The main objective in this study is to minimize the magnetic fields at the edge of the ROW for every critical power line without penalizing excessively the generation and transmission costs. In a general sense, this goal can be achieved by solving a MO-OPF problem that minimizes simultaneously p objective functions on X , a set of feasible solutions, and that may be formulated as follows [4]: minimize f (x) h(x)0 subject to x X x g(x)5 0
(3:1)
where the vector x consists of a set of controllable quantities and dependent variables. The controllable quantities can be assumed to include generating unit outputs, company transactions, phase shifter angles, generator bus voltage magnitudes, and transformer tap positions. The vector f(x) of the objective functions includes multiple objective associated with economy, and magnetic fields protection. The equality constraints h (x) represent the static load flow equations and the inequality constraints g (x) consist of the limits on the controllable quantities and the operating limits of power system [4]. For the formulation of the MO-OPF problem, in this paper, jointly with the conventional economy evaluation index,
For a better balancing of the magnetic fields from power lines and to emphasize the minimization of only those magnetic flux densities which exceed their predefined reference value, a function j(x) is introduced in formalizing the objective function in the f2(x). Moreover, since an OPF solution must satisfy power flow equations at each node, the following equality constraints are assumed: P(x)PS 0 Q(x)QS 0
(3:4) (3:5)
As inequality constraints, in the present study, voltage magnitude at each node, active and reactive generator output, line flows, and phase shifter angles are used. To be specific, the following constraints have been defined: Pi;min 5 Pi 5Pi;max Qi;min 5Qi 5Qi;max PLij 5PLmax ij
i 1; 2; . . . ; NG i 1; 2; . . . ; NG
i; j 1; 2; . . . ; n
(3:6) (3:7) (3:8)
½Vj;min ½ 5½Vj ½5 ½Vj;max ½ i 1; 2; . . . ; n
(3:9)
i 1; 2; . . . ; Nphs
(3:10)
fi;min 5fi 5fi;max
where Pi , active power at the ith generator busbar; Pi, min, Pi, max, minimum and maximum active power at the ith generator busbar; Qi , reactive power at the ith
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generator busbar; Qi, min, Qi, max, minimum and maximum reactive power at the ith generator busbar; PLmax ij ; maximum transmission capacity; Vj , voltage at the j th load or generator busbar; Vj ,min, Vj ,max, minimum and maximum voltage at the jth busbar; fi , angle of the i th phase shifter; fi ,min, fi ,max, minimum and maximum angle of the ith phase shifter; NL, total number of the load busbars of the system; Nphs, total number of the phase shifters. As evident, in the previous formalized optimization problem there exists two of objectives to be achieved which inherently have different characteristics, and are non-commensurable in their nature. These objectives are in trade-off relations, and with no invariant priority order amongst them. These inherent characteristics of the specific optimization problem suggest the use of the goal programming method [5 /7], which pursuits the satisfaction of objectives rather than their optimization, as solving methodology. Solving the previous formalized multi-objective problem means to find a non-inferior solution, for that any further improvement in one objective would result in a degradation in another objective. In this paper, the goal attainment method of Gembicki [8] is used for computing non-inferior solutions. With such a method, as it can be easily proved, the best-compromise solution, if it exists, is a Pareto optimal solution for the problem. Moreover, the method does not suffer of any convexity limitations, and the parameters that it uses have a convenient intuitive meaning. The goal attainment method involves expressing a set of design goals, f ff 1; f 2; . . . ; f m g; which is associated with a set of objective functions, f (x) ff1 (x); f2 (x); . . . ; fm (x)g: The degree to which these goals are met is adjusted using a vector of weighting coefficients, w fw1 ; w2 ; . . . ; wm g; and the optimization problem is expressed using the following formulation:
