Scheduling of radio-controlled heating load

B. Fox A.I. McCartney B.M.McCann

Indexing terms: Dynutnic programming, Radio teleswitcliing, Hecit energy cost nzinitnisution, 0fl:peuk electricity

Abstract: An economic loading program has been adapted to enable it to obtain an optimum heat- load profile to meet the forecast heat requirement. The heat load is represented by a ‘generator’ whose load is constrained to be negative. The incremental cost of this unit is a heat energy price. This is adjusted to obtain a heat profile containing the requisite energy. The profile is then used by a dynamic programming algorithm to derive a commitment pattern for each block. A case study is presented which shows that the procedure can minimise heat energy cost. It is also shown that use of the proposed method results in less generator load cycling. This reduced regulation duty should improve reliability.

1 Introduction

Most utilities provide cheap electrical energy for off- peak load. In the UK, these loads are now controlled mainly by radio teleswitching [l]. The system relies on phase modulation of the BBC’s 198-kHz Radio 4 sig- nal. The facility to control part of the demand has pro- vided an incentive for more flexible off-peak tariffs. Thus the traditional Economy 7 tariff is no longer restricted to seven hours of night-time supply. This has been made possible by providing separately metered supplies for heating and nonheating load. The nonheat- ing load is supplied within a fixed eight-hour period. The heating load, however, may be supplied for any seven hours over a 24-h period. The more recent ‘Auto- matic’ tariff, designed specifically for heating load, has, in addition, removed the seven-hour restriction. The period of supply is set to achieve a satisfactory temper- ature in a typical home.

The work described reflects certain conditions pecu- liar to Northern Ireland. The local utility, Northern Ireland Electricity (NIE), is responsible for scheduling generation. NIE also interacts directly with consumers through the radio teleswitching system. The decision to

0 IEE, 1998 IEE Proceedings online no. 19982356 Paper first received 18th March and in revised form 24th June 1998 B. Fox is with the Department of Electrical and Electronic Engineering, The Queen’s University of Belfast, Ashby Building, Stranmills Road, Belfast BT9 SAH, Northen Ireland A.I. McCartney and B.M. McCann are with the Cdstlereagh House Control Centre, 12 Manse Road, Castlereagh, Belfast BT6 9RT, Nothern Ireland

commit or decommit one of a small number of genera- tors can have a significant bearing on operating cost. It is therefore important for NIE to achieve good co-ordi- nation of the heating load control with generator scheduling. Effective procedures have been developed and are described in [2]. The controlled heat load has been divided into a number of Economy 7 and Auto- matic blocks. An essential requirement is knowledge of the power/time profiles of the blocks. These will depend on the tariff, the number of consumers in the block, and the weather. Many measurements were taken and analysed to model the load blocks, in partic- ular their response to ambient temperature [3]. Typical profiles are shown in Fig. 1. The first step of the cur- rent procedure is to predict the underlying demand. The heat load blocks are then placed on this demand curve in such a way as to fill the night-time ‘trough’. The objective of an almost flat off-peak demand has been largely achieved [2].

l2 t \

time, h

Fig. 1 Loud block profiles ~ E l - Automatic

The current approach is essentially heuristic. It is therefore difficult to show that the solutions are the cheapest possible. The present work aims to determine the most economic load block switching pattern auto- matically. A related exercise has been reported for the Taiwan power system [4]. Dynamic programming was used to remove blocks of air conditioning load from the demand peak. The task was made easier by assum- ing all blocks to be identical. This greatly reduced the dimension of the problem. It was therefore neccessary to develop a new approach for the heating load control problem, where the load blocks have individual pro- files.

The method contains two steps. First, an economic loading program is used to obtain an optimum profile for the entire heating load. Then dynamic program-

64 1 IEE Proc.-Gener. Trunsm. DiJtrib., Vol. 145, No. 6. November 1998

ming is used to fit the individual block profiles into the overall profile. A case study is presented to demon- strate the effectiveness of the new method. The main advantages over the existing approach are that the process is formal and repeatable.

2 Load scheduling problem

The load profiles shown in Fig. 1 are for continuous supply for a given period. In the case of the Economy 7 load, the demand is at the maximum for several hours. Then, when the associated water or space heaters reach their target temperatures, thermostats disconnect increasing amounts of load. This fall off in demand is less pronounced for the Automatic load, because the period of supply has been set to provide just enough energy to reach a given temperature.

