DETERMINATION OF OPTIMAL DIRECT LOAD CONTROL STRATEGY USING LINEAR PROGRAMMING

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1 DETERMINATION OF OPTIMAL DIRECT LOAD CONTROL STRATEGY USING LINEAR PROGRAMMING Zelko Popovic Distribution engineer Rae Koncara 57, Backa Topola, Yugoslavia Phone: Fax: SUMMARY Direct loa control (DLC) is one of the most wiely use techniques of loa management. This technique is usually applie at resiential sector. In most cases DLC is being applie to minimize peak loa or prouction costs in power system. In this paper is expose a moel for etermining an optimal DLC strategy of evices in househols with the aim of reucing system peak loa. The propose moel enables optimal (maximal) reuction of the system peak loa in cases when various types of evices in househol are controlle at the same time. The presente moel is base on linear programming an teste on a real power system where water heaters an air conitioners are controlle at the same time. INTRODUCTION Many loa management (LM) programs have been applie by the utilities in recent years. Loa management obectives for ifferent utilities may be ifferent. In the power systems with a little reserve in power prouction an transmission loa management is primarly applie for reucing system peak loa. Thus, the nees for builing new capacities are postpone (reuce). Systems with relatively high reserve are applying loa management for reucing prouction costs an epenence of expensive fuels [,2]. For achieving loa management obectives, it is usually use DLC of some evices in househols. Commonly controlle evices are the following: water heaters, air conitioners, electric space heaters an storage heaters. In the literature, a consierable attention is eicate to DLC an especially to etermination of optimal DLC strategy. Direct loa control strategy, which etermines turning on/off iary scheule of controlle evices, is compose of several elements, the most important of which are: (i) type of controlle evices an the way of their control, (ii) beginning an uration of control perio uring a ay, (iii) number of groups of controlle evices an their mutual time shift, i.e. turning on/off moments of some groups an (iv) number of evices accoring to groups. There is a series of moels for etermining an optimal DLC strategy [2-3]. Maority of the propose moels is evelope with the aim of reucing prouction costs of the power system [2-9], an a smaller part with the aim of reucing system peak loa [0-3]. This paper is eicate to etermination of optimal DLC strategy with the aim of reucing system peak loa. Cohen [0] has evelope a moel for etermining elements of DLC with the aim of reucing system peak loa. The moel is base on ynamic programming. In that moel, total number of controlle evices is in avance ivie in etermine, beforehan known (assigne), number of groups of the same size (the same number of evices in each group). During optimization proceure, the expose moel treats each group of evices separately. In this way, proper (optimal) coorination of loa control (connection an isconnection) between ifferent evice groups is missing an maximum reuction of the system peak loa is not obtaine. Lee [] has evelope a moel base on linear programming which enables etermination of number of controlle groups an number of evices in each group as a result of optimizing proceure. In the moel, turning on/off moments of some groups of evices are etermine in avance. A set of these moments forms so-calle control shame. Thus, the problem of coorination of loa control between some groups of evices is partially surpasse, but maximal reucing of system peak loa is not possible because of limits impose by the metho of forming control scheme. We can fin similar rawbacks in the moel expose in the literature [2], which is base on the combination of linear an ynamic programming. The moel propose in the literature [3] is base on linear programming. This moel enables etermination of optimal DLC strategy when evices, which can be turne off for a long perio of time (for example, water heaters), are controlle, while it is not possible for evices which can be turne off only for a brief perios of time (perioically) (for example, air conitioners). In this paper is evelope a moel, base on linear programming, which enables etermination of most important elements of DLC strategy, an consequently realizing an appropriate (optimal) coorination of loa control (turning on/off moments) between ifferent groups of controlle evices. In this way, propose moel ensures maximal reuction of system peak loa (maximal utilization of available DLC resources) even in case of applying DLC to various types of evices at the same time. This paper is compose of five parts. In the secon part is expose mathematical moel for etermining an optimal DLC strategy. Appliance of the propose moel to peak loa reuction of the real power system, when water heaters an air conitioners are controlle simultaneously is expose in the thir part. The most important conclusions are given in the fourth an the list of cite works in the fifth part. MATHEMATICAL MODEL Direct loa control is generally realize in two ways:

2 with perioical turn off of evices (several times uring control perio (cycling)) or with long-term turn off of evices (once uring control perio). The first way is usually applie to evices that can accumulate a little quantity of energy (air conitioners, space heaters, etc.), an the secon to evices that are capable of consierable accumulation of energy (water heaters, storage heaters, etc.). For that reason, in the propose moel will be mae ifference between these two ways of controlling of evices. Mathematical moel, base on linear programming, for etermining an optimal DLC strategy when are simultaneously controlle ifferent types of evices which can be turne off only for a long perio of time an ifferent types of evices which can be turne off only for a brief perios of time (cycling), is the following: Minimize P p () Subect to: m n p = = p x - l= nl P A B, i=,..,k, (2) = l l y i n x M, =,..,m, (3) = [C l ] [Y l ] [M l ], l=,..,p, (4) nl l l l y z M, l=,..,p, (5) = x 0, =,..,n, =,..,m, l y 0, =,..,n l, l=,,p, where: P p unknown system peak loa (after DLC), m number of ifferent types of evices which can be turne off only for a long perio of time, n number of observe 5-minutes lasting perios for -type evices (evices that are capable of a long-term turn off); ue to the fact that peak loa is being registere in 5-minutes lasting intervals for the perio of 24 hours, it is usually assume that n 96, A known value which escribes a loa control influence of -type evices on the power system loa curve, A = { -P u (i), uring isconnection perios, P pb (i), uring payback perios, (6) 0, in other cases, P u (i) iversifie loa of -type evice, P pb (i) net restore eman of -type evice, x unknown number of -type evices that shoul be turne off in -th perio, p number of ifferent types of controlle evices which can be turne off only for a brief perios of time (cycling), n l number of observe 5-minutes lasting perios for l-type evices (evices that are capable of perioical turn off); in this case, for the above mentione reasons, it is usually assume that n l 96, B l known value which escribes a loa control influence of l-type evices on the power system loa curve, B l = { -P u l (i), uring isconnection perios, P pb l (i), uring payback perios, (7) 0, in other cases, P u l (i) iversifie loa of l-type evice, P pb l (i) net restore eman of l-type evice, y l unknown number of l-type evices that shoul be turne off in -th perio, i forecaste 5-minute loa of the power system (before DLC), k number of 5-minute lasting intervals when is observe DLC influence of all controlle evices on system peak loa reuction, M available number of controlle -type evices, M l available number of controlle l-type evices, z l maximal number of turning off of l-type evices uring control perio. Matrix [C l ] has the following form:..... c l c l+... n l [C l ]= (8) n l+-c l With c l is marke control cycle of l-type evices. It is compose of turning off time length (number of 5-minutes lasting intervals) an minimal turning on time length (number of 5-minutes lasting intervals) of evices. For example, if control cycle is 2 hours (30-minutes lasting turning off perio an minimum 90-minutes lasting turning on perio) c l =8. Control cycle for the same evices can have ifferent lengths (values) uring control perio, making possible to be combine ifferent cycling scenarios within one control perio. Colum vector [Y l ] has imension n l x an colum vector [M l ], whose elements are the same (available number of controlle evices), has imension (n l + - c l ) x. The aim of the propose moel, expose with expression (), is to reuce a system peak loa. Constraints given with expression (2) (one for each i-th time interval) ensure that the new peak loa is not going to be smaller that new loa for each i-th time interval. The new loa for each i-th time interval is calculate as a sum of original forecaste loa an a total of loa change ue to loa control of all types of evices. Total of change loa uring i-th time interval, ue to control of all types of l evices, is given by l l A x + B y. m n = = p l= n = Constraints (3) ensure, for each -type evice, that number of controlle evices in not going to surpass a total number available for control. Constraints (4), for each l-type evice, ensure cycling (controlling) accoring to fixe scenario without isturbing consumer s comfort.

3 Constraints (5) ensure, for each l-type evice that, if there are such emans, turning off number of evices uring control perio is not going to be more than it is maximally allowe. Number of examine perios n an n l epens on power system aily loa curve shape, that is on a peak loa time length. It coul be usually assume that n an n l are less than 96, what reuces number of variables in the moel. Also, in such cases, can be observe a perio shorter than 24 hours (k < 96), what reuces number of constraints in the moel. Interval (k) where is observe DLC influence on system peak loa reuction, must inclue payback influence of all controlle evices on aily loa curve shape of the observe power system. In escribe way, imensionality of linear programming problem (relations ()-(5)) is being reuce an settling of the problem is being accelerate. RESULTS The propose moel is applie to the peak loa reuction of a real power system, where are simultaneously controlle water heaters (evices capable of a long-term turn off) an air conitioners (evices capable of a perioical turn off). Daily loa curve shape of the observe power system is taken from the literature [4] an is expose in Figure 3. Daily loa curves of controlle evices, the way of its acquiring as well as effects of control of their work (payback) are escribe in the literature [6,, 4-9]. Daily iversifie loa curves of uncontrolle an controlle water heaters an air conitioners are shown in Figure an Figure 2, respectively [4]. Shape an values of payback for water heaters an air conitioners are escribe in the Table [] an Table 2 [6], respectively, where E stans for energy (in kwh) that evices were eprive uring turning off perio, an P pb stans for net restore eman. Maximally allowe turning off time length for water heaters is 3 hours [4]. Air conitioners are turne off for 30 minutes an after that they shoul be turne on for hour an 30 minutes [6]. Accoring to this scenario, they can be turne off several times uring control perio. Figure 2. Daily iversifie loa curve of controlle an uncontrolle air conitioner Table. Relations for net restore eman calculation of water heater Time elapse after restoration of service 0:5 0:30 0:45 :00 :5 :30 :45 2:00 P [kw] for E < 3.6 kwh P pb (kw) E E Max(0, E ) Max(0, E ) Max(0, E ) Max(0, E ) Max(0, E ) Max(0, 0.60 E ) Max(0, 0.23 E ) for E > 3.6 kwh Note: Expression Max(0, ) means that for small values of E, for which secon part of expression in parenthesis becomes negative, the net restore eman is zero i. e. payback perio is finishe. Table 2. Relations for net restore eman calculation of air conitioners Time elapse after restoration of service 0:5 0:30 0:45 :00 :5 :30 uncontrolle controlle P pb (kw) 0.6 E / E / E / E / E / E / 0.5. uncontrolle controlle Figure. Daily iversifie loa curve of controlle an uncontrolle water heater Results of appliance of the moel, for ifferent number of controlle evices, are shown in Table 3. In the Table 3 we can see work scheule for each type of evice separately, that is for each type of evice is given number of groups an number of evices accoring to the groups, initial time an length of control perio as well as time shift between the groups. For example, x 43 =5252 means that 5252 water heaters shoul be turne off in the 43 r 5- minutes lasting interval, that is at 6 hours an 45 minutes. Those water heaters will be turne off for the 3 hours. It

4 also can be conclue from the Table 3 that the total number of air conitioners accoring to the groups is greater than the total number of evices available for control, what means that some air conitioners will be turne off several times uring a ay, i.e. control perio. In the first case, when are controlle air conitioners an the same number of water heaters, system peak loa, which is 8000 MW, is reuce for 263,8 MW (3,3 %). In the secon case, when are controlle air conitioners an water heaters, this reuction is 438 MW (5,5 %), an in the thir case, when are controlle air conitioners an water heaters, the reuction amounts to 557,4 MW (7,2 %). Power system aily loa curve shape, when are controlle air conitioners an water heaters, is shown in Figure 3. Table 3. Results of DLC appliance for ifferent number of controlle water heaters an air conitioners Total number of controlle evices M WH = M AC = M WH= M AC = M WH= M AC = Water heaters x 43= 5252 x 44= 2677 x 45= x 48= 387 x 44= 5206 x 45= 6744 x 46= x 47= x 43= 7483 x 44= 6586 x 45= x 46= x 47= 4676 x 48= x 49= 5464 x 50= 7499 Working scheule Air conitioners y 48= 8405 y 49= 6667 y 50= 6359 y 5= 4706 y 52= 2696 y 53= 3995 y 54= 3670 y 58= 343 y 46= 6292 y 47= 5360 y 48= y 49= 2777 y 50= y 5= 2699 y 52= y 53= 2964 y 54= 3523 y 55= 6878 y 56= y 57= 2777 y 58= y 59= 2532 y 43= 675 y 44= 5239 y 45= 78 y 46= y 47= 9938 y 48= y 49= 2098 y 50= y 5= y 52= y 53= y 54= y 55= 9938 y 56= y 57= 2098 y 58= y 59= y 60= 575 y 6= y 62= 5678 y 63= 3668 y 64= y 65= 2098 y 66= P [kw] CONCLUSION Figure 3. Power system aily loa curve when are simultaneously controlle water heaters an air conitioners. In this paper is evelope a moel for etermining an optimal irect loa control strategy of evices in resiential sector with the aim of reucing system peak loa. It is base on linear programming. The propose moel enables etermination of the most important elements of irect loa control strategy, an consequently obtaining appropriate (optimal) coorination of loa control (turning off/on moments) between ifferent groups of evices. With this is ensure a maximal (optimal) reuction of power system peak loa (maximal utilization of available resources of controlle evices). Develope moel ensures etermination of optimal irect loa control strategy when are observe iniviual types of evices, which are controlle either by perioical turn off (cycling) or by long-term turn off, as well as ifferent types of evices simultaneously. Moreover, the propose moel can be, in an appropriate manner, applie to other loa management techniques that are being use in resiential sector. REFERENCES before DLC after DLC [] S. Talukar an C. W. Gellings, Loa Management, The Institute of Electrical an Electronics Engineers, Inc., New York (987). [2] M. A. Kuliasha, Dynamic Moel of Power System Operation Incorporating Loa Control Stuy, Oak Rige National Laboratory, Oak Rige (980). [3] K. D. Le, et. al., A Proceure for Coorinating Direct Loa Control Strategies to Minimize System Prouction Costs, IEEE Trans. on PAS, Vol. PAS 02, No. 6, pp [4] F. N. Lee an A. M. Breipohl, Operational Cost Savings of Direct Loa Control, IEEE Trans. on PAS, Vol. PAS-03, No. 5, pp

5 [5] R. Bhatnagar an S. Rahman, Dispatch of Direct Loa Control for Fuel Cost Minimization, IEEE Trans. on PWRS, Vol. PWRS-, No. 4, pp [6] H. Salehfar an A. D. Patton, A Prouction Costing Methoology for Evaluation of Direct Loa Control, IEEE Trans. on PWRS, Vol. 6, No., pp [7] Y. Y. Hsu an C. C. Su, Dispatch of Direct Loa Control Using Dyinamic Programming, IEEE Trans. on PWRS, Vol. 6, No. 3, pp [8] D. C. Wei an N. C. Chen, Air Conitioner Loa Control by Multi-Pass Dynamic Programming, IEEE Trans. on PWRS, Vol. 0, No., pp [9] J. Chen, et. al., Scheuling Direct Loa Control to Minimize System Operational Cost, IEEE Trans. on PWRS, Vol. 0, No. 4, pp [0] A. I. Cohen, An Optimization Metho for Loa Management Scheuling, IEEE Trans. on PWRS, Vol. 3, No. 2, pp [] S. H. Lee an C. L. Wilkins, A Practical Approach to Appliance Loa Control Analysis: A Water Heater Case Stuy, IEEE Trans. on PAS, Vol. PAS-02, No. 4, pp [2] J. C. Laurent, et. al, A Column Generation Metho for Optimal Loa Management via Control of Electric Water Heaters, IEEE Trans. on PWRS, Vol. 0, No. 3, pp [3] C. N. Kurucz, et. al., A Linear Programming Moel for Reucing System Peak Loa Through Customer Loa Control Programs, IEEE Trans. on PWRS, Vol., No. 4, pp [4] M. W. Davis, et. al, The Economics of Direct Control of Resiential Loas on the Design an Operation of the Distribution System, Part II, IEEE Trans. on PAS, Vol. PAS-02, No. 3, pp [5] R. F. Bischke an R. A. Sella, Design an Control Use of Water Heater Loa Management, IEEE Trans. on PAS, Vol. Pas-04, No. 6, pp [6] J. J. Bzura, Raio Control of Water Heaters in Rhoe Islan, IEEE Trans. on PWRS, Vol. 4, No., pp [7] H. Ree, et. al, Analysis of Water Heater Data from Athens Loa Control Experiment, IEEE Trans. on PWRD, Vol. 4, pp [8] J. D. E. Norell, Force Duty Cycling of Air Conitioning Units for Loa Management, IEEE Trans. on PWRS, Vol. 2, No. 4, pp [9] J. J. Bzura, Raio Control of Air Conitioning in Rhoe Islan, IEEE Trans. on PWRS, Vol. 5, No. 2, pp