where fimin and fimax are the minimum and maximum values of ith objective function in which the solution is expected. The value of the membership function indicates, how much (in scale from 0 to 1) a solution is satisfying the fi objective. To search for the optimal weight pattern in the noninferior domain an evolutionary optimization technique, proposed in Ref. [13] and summarized below, is used. Such optimization technique starts defining a hypercube of weight combination around an initial search point, wci of the hypercube i 2; 3; . . . ; L; where L is the number of objectives. Then 2L1 weight combinations at the edges of the hypercube are generated as given below:
min g
wji wci Gji ;
such that
g R;x V
fi (x)wi g5f i (3:11)
i 1; . . . ; m In the previous formulation the scalar g is a measure of how far away a solution point is from the goals, the weighting vector, w, enables the designer to choose the relative tradeoffs between the objectives. By varying the weighs, the set of non-inferior solutions (Pareto optimal) for a problem is generated [9,10]. In the present study the trade-off analysis and the decision-making problem for the ‘best’ compromised solution are made by applying Bellman and Zadeh’s maximizing decision [11]. The essence of the maximizing decision lies in taking the minimum value of the performance indices, and finding the maximum value of such minimum perfor-
mance indices to represent the best option for the decision. Moreover, due to the imprecise nature of the decision maker’s judgement, it is assumed that the decision maker has fuzzy goals for each of the objective functions. The fuzzy goals are quantified by defining their corresponding membership functions. These functions represent the degree of membership in certain fuzzy sets using values from 0 to 1 [12]. The membership value 0 indicates incompatibility with the sets, while 1 means full compatibility. Taking account of the minimum and maximum values of each objective function together with the rate of increase of membership satisfaction, the decision maker has to determine the membership function m(fi ) in a subjective manner. In the present research it is assumed that m(fi ) is a strictly monotonic decreasing and continuous function defined as: 8 1 fi 5fimin > > < f max f i i (3:12) m(fi ) max fimin 5 fi 5 fimax min > f f > i i : 0 fi ]fimax
i 2; 3; . . . ; L; j 1; 2; . . . ; 2L1 wj1 1
L X
wji ;
j 1; 2; . . . ; 2L1
(3:13)
(3:14)
i2
where G is the distance of the corners of the hypercube from the point around which hypercube is generated. The membership functions for each objective, m(fi )k ½i1;2;...;L and the intersection of membership funck tions, mmin k Min(m(fi ) ½i1;2;...;L ) are then computed for each weight combination. The decision regarding the ‘best’ solution is made by the selection of minimax of membership function as defined below [14]: L1 m0 Maxfmmin 1g k ; k 1; 2; . . . ; 2
(3:15)
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where L is the number of objectives, 2L1 is number of corner points of an (L /1)-dimensional hypercube. Then to continue the iterative process, another hypercube is formed around the combination wc0 i having maximum satisfied membership function m0 as compared to the previous one and the procedure is repeated until the solution criteria for ‘best’ compromised solution is met.
4. Choosing line magnetic flux density weighting factors As previously written, in the formulation of the MOOPF problem a magnetic flux density index for the magnetic fields is introduced. This evaluation index permits to assess the safety of human beings under ELF magnetic field, driving the solution towards a current distribution which minimizes the magnetic fields generated from those lines to which correspond more potential hazards. As evidenced in Eq. (3.3), weighting factors (wBij ) are used for varying the relative emphasis of each line in the magnetic flux density index. Through the selection of these factors the relative emphasis of the power lines can be varied depending on the effective dangerousness of the specific line for the population. In the present work, it is assumed that the degree of hazardousness is influenced by various variables as: . line current; . conductor sag (which changes with ambient temperature, wind speed, solar irradiation, etc.); . line edification distance from power line; . kind of edification in vicinity of existing power lines (schools, houses, hospitals, amusement parks, etc.); . density of population; . daily hour, week day, weekend; . health risk factor (HRF) for each power line; . acceptable risk for each power line; . human perception threshold of EMFs.
Fig. 2. RF vs. magnetic flux density.
