Scheduling Smart Home Appliances Using Mixed Integer Linear Programming

Transcription

1 211 5th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, Deceber 12-15, 211 Scheduling Sart Hoe Appliances Using Mixed Integer Linear Prograing Kin Cheong Sou, Jaes Weier, Henrik Sandberg, and Karl Henrik Johansson Abstract This paper considers the iniu electricity cost scheduling proble of sart hoe appliances. Operation characteristics, such as expected duration and peak power consuption of the sart appliances, can be adjusted through a power profile signal. The optial power profile signal iniizes cost, while satisfying technical operation constraints and consuer preferences. Constraints such as enforcing uninterruptible and sequential operations are odeled in the proposed fraework using ixed integer linear prograing (MILP). Several realistic scenarios based on actual spot price are considered, and the nuerical results provide insight into tariff design. Coputational issues and extensions of the proposed scheduling fraework are also discussed. I. INTRODUCTION Electricity consuption varies between different hours of the day, between days of the week, and between seasons of the year, where the highest power deand typically occurs when the outdoor teperature drops. In recent years, the power deand has reached new peak levels and created extra stress to balance deand and generation. Environental and econoical reasons will, in the near future, require distribution copanies to consider ore coplex power balance scenarios based on the introduction of large scale renewable electricity generation, personal electrical vehicles (PEVs) and distributed electricity generation in residential areas. Interittent renewable energy sources, such as wind, are dynaic by definition and will require additional balancing power to aintain quality of electrical supply to consuers. Additionally, an increasing nuber of PEVs will introduce high electricity consuption that is not always predictable. Both the wind power s dynaic contribution to electricity generation and the PEVs rando deand of electricity require a balancing force in the electricity grid. Load balancing of urban electrical loads, such as residential/industrial electricity consuption, can be accoplished by iniizing the usage of non-renewable generation and scheduling controllable loads to ties when renewable energy generation is high. Particular ways to engage the consuers in participating in load balancing is achieved through econoic incentives such as tie-varying electricity tariff (e.g. spot pricing [1]), or CO 2 footprint for environentally concerned consuers (e.g. the Stockhol Royal Seaport The authors are with the ACCESS Linnaeus Center and the Autoatic Control Lab, the School of Electrical Engineering, KTH Royal Institute of Technology, Sweden. {sou,weierj,hsan,kallej}@kth.se This work is supported by the Swedish Energy Agency, the Swedish Governental Agency for Innovation Systes (VINNOVA), the Swedish Foundation for Strategic Research (SSF) and the Knut and Alice Wallenberg Foundation. project [2]). An illustration of spot price is shown in Fig. 1. Tariff (USD/kW*h) Hour Fig. 1. Electricity tariff (spot price) for New York City on February 15th, 211. Data taken fro NYISO. The web address is References such as [3] [5] have deonstrated the value of tie-varying electricity tariff in the anageent of the power grid, especially in the reduction of peak power consuption; however, such load balancing is feasible only if the consuers are both able and willing to consider tariff inforation. For instance, it is unrealistic to expect ost consuers to identify the ost econoical operation of their appliances in the presence of dynaic tariff prices and peak consuption penalties. Hence, an autoatic decision support syste is highly desirable, that either directly takes control of the appliance operation or provides siple advice to the consuers. Prior works exist for load balancing fro both industrial and residential consuers perspective. For exaple, [6] describes the use of linear prograing to schedule the electricity use of an industrial consuer for product anufacturing. Under various assuptions and odel restrictions, [7] derives analytical forulas and scheduling strategies for an industrial consuer. A reinforceent learning based appliance scheduling fraework for residential consuers is introduced in [8], assuing that both the tariff and the consuers requests for appliances operation follow soe Markov decision odel. In this paper, we also consider the proble of residential appliances scheduling; however, unlike [8], the proposed scheduling fraework is deterinistic and the cost calculation is based on the tariff, typically known 24 hours in advance. In addition, this work ais to odel the decision proble as realistically as possible. Hence, siplified analytical and linear prograing based analysis tools are not sufficient for the purpose of this work. Instead, we intention /11/$ IEEE 5144

2 ally explore the full odeling power of ixed integer linear prograing (MILP). While the planning is 24 hours ahead, in practical household applications real-tie adjustents are often required. Thus, this paper conducts a nuerical study and deonstrates that in a typical scheduling scenario, preaturely terinating the MILP solving can result in very good suboptial schedules. At the sae tie, the solve tie can be reduced draatically and is shown to be viable even for real-tie household applications. The reainder of this paper is organized as follows. Section II otivates and describes the appliance scheduling proble. Section III forulates the appliance scheduling proble into a MILP proble. Nuerical studies using the MILP fraework is provided in Section IV. II. SCHEDULING HOME APPLIANCES In this section, we provide a written description of the appliance scheduling proble which will be atheatically forulated as a MILP proble in the following section. In the proposed scheduling fraework, an appliance operation process is divided into a set of sequential energy phases. An energy phase is an uninterruptible sub-task of the appliance operation which uses a pre-specified aount of electric energy. The energy phases are sequential since the next appliance sub-task cannot begin until the previous sub-task is copleted (e.g. the washing achine agitator cannot start until the basin is filled with water). In addition to being uninterruptible and having a specified energy usage, each energy phase ay have additional anufacture-defined constraints. These constraints for a specific energy phase include bounding the instantaneous power consued (corresponding to the axiu operating power and idle power) and achieving a axiu execution tie. While all energy phases associated with a single appliance ust be run sequentially, there can be delays between the energy phases so long as the energy phase ordering is preserved (e.g. the washing achine agitator ust start within ten inutes of the basin being filled). Besides the technical requireents for an energy phase specified by the anufacturer, there exist additional appliance-level operational constraints. For instance, a certain appliance cannot start before soe other appliance finishes (e.g. washing achine and dryer). Moreover, for safety reason the total power assigned to all appliances at any oent cannot exceed a liit called peak signal. Finally, there ight be user specified tie preferences, requiring that certain appliances should be run within soe particular tie intervals (e.g. washing dishes with the dishwasher between 4p and 6p). To eet the appliance energy phase and operational constraints, the proposed appliance scheduling fraework deterines the power assignents, as functions of tie over the execution period (e.g. a day), to all energy phases of all appliances. The tie-dependent power assignents are called power profiles and each corresponds to an appliancespecific energy phase. The objective of the proposed scheduler is to find the least expensive set of power profiles, while satisfying the necessary operational constraints. To atheatically design the appliance scheduler, the following section introduces a MILP proble based on the appliance scheduling proble described in this section. III. MIXED INTEGER LINEAR PROGRAM FORMULATION The power profile scheduling decision proble in Section II can be odeled as a MILP proble (e.g. [9]). To define a MILP instance, a atheatical description of the proble setup, decision variables, cost function and constraints is provided below. A. Proble setup The appliances execution period is discretized into unifor tie slots (e.g. 5 inutes per slot). The nuber of appliances considered for scheduling is denoted N, and the nuber of uninterruptible energy phases for each appliance is denoted n i for i=1,2,...,n. Note that in this paper appliance and energy phase are abstractions. For instance, a single oven for lunch and dinner can be treated as two separate appliances. B. Decision variables The discretized power profiles are the output of the proposed scheduler and are denoted p k i j, corresponding to the energy assigned to energy phase j of appliance i during the whole period of tie slot k. The typical unit for p k i j is kwh. The power profiles p k i j are real (i.e., continuous) decision variables. In addition to p k i j, auxiliary binary decision variables are required to indicate whether a particular energy phase is being processed or not. These binary decision variables are denoted x k i j {,1}. xk i j = 1 if and only if for appliance i, energy phase j is being processed during tie slot k. The decision variables x k i j are required, for instance, to odel the energy phase sequential operation constraint described in Section II. In addition, two other sets of binary decision variables are needed to odel the decision proble. One set is denoted as s k i j, with a value of one indicating that, in appliance i, energy phase j is already finished by tie slot k. The other set of auxiliary binary decision variables is denoted as k j. These decision variables are used to indicate whether at tie slot k, appliance i is aking a transition between running phase j 1 to j. Hence, the index j ranges fro 2 to n i only (i.e., the nuber of energy phases in appliance i inus one). More explanation regarding the use of the auxiliary binary decision variables will be given in Section III-D. C. Cost function The objective of the proposed scheduler is to iniize the total electricity cost for operating the appliances. In this paper the cost calculation is based on a given 24-hour ahead electricity tariff (e.g. USD per kwh). The tariff curve is piecewise constant, with possible jups at the start of each hour. Fig. 1 shows an exaple of a typical tariff curve. 5145

3 Let c k denote the electricity tariff for tie slot k. The total electricity cost for running all appliances is ) D. Constraints k=1c k( N i=1 n i p k i j j=1 To ease the description, the constraints are organized into two groups energy constraints and tiing constraints. 1) Energy Constraints: Energy phase energy requireent: To ake sure that the energy phases fulfill their energy requireents, the following constraint is iposed: (1) p k i j = E i j, i, j (2) k=1 where E i j is the energy requireents for energy phase j in appliance i. These are technical specifications fro the appliances. Instantaneous energy phase power assignent bounds: To odel whether an energy phase is being processed during tie slot k, as well as the lower and upper liits of power assignent to the phase, the following constraint is iposed: P k i j x k i j p k i j P k i j x k i j, i, j,k (3) where P k i j and P k i j are appliance specific data characterizing the lower and upper liits of power assignent to the energy phases. Note that if x k i j =, then the inequalities above collapse to a single condition p k i j =. Power safety: The power safety constraint (i.e., upper liit of the total energy assigned in any tie slot) can be odeled as N i=1 n i j=1 p k i j PEAK k, k (4) where PEAK k is the peak signal (i.e., total slot energy upper bound) at tie slot k. The peak signal is provided by the external power grid operator, which can be a deand response signal. 2) Tiing Constraints: Energy phase process tie liits: To odel the liits on energy phase process tie, the following constraint is enforced: T i j k=1 x k i j T i j, i, j (5) where T i j and T i j are the lower and upper liits of the nuber of tie slots for energy phase j in appliance i to be processed. Uninterruptible operation: An energy phase being uninterruptible eans that it cannot be resued. This can be odeled by the constraint that, for all i and j, x k i j = if there exists an earlier tie slot k<k such that x k i j = 1 and x k+1 i j =. An alternative constraint can be iposed with the aid of the auxiliary decision variables s k i j introduced in Section III-B: x k i j 1 s k i j i, j,k (6a) x k 1 i j x k i j s k i j i, j, k=2,3,..., (6b) s k 1 i j s k i j i, j, k=2,3,..., (6c) In constraint (6a), if s k i j = 1, then during tie slot k energy phase j in appliance i is already finished. Hence, the corresponding x k i j ust be. The condition triggering s k i j = 1 is that x k i j switch fro 1 to (i.e., the phase is just finished). This is the situation in (6b). Then s k i j should reain unity, as (6c) iposes. Sequential Processing: Sequential processing of the energy phases of an appliance eans that an energy phase cannot be processed unless its preceding phases have finished. This condition can be conveniently described using the auxiliary decision variables s k i j as follows: x k i j s k i( j 1), i,k, j = 2,3,...,n i (7) Siilarly, to odel the sequential operation between appliances, a constraint siilar to above can be iposed: x k i1 sk ĩnĩ, k (8) with ĩ being the index of the appliance which ust be finished before i can start. In above, s k pertains to appliance ĩnĩ ĩ, energy phase nĩ (i.e., the last phase of appliance ĩ) and tie slot k. Between-phase delay: To count the nuber of tie slots spent between the energy phases in an appliance, the auxiliary -1 decision variables ti k j defined in Section III-B can be utilized. During any tie slot k, ti k j = 1 if and only if that appliance i has finished processing energy phase j 1, and it is waiting to process the phase j (i.e., phase j is not being processed or finished). The corresponding constraint is t k i j = sk i( j 1) ( x k i j + sk i j), i,k, j = 2,3,...,ni (9) Note that x k i j + sk i j 1 because an energy phase cannot siultaneously be processed and finished (cf. (6a)). Hence, the equality in (9) is valid. With ti k j defined, the constraint enforcing the lower and upper liits of the nuber of transition tie slots (i.e., delay between energy phases) can be written as D i j k=1 t k i j D i j, i, j = 2,3,...,n i (1) In above, D i j and D i j are appliance technical specifications describing the between-phase delay bounds (lower and upper, respectively) in the nuber of tie slots. User tie preference: The household user can set up the tie preference constraints, specifying the tie interval a particular appliance ust be finished within. Alternatively, this eans that the appliances cannot be run outside of the tie preference interval. The constraints are written as x k i j TP k i, i, j,k (11) 5146

4 where TP k i characterizes the tie preference interval. That is, TP k i = if and only if none of the energy phases of appliance i can be processed during tie slot k. E. MILP forulation To su up, the proposed iniu electricity cost appliance scheduling proble can be suarized into iniize p, x, s, t cost function (1) subject to constraints (2) (11) p k i j x k i j s k i j R, i, j,k {,1}, i, j,k {,1}, i, j,k t k i j {,1}, i,k j = 2,...,n i (12) The MILP proble such as (12) is a classical optiization paradig. It can be solved using algoriths such as branchand-bound and cutting-plane ethod (e.g. [9], Chapter 11). Coercial (and acadeic) ipleentations of these algoriths are available (e.g. CPLEX and Gurobi). IV. NUMERICAL STUDIES All experients in this paper are perfored on a laptop with an Intel Core i5 2.53GHz CPU and 4GB of eory. A. Two case studies based on tariffs in Sweden and NYC In this experient, two instances of the MILP scheduling proble in (12) are solved using CPLEX (using the YALMIP MATLAB interface [1]). In the first instance, the electricity tariff (i.e., c k in (1)) is taken to be the spot price on Feb 15th, 211 in Sweden. Fig 2 shows the Swedish tariff. On.61 finished before the dryer can start. The above specifies the constraints in (8) and (11). The values of the technical specifications in reaining constraints can be found in the Appendix. Finally, the peak signal in (4) is assued to be constant, and is always equal to 55 Wh. After solving the first instance (with c k being the Swedish spot price), the su of energy assigned to all appliances in each tie slot, as well as the electricity tariff, are plotted in Fig. 3, which verifies that the proposed scheduler assigns electricity usage when it is cheap. Electricity price (SEK/(kW*h)).65.6 price power Min cost schedule hour Fig. 3. Total energy assignent (for each tie slot) and the electricity tariff, in the Sweden case. The iniu cost schedule assigns energy only when the tariff is low, subject to the constraints fro (2) through (4). The analogous result for solving the instance with the NYC tariff are shown in Fig. 4. Fig. 5 also shows the detailed.1 price power Min cost schedule 4 4 Total assigned power (W) Tariff (SEK/kW*h) Electricity price (USD/(kW*h)).5 Total assigned power (W) Hour Fig. 2. Electricity tariff (spot price) for Sweden on February 15th, 211. Data are taken fro Nord Pool Spot. The web address is the other hand, the tariff in the second instance is the spot price of New York City on Feb 15th, 211 in Fig. 1. The (planned) execution period is fro 9a to the end of the day. The length of the tie slots is 5 inutes. There are three controllable sart appliances including a dishwasher, a washing achine and a dryer. The household user iposes the following tie preference (cf. (11) in Section III-D). The dishwasher is run between the beginning of 7p and the end of the day. The washing achine and dryer can be run anytie between the beginning of 9a and the end of 11p. However, the washing achine phases ust be hour Fig. 4. Total energy assignent (for each tie slot) and the electricity tariff, in the NYC case. The iniu cost schedule assigns energy only when the tariff is low, subject to the constraints fro (2) through (4). power profiles achieving the iniu cost in the NYC case. The optial electricity cost in the Sweden case is SEK (about 15 SEK a onth if the schedule is applied every day). On the other hand, the optial electricity cost in the NYC case is.3256 USD (about USD a onth if the sae schedule is applied every day). To understand how uch cost saving the proposed scheduling fraework can achieve, the optial scheduling proble in (12) can be turned into the worst case scheduling proble by changing the cost function in (12) fro iniization to axiization. 5147

5 power (W) power (W) power (W) 3 1 dishwasher washing achine dryer hour Fig. 5. The iniu cost power profiles for the appliances in the NYC tariff case. Different energy phases are painted with different colors. For the Sweden case, the worst cost is SEK (about 2.5% ore than the optial cost). The saving is quite insignificant in this case. On the other hand, for the NYC case, the worst cost is.4781 USD (about 47% ore than the optial cost). The case studies here confirs the intuition that tariff fluctuation needs to be large enough to otivate changes of power consuption behavior. In the NYC case, the ratio of the axiu and iniu tariff is On the other hand, in the Swedish case the ratio of the axiu and iniu tariff is only With the axiu saving of only 2.5%, there is no otivation for the consuers to change their behaviors. B. Coputation tie experient For real-tie household application, the solving of the scheduling proble in (12) cannot be too tie-consuing. In (12), the tuning paraeter which is responsible for the tradeoff between coputation tie and odel fidelity is the length of the tie slot. Table I shows the statistics of solving the NYC instance with three different values of tie slot lengths. Table I suggests that while the tie TABLE I SOLVING THE NYC INSTANCE WITH DIFFERENT TIME SLOT LENGTHS tie slot length in cost ax cost ax saving solve tie 3 in $.3257 $ % 86 sec 5 in $.3256 $ % 83.6 sec 1 in $.3251 $ % 15.4 sec slot length has a significant ipact on coputation tie, its effect on the optial cost is not very obvious. This justifies the use of lower fidelity optiization odels, so long as the teporal constraints (e.g. process tie bounds) are reasonably captured. The MILP solver CPLEX has the ability to preaturely terinates the optial solution search. In particular, CPLEX can stop as soon as it finds a feasible solution of (12). Also, CPLEX allows the user to specify a tie liit upon which the solving is terinated, irrespective of whether a feasible/optial solution has been found or not. In the following experient, CPLEX is used to first find a feasible solution (this is the iniu requireent of using CPLEX in the appliance scheduling proble). Then the feasible solution is used as an initial guess for the next CPLEX runs, with variable solve tie liits. The costs of these runs are then copared against the optial cost obtained by a single CPLEX solve (i.e., the second and third rows in Table I). The results of the experient with the NYC case with the tie slot lengths being 1 and 5 inutes are shown in Table II. TABLE II RELATIVE ERROR VS CPLEX ALLOWED RUNTIME. NYC CASE WITH DIFFERENT TIME SLOT LENGTHS d (IN MINUTES). IN THE d = 1 AND d = 1 d = 5 d = 5 CASES THE FIRST FEASIBLE SOLUTIONS ARE OBTAINED, RESPECTIVELY, AFTER 1.19 AND 6.94 SECONDS runtie (s) rel. err. % runtie (s) rel. err. % Table II suggests that the suboptial power profiles returned by preaturely terinating CPLEX are acceptable substitutes for the true optial profiles which are uch ore tie consuing to find. In particular, in the 1-inute tie slot case the 1-second feasible solution incurs no ore than.5% of relative error (w.r.t. the true optial cost.3251 in Table I). In the 5-inute tie slot case the 7-second feasible solution incurs no ore than.35% of relative error. The above experient deonstrates that it is proising to apply the proposed fraework in the real-tie household appliances scheduling scenario. C. Scalability Test To test the liit of coputation tie and eory requireents for solving (12), hypothesis scenarios are considered with increasing nuber of appliances (each appliance has 6 energy phases). In these scenarios, the energy and tiing constraints in Section III-D are randoly specified. The NYC tariff is chosen, and the length of the tie slots is 1 inutes. The rando scenarios are solved using (a) CPLEX to optiality, (b) CPLEX for the first feasible solution as in Section IV-B, (c) a siple strategy to run the appliances as soon as possible (ASAP), provided that the constraints are satisfied, and (d) the analogous as late as possible (ALAP) strategy. CPLEX runs out of eory for solving the scenario with 1 appliances. The ethod for finding the first feasible solution fails with 2 appliances. This verifies the intuition that (12) does not adit scalable solution algoriths, and it should be restricted to the case of a single household with a few appliances (e.g. less than five). Figure 6 shows the experient results for the scenarios with less than 9 appliances. The solve tie to optiality increases rapidly as the nuber of appliances increases. On the other hand, for the first feasible solution only the solve tie reains reasonable (about 4 sec for 9 appliances) with good approxiation quality (less than 5% of relative increase in electricity cost). Note also that the ASAP strategy fares well, but the ALAP one is inaccurate. Since the accuracy of the ASAP and ALAP schees depends sensitively on the tariff, the validity of these schees are questionable. 5148

6 Solve tie (sec) # of appliances Fig. 6. Coputation cost and relative error for solving randoly generated scenarios with different nuber of appliances. Blue solid line with circles: solve tie for CPLEX optial; Blue dashed line with x : solve tie for CPLEX first feasible solution; Red dotted line with dots: CPLEX first feasible solution relative error; Red dotted line with downward pointing triangles: ASAP relative error; Red dotted line with upward pointing triangles: ALAP relative error. CONCLUSION In this paper a MILP based sart appliance scheduling fraework is proposed, capturing all relevant appliance operations. With appropriately defined tariff (e.g. the NYC case), the proposed fraework can result in a scheduling achieving about 47% of axiu cost saving. In addition, it is deonstrated that good quality approxiate solutions can be obtained in a reasonable aount of coputation tie (e.g. in about 1 second an approxiate solution with relative error less than.5% can be obtained). Finally, the proposed fraework can be extended to incorporate renewable energies, battery and the ulti-objective optiization with respect to energy consuption and CO 2 footprint. V. ACKNOWLEDGEMENTS The authors are grateful to the Stockhol Royal Seaport project collaborators including Electrolux and ABB. REFERENCES [1] F. Schweppe, M. Caraanis, R. Tabors, and R. Bohn, Spot pricing of electricity. Boston, MA, USA: Kluwer Acadeic Publishers, [2] Stockhol royal seaport project, [3] A. Sanghvi, Flexible strategies for load/deand anageent using dynaic pricing, Power Systes, IEEE Transactions on, vol. 4, no. 1, pp , Feb [4] J. Pyrko, Load deand pricing - case studies in residential buildings, in International Energy Efficiency in Doestic Appliances and Lighting Conference, 26. [5] P. Fritz and E. Jörgensen and S. Lindskoug, Elforsk technical report 9:7, Elforsk, Tech. Rep., 29, Report written in Swedish with English suary. Available online fro [6] B. Daryanian, R. Bohn, and R. Tabors, Optial deand-side response to electricity spot prices for storage-type custoers, Power Systes, IEEE Transactions on, vol. 4, no. 3, pp , Aug [7] J. Roos and I. Lane, Industrial power deand response analysis for one-part real-tie pricing, Power Systes, IEEE Transactions on, vol. 13, no. 1, pp , Feb [8] D. ONeill, M. Levorato, A. Goldsith, and U. Mitra, Residential deand response using reinforceent learning, in IEEE SartGrid- Co, 21. [9] J. Tsitsiklis and D. Bertsias, Introduction to Linear Optiization. Athena Scientific, Relative error (%) [1] J. Löfberg, Yalip : A toolbox for odeling and optiization in MATLAB, in Proceedings of the CACSD Conference, Taipei, Taiwan, 24. [Online]. Available: [11] A. Rugo, Power profiles for sart appliances, Private counication, ELECTROLUX ITALIA S.P.A. APPENDIX A. Technical specifications of the sart appliances The nuber of energy phases in the dishwasher, washing achine and dryer processes are, respectively, 6, 8 and 1. The data of the scheduling proble are derived fro the technical specifications of the three appliances, which are listed in Table III, Table IV and Table V respectively. TABLE III DISHWASHER TECHNICAL SPECIFICATIONS [11] Energy phase Energy Min power Max power Noinal op. (Wh) (W) (W) tie (in) pre-wash wash st rinse drain nd rinse drain & dry TABLE IV WASHING MACHINE TECHNICAL SPECIFICATIONS [11] Energy phase Energy Min power Max power Noinal op. (Wh) (W) (W) tie (in) oveent pre-heating heating aintenance cooling st rinse nd rinse rd rinse TABLE V DRYER TECHNICAL SPECIFICATIONS [11] Energy phase Energy Min power Max power Noinal op. (Wh) (W) (W) tie (in) drying In ters of the optiization proble in (12), the energy requireents E i j are listed in the Energy colun in the tables. The lower and upper liits for energy assignent in each tie slot, P k i j and P k i j in (3), are listed in the Min power and Max power coluns in the tables, respectively. The last colun in the above tables list the noinal operation tie of all phases. In this paper, it is assued that the operation tie of the energy phases can be between 8% and 12% of the noinal tie. The tie slot liits T i j and T i j in (5) are obtained by dividing the operation tie liits (in inutes) by the tie slot length (i.e., 5 inutes), and rounding is perfored where necessary. Finally, the lower liit for between-phase delay D i j in (1) for all phases is assued to be zero, while the upper liits D i j are 5, 1 and inutes for the energy phases in the dishwasher, washing achine and dryer respectively. 5149