The radio teleswitching scheme provides the opportu- nity to supply the load blocks in a number of discrete episodes. In practice, the amount of on/off switching which the scheme can apply to the load blocks is lim- ited. In the present work, a loading resolution of 0.5h has been adopted. To avoid frivolous switching, mini- mum ‘up’ and ‘down’ times of two loading intervals, or 1.0 h, have been applied. It has been found from expe- rience that these constraints avoid undue switching activity, while providing the flexibility needed to reduce the operating cost. The method used for the load block scheduling, dynamic programming, can of course han- dle other constraints if required.

Load or energy ‘payback’ is not a major concern in the present work. Payback in this context refers to the fact that when the controlled load has been offline, there is an enhanced demand when supply is restored [4]. In the load scheduling problem dealt with here, the energy requirement of the controlled load is predicted for the period of interest. The scheduling process is concerned only with the timing of the supply, not with the quantity. The heating load has a large thermal iner- tia. Any enhancement of the load following an offline period is therefore small and may be neglected in this application.

The problem of load block scheduling may now be posed as follows:

What is the block commitment pattern which satisfies a predicted energy requirement, subject to the switching constraints, at least generating cost?

In principle, all commitment patterns could be exam- ined by adding the resulting load block profiles to the basic demand profile. An economic loading calculation would then obtain the cost of each pattern. The opti- mum block commitment pattern would be the one resulting in the cheapest loading solution. However, this complete enumeration approach is not a practical proposition. One need only consider the number of possible commitment patterns for just one load block over, say, 12 hours (24 intervals). Ignoring constraints, there are about two million patterns for a 7-h block of 14 load values.

Two possible alternatives to complete enumeration have been considered: - Define a ‘heat load’ sink within a standard loading algorithm, and adjust the incremental price to match the heat energy to the forecast; then fit the individual block profiles into the overall heat energy profile.

642

- Define multiple heat load sinks corresponding to the load blocks and compute the block commitments directly. It is the first alternative which forms the basis of the method presented here. However, work is in progress on the second approach within the framework of a loading program based on Lagrangian relaxation [5].

The method adopted requires the determination of a heat load profile prior to the block scheduling process itself.

3

The first requirement is to determine how best to sup- ply the forecasted heat energy. The inputs for this phase of the process are

the base load profile for the next 24h, with managed heat load removed

the managed heat energy requirement for the next 24h

generation parameters An economic loading algorithm has been adapted to include a ‘negative generator’ or heat sink [2]. The basic algorithm is first summarised.

Determination of heat load profile

We wish to minimise the operating cost C, where P N

c [xi, (ai + bixij)Tj + 4ijsij1 (1) ,=1 i=l

a; = unit i no-load cost bi = unit i incremental cost x, = unit i power output in interval j S, = cost of starting unit i in intervalj N = number of generating units P = number of intervals TJ = interval j duration A, = 1 if unit i is on in intervalj, otherwise 0 @G = 1 if unit i starts in intervalj, otherwise 0 The start-up costs included in C are given by

where SCi = unit i cold-start cost t

ti

The cost minimisation is subject to unit loading con- straints

where Zi, ui = unit i lower and upper limits, respectively. There are demand constraints at each interval

Si, = SC, [1 - e x p ( - t / ~ ~ ) ] ( 2 )

= the time unit i has been shut down before the

= unit i cooling time constant current interval j

X i j l , 5 xzj 5 X i j U i , v j ( 3 )

N

Cxij~ij = D ~ , v j (4) i= 1

where D. = demand at interval j. The formulation is essentialfy complete when reserve constraints are added. In the present application, these are designed to ensure that a proportion R of any infeed loss following a generator outage is covered by emergency reserve from healthy machines. The reserve from a generator i at interval j is given by

rtj = min [Xijkiui , Xij(ui - xij)] , Yi,j (5)

IEE Proc-Gmer. Transin. DbWib., Vol. 145, No. 6, November I998

where kp, is the maximum reserve for the unit. The system reserve constraints may then be stated as fol- lows for the loss of unit n during intervalj:

N

&r,g - &r?lJ) 2 R L J X , J , Vn ,g (6) z = 1

The loading at each interval is a mixed integer-linear problem. The algorithm uses a search truncation method [6] to select unit combinations for costing. Lemke’s algorithm [7] is then used to solve the linear programming problem posed by each combination. The interval solutions are linked by dynamic features such as start-up costs. Forward dynamic programming [5] has been used to determine a limited number of ‘best’ paths. These paths are developed by considering all transitions to combinations in the current interval. The cheapest path when all intervals have been considered is the overall optimum.