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As it concerns to HRF, it has been defined the following risk function (RF), depicted in Fig. 2, in which the magnetic flux density (B ) is expressed in microtesla: 8 2 < RF k 1tanh k2 B k3 : 1 k4
B [0:5; ] B [0; 0:5]
(4:1)
To obtain the RF, depicted in Fig. 2, it was assumed k1 /1, k2 /25, k3 /5, k4 /2.8. For each power line, by computing the difference between the value of the RF and the value associated to the acceptable risk, the residual risk (RR) is obtained. Because many parameters, influencing the safety assessment estimation, are not directly measurable or are time-variant, a fuzzy approach is applied for assessing the safety of human beings. With the proposed fuzzy approach, described also in Ref. [15], the power lines to which correspond more potential hazards are identified and values are assigned to the weighting factors (wBij ): The weight’s assigning reflects the aim to reduce the level of magnetic flux density for critical transmission lines for which a great health risk exists. In the problem under examination a set of fuzzy rules could have the following structure: . ‘‘IF (daily hour is Morning) AND (Day is monday) AND (kind of edification is School) AND (health risk factor is High) AND (line current is High) THEN (Weight is VeryHigh)’’; . ‘‘IF (kind of edification is Hospital) OR (population density is VeryHigh) AND (health risk factor is High) AND (line current is High) THEN (Weight is VeryHigh)’’; . ‘‘IF (kind of edification is not Hospital) AND (population density is VeryLow) AND (health risk factor is High) AND (line current is VeryHigh) THEN (Weight is Low)’’. The tuning of the fuzzy model, including the rules and the term sets of the variables with their related fuzzy sets, was obtained through a process which started from a set of initial insight considerations and progressively modified the parameters of the system until it reached a level of performance considered to be suitable. In the definition of the inputs and output membership functions Gaussian membership function have been chosen, moreover, as concerns the control architecture, a Mamdani based system architecture has been implemented. An example of membership functions for some input variables and for the normalized output variable is presented in Fig. 3. Introducing such a fuzzy-based procedure into the MO-OPF routine, the weights, (wBij ); are calculated on a iteration by iteration basis.
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Fig. 3. Membership functions for input/output variables.
5. Numerical results In order to verify the effectiveness of the previously described algorithm it is applied to a modified 30-bus IEEE system, schematically reported in Fig. 4. The power system comprises six generators, eight phase shifters, and 20 loads. The best number and the best location of phase shifters is identified through a genetic algorithm-based approach, discussed also in Ref. [16], while their ratings is calculated directly in the MO-OPF routine. Briefly, the objective function of the implemented GA regards different design indices, such as the generation cost, transmission losses cost, magnetic flux index, and the cost of installation and maintenance the devices. In this case study, the goals used for the generation of the non-inferior solutions are computed by solving the single objective problem for each single objective function and it is reasonably assumed that they coincide with the minimum values of the objectives used to determine the membership functions. Moreover, owing to the
conflicting nature of the objectives, each objective function will have its maximum value when the other is minimum. Generation costs, including transmission losses (f1), and magnetic flux density function (f2) are the two objectives which have weightings, w1 and w2, respectively. The minimum and maximum values of these objectives are given in Table 1. The initial centre is wci (0:5; 0:5); since L /2 in this case an hypercube is formed around weight w2 only. The one binary bit can be represented in two possible different combinations. These two binary bit combinations are the corners of hypercube away from the centre of hypercube. Table 1 Minimum and maximum values of the objectives
Min Max
f1 [$/h]
f2 [mT]
1056.2085 1071.4860
943.2032 1501.0611
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Fig. 4. Modified IEEE 30-bus system.
Table 2 Variation in weight search K
w1
w2
f1 [$/h]
f2 [mT]
m(f1)
m(f2)
1 2 3 4 5
0.6 0.5 0.4 0.3 0.2
0.4 0.5 0.6 0.7 0.8
1068.9110 1067.0949 1064.1708 1061.7618 1061.3654
1033.3364 1038.2261 1052.4091 1062.3112 1158.9850
0.1685 0.2874 0.4788 0.6365 0.6625
0.8384 0.8296 0.8042 0.7865 0.6132
The values of the two objective functions, the membership functions, the minimum membership function values from two objectives in each iteration of search are tabulated in Table 2. The ‘best’ weights: w1 / 0.3, w2 /0.7 are obtained after five iterations. With the identified ‘best’ weighting factors combination the MO-OPF is solved and the results are compared with those obtained solving a conventional OPF with the unique objective of minimising the total generation costs, including losses. The results of these two different approaches are shown in Tables 3 and 4. The analysis of the results reveals the feasibility of the proposed approach, achieving a good balancing of the
Table 3 Comparison between conventional and MO-OPF approach
Total generation costs (including losses) [$/h] Magnetic flux density function [mT]
Conventional OPF
MO-OPF
1056.2085
1061.7618
Not defined
1062.3112
/
mmin k
m0
0.1685 0.2874 0.4788 0.6365 0.6132
0.1685 0.2874 0.4788 0.6365 0.6365
magnetic fields generated from the power lines, and calculated at given distances. The MO-OPF approach aims to reschedule the current flows along the lines in order to reduce the flows for those lines characterized by values of the magnetic flux density higher than the reference values */0.5 or 0.2 mT*/and to increase the flows for those lines with values of the magnetic flux density lower than the reference values. Moreover, from the same analysis it can be noted as the total generation costs, including losses, are practically unchanged. Finally, it is worth to note that in the simulation to vary the relative emphasis of power lines in the magnetic flux density index some ‘zones’ have been considered around the electrical installations where allowed EMFs limits are more restrictive. In practice, using the previous described fuzzy approach, for each line the input variables influencing the degree of hazardousness are set. Making some assumptions, a social impact factor was derived for each power line, which together with the remaining input variables, permitting to induce through the fuzzy sets the weighting factors, (wBij ): For example, in the presented simulation it was assumed that
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Table 4 Comparison between magnetic flux density with conventional OPF (BC OPF) and MO-OPF (BMO OPF) approach From bus
To bus
BC OPF [mT]
BMO OPF [mT]
Variation (%)
1 1 2 3 2 2 4 5 6 6 6 6 9 9 4 12 12 12 12 14 16 15 18 19 10 10 10 10 21 15 22 23 24 25 25 28 27 27 29 8 6
2 3 4 4 5 6 6 7 7 8 9 10 11 10 12 13 14 15 16 15 17 18 19 20 20 17 21 22 22 23 24 24 25 26 27 27 29 30 30 28 28
0.4588 0.3137 0.5285 0.3813 0.5065 0.4107 0.4727 0.2182 0.3468 0.1049 0.1965 0.3685 0.3769 0.5126 0.1745 0.5342 0.3386 0.4913 0.2295 0.1205 0.2663 0.2432 0.3228 0.2273 0.3504 0.0967 0.2352 0.3357 0.0813 0.2992 0.3792 0.2076 0.096 0.1714 0.1861 0.5129 0.2543 0.3819 0.2118 0.1629 0.2903
0.4313 0.2837 0.4659 0.3433 0.4991 0.4391 0.321 0.2022 0.2818 0.1057 0.1799 0.3517 0.3744 0.4998 0.2843 0.4997 0.3411 0.4976 0.2668 0.1237 0.3386 0.2429 0.3232 0.2279 0.351 0.0817 0.2394 0.3267 0.0929 0.311 0.3869 0.222 0.1316 0.1714 0.1857 0.5 0.2543 0.3818 0.2118 0.1574 0.2811
/5.99 /9.56 /11.84 /9.97 /1.46 6.92 /32.09 /7.33 /18.74 0.76 /8.45 /4.56 /0.66 /2.50 62.92 /6.46 0.74 1.28 16.25 2.66 27.15 /0.12 0.12 0.26 0.17 /15.51 1.79 /2.68 14.27 3.94 2.03 6.94 37.08 0.00 /0.21 /2.52 0.00 /0.03 0.00 /3.38 /3.17
the hour of the day was 12 am and the acceptable risk was 2.0.
6. Conclusions As concern over exposure to low frequency AC magnetic fields has grown over the past several years, the knowledge of field characterization and effective control strategies has grown concurrently. In the technical literature, mitigation of the magnetic field near power lines has been approached in a number of ways, but the method described in this paper has its own merit due to the simplicity, the resulting low cost of implementation, and the applicability to existing trans-
mission/distribution lines. In the proposed method, the generation rescheduling is used for adjusting line flows and consequently to reduce, where it is required, the magnetic fields. The proposed methodology to implement EMF frequency limits has revealed its effectiveness, achieving an acceptable balancing of the magnetic fields generated from the power lines without producing a great change of the total generation costs and transmission loss. It could result useful in a number of practical application assisting the ‘decision makers’ into identifying a suboptimal solution, which takes into account directly the social impact of the magnetic fields. Obviously, the choice of the modified 30-bus IEEE system as test network derives only from the need to prove the efficiency of the proposed methodology. In order to demonstrate its scalability to large power systems, some further hardware-in-the-loop simulation, based on a real time digital simulator, are carrying on. At the moment with the proposed algorithm any bottleneck has not been evidenced. Moreover, to reduce the global computational time, a multi agents system is implementing.
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[15] L. Ippolito, P. Siano, A fuzzy decision support system for multiobjective optimal power dispatch to reduce magnetic fields from power lines, Proc. of 10th International Fuzzy Systems Association World Congress, 2003. [16] L. Ippolito, P. Siano, Optimal location of thyristor-controlled phase shifting transformers in a power system by means of genetic algorithms, submitted to Electric Power Syst. Res. 2003.