The implementation of this algorithm contains three refinements:

units may have two incremental costs rate limits on unit run-up may be imposed unit minimum ‘up’ and ‘down’ time constraints may

be included The algorithm can be used to represent a heat load as a ‘negative generator’. Denoting this heat unit by the index h, the relevant unit parameters will be a/, = 0 lh 1 - H L uh = o SC, = 0 q* = 1 .0 k,, 0 Parameter H L is a positive number corresponding to the maximum heat load. If there is no restriction on this load, it may be set to an arbitrarily high value. The ‘start-up’ cost is of course zero. The cooling time con- stant is set to a nonzero value to avoid division by zero. The unit reserve kh is set to zero. However, if it is feasible to disconnect heating load rapidly, then it may supplement emergency reserve. This could be included by setting kh to that fraction of heat load which is dis- connectable.

8000 7000

r 6000 5000

$ 4000 3000 2000 1000

0

c-

13

2.2 2.3 heat price, $/MWh

Fig.2 Vuriation of heat energy with heat price

2.4

Because the heat unit outputs XI,, are negative, the demand will be supplemented by this amount. Conven- tional generators must therefore generate extra power to satisfy the demand. The heat unit’s incremental cost bh is actually a price. Increasing this price leads to a sit- uation where it is economic for the generators to sup- ply the corresponding energy. The way in which the

IEE PIOL -Gener Trunsm Disrrrh , Vol 145. No 6, Noceinher 1998

heat energy increases with heat price is shown in Fig. 2; this graph applies to the case study presented later. The heat price may be increased until the heat energy sup- plied equals or exceeds the forecast heat requirement HE:

xhj = -xh 3 , ‘dj P

X X h j T j 2 H E (7) j=1

The heat energy profile might be expected to fill the troughs in the demand curve. Fig. 3 shows the basic demand curve and a heat profile for the case study. It may be seen that some of the heat energy is supplied outside the off-peak period. This is because the loading algorithm will sometimes exploit unused capacity at a higher demand rather than commit an extra unit at a lower demand.

4500 i 4000 3500

3 3000 5 2500 - 1500

1000 500

n

g 2000

0 6 12 time, h

Fig. 3 Q heat load 0 demand

Demand and heat load pro$les

24

It has been shown how a conventional loading algo- rithm may be adapted to calculate an optimum heat load profile. The following Section considers how block load profiles may be scheduled to fit the overall profile.

4 Load block scheduling

A heat load profile over, say, 24h is now available. The problem is to match the individual block profiles to the overall profile. The block profiles consist of ordered 0.5-h load levels.

Consider some arbitrary interval j during the sched- uling process. In general, some of the energy within each block will have been scheduled already. The next available power level in each block 6, xb, is considered. All block combinations are examined. The combination which minimises

( g J b X b - XhJ ) 2 ( 8 )

where B = number of blocks Qlh = 1 if block b is selected, 0 otherwise

without violating the block minimum ‘up’ and ‘down’ times is chosen. The procedure results in the creation of a single path, from start to finish, consisting of the best feasible combination at each interval. This method has been implemented and works quite well. However, it suffers from a lack of flexibility. The best combination at a particular interval may be a poor basis for the next interval when constraints are taken into account. It was therefore decided to use dynamic programming (DP) [ 5 ] to provide the required flexibility.

The current version of the software develops ten paths. These emanate from the ten best combinations

643

at the first interval with a significant heat load. The paths are then developed on the basis of the smallest accumulated squared errors to the current interval. When the end of the period of interest is reached, the path with the smallest error is chosen. The approach is very similar to the generator loading process, a combi- natorial search at each interval and dynamic program- ming to handle the co-ordination of the interval solutions over time.

The DP-based algorithm generally produces a smaller squared error than the single-path method out- lined. It suffers from a tendency for the paths to coa- lesce over rime. There is scope for further improvement of the algorithm by including a mechanism to maintain path diversity.

The multipath load block scheduling method has been applied successfully to actual loading scenarios. The process is now demonstrated for a 16-unit thermal power system [SI.

5 Casestudy

The data for the 16-unit system have been modified to suit the loading software. The weakly quadratic cost curves of the original are replaced with two incremental costs. The lower cost b, applies up to a breakpoint m and the higher cost b , above m. The way in which these modified costs are obtained is described in [9]. The effect on the loading results is negligible.

Generating unit operating data are given in Table 1. Data related to the operating costs are shown in Table 2. Note that an extra unit h has been included to represent the heat load. The reserve cover R has been taken as 0.15.

Table 1: Unit operating parameters

Min. Time Unit Status k upldown

times, h

off-line at start, z, h

M W

1 must run 250-550 0.2 30115 0 10 2

3

4

5

6

7

8

9

10

11

12

13

14

15 16

h

must run

must run

must run

must run

must run

must run

must run

must run

available

available

available

available

available

available

available

available

250-550

250-520

250-520

125-443

125-443

120-320

120-320

75-280

75-280

50-1 48

50-1 48

25-1 18

25-1 18

30-100

30-1 00

-1 0000-0

0.2

0.2

0.2

0.226

0.226

0.219

0.219

0.214

0.214

0.608

0.608

0.763

0.763

0.7

0.7

0

3011 5

3011 5

3011 5

20110

2011 0

512

512

512

512

511

511

010

010

010 010

010

0 10

0 10

0 10

0 5

0 5

0 5

0 5

0 10

0 10

4 10

4 10

5 10

4 10

5 I O

6 10

0 1

The 24 1-h load levels of the original study have been replaced by 48 0.5-h load levels for the present work. This accords with normal commercial practice in the UK, and allows more scope for load block switching than is possible with a resolution of 1 h. The demand data are given in Table 3.

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Table 2: Unit cost parameters

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

h

Unit SC, $ a, $/h b,, $/MWh m, M W b,, $/MWh

282 61.3 1.682 341 2.305

262

2678

227

227

207

187

157

176

156

113

103

94

99

114

114

0

72.1

71.2

81.8

60.6

71.7

23.9

44.7

57.2

35.6

25.3

54.0

28.5

39.2

37.0

37.0

0

1.668

1.697

1.679

1.644

1.521

1.916

1.876

2.010

2.265

2.017

1.988

1.977

1.828

2.114

2.117

variable

332 2.237

317 2.21 1

320 2.193

266 2.295

266 2.137

202 2.832

202 2.736

167 2.736

167 3.074

92 2.328

92 2.401

68 2.425

68 2.182

61 2.504

61 2.509 - -

Table 3: Demand profile

Load, M W

Load, Interval Load, MW Interval MW I nterva I

1 4200 17 3632 33 3210

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

4195

4185

4170

4150

4130

41 10

4060

3980

3923

3887

3858

3832

3790

3730

3678

18

19

20

21

22

23

24

23

26

27

28

29

30 31

32

3585

3535

3488

3442

3430

3450

3420

3340

3290

3270

3230

3170

3120

3080

3110

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

3340

3500

3600

3640

3695

3765

3830

3890

3965

4055

4125

4175

4212

4238

4250

Duration of all intervals is 0.5h

It is now assumed that the block load profiles over the next 24h may be forecast. Profiles for the purposes of this study are given in Table 4. The first five profiles are typical of the Economy 7 tariff, the remainder fol- lowing the Automatic pattern.

The energy contained in the block loads is 4160MWh. The next step is to find the heat energy price which will result in a heating profile which can supply this energy. In practice, it is useful to overshoot the target by at least 5% to give the block scheduling process some leeway. The loading algorithm was run for heat energy prices in the range 0 to $ 2.40lMWh. The corresponding variation in heat profile energy is shown in Fig. 2. Using a resolution of $ O.Ol/MWh, it was found that the lowest price to give a heat energy greater than 1.05 x 4160MWh was $ 2.34lMWh (see arrow). The way in which the loading algorithm has placed this energy on the underlying demand is shown in Fig. 3.

1EE Proc -Genet Trantm Distttb , Vol 145, No 6, Noventher I998

Table 4: Block load profiles ~~

Interval Block load, MW

1 2 3 4 5 6 1 8 9 1 0 I nterva I

1 60 80

2 60 80

3 60 80

4 60 80

5 60 80

6 60 80

1 51 16

8 51 68

9 45 60

10 39 52

11 33 44

12 30 40

13 30 40

14 30 40

100

100

100

100

100

100

95

85

15

65

55

50

50

50

120

120

120

120

120

120

114

102

90

18

66

60

60

60

140

140

140

140

140

140

133

119

105

91

11

10

70

10

33 55 71 99 121

33 55 7 1 99 121

33 55 17 99 121

33 55 11 99 121

33 55 11 99 121

33 55 7 1 99 121

33 55 17 99 121

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Duration of all intervals is 0.5h

This stage of the overall process is user-driven at the moment. It could be automated, using previous experi- ence to guide the initial choice of heat energy price.

700 r 600 5 500

=- 400 - 300

200 100

n ” 0 6 12 18 24

time, h

Fig.4 profile

0 block loads

Heat and overall block loudprofiles

The heat energy profile represents 4500MWh. This exceeds the heat requirement of 4160MWh by at least 5%, as required. The block scheduling algorithm was run to obtain a schedule for the individual load blocks. The algorithm’s success in matching the total block load to the overall profile may be gauged from Fig. 4. The switching pattern for block 3, represented by 1’s for ‘on’ and 0’s for ‘off over the 48 0.5-h intervals, is typical:

000000000000000001100001110011111111100000000000

To estimate the value of the load scheduling process, three cases were costed: (i) Underlying demand C1 (ii) Underlying demand with scheduled block loads C, (iii) Underlying demand with unscheduled block loads c3

Case (iii) considered placement of the load blocks in the period of least demand, from the end of hour 10 until the end of hour 17 (intervals 21 to 34 inclusive). It was assumed that all blocks were simply switched on at the beginning of this period and supplied continuously for the times given in Table 4. The costs for the three cases were C, $ 184,498

IEE Proc.-Gener. Trunsm. Dislrib.. Vol. 145, No. 6, November 1998

C, $ 194,037 C, $ 194,494 The heat cost (C, - C,) is $ 9,539. The additional cost when the load blocks are not scheduled (C, - C,) is $ 457, or 4.8% of the minimum heat energy cost. This is in line with the result of studies based on actual operat- ing data for a medium-sized power system.

A further benefit of scheduling the heat load blocks is that generator power outputs change less. This may be quantified as a ‘control ratio’, defined [lo] as

control ratio= (sum of unit output change moduli) /(sum of demand change moduli)

The control ratios were load blocks scheduled (case (ii)) 1.597 load blocks unscheduled (case (iii)) 2.354 The lower control activity for the scheduled load blocks improves generator reliability. There will also be small improvements in thermal efficiencies which are not reflected in the costs calculated by the loading algorithm.

6 Conclusions

A procedure for scheduling radio-controlled heating load has been proposed. There are two main stages. First, an economic loading algorithm was used to determine how best to place the required heating energy on the basic demand profile. It was shown how a standard loading algorithm based on mixed integer- linear programming at each interval and co-ordinated over time by dynamic programming may be adapted to this purpose. An extra unit has been included to repre- sent the heating load. This unit consumes power, and its incremental cost is a heat energy ‘price’. The price is adjusted to achieve a target heat energy. The second stage of the procedure was to match a number of block loads, each with a predicted profile, to the overall heat load profile. A special purpose algorithm, also based on dynamic programming, was developed for this pur- pose.

A case study for a 16-unit power system with ten load blocks was presented. It was shown that the cost of the heat energy can be reduced by about 5% with the proposed load block scheduling method. Also, gen- erator load changes, as measured by control ratio, are reduced. Similar results have been obtained with actual data for a medium-sized power system.

It is intended to improve the load block scheduling algorithm. In particular it is hoped to find a way to maintain path diversity for the whole scheduling period. In parallel with work on the existing procedure, a more radical approach is being developed. It is pro- posed to represent each load block within a Lagrangian relaxation framework. This would result in complete integration o f controlled heating load with generation scheduling.

7 Acknowledgment

B. Fox is grateful to the Royal Academy of Engineer- ing and Northern Ireland Electricity for providing sup- port under the Industrial Secondment Scheme.

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8 References

1 EDWARDSON, S.M., EYRE, B.E., HENSMAN, G.O., and WRIGHT, D.T.: ‘A radio teleswitching system for load manage- ment in U K . Proceedings of the IEE Conference on Metering, appcirutus and tarijjjs f o r electricity supply, London, 1982, pp. 40- 46

2 McCARTNEY, A.I.: ‘Load management using radio teleswitches within NIE’, Power Eng. J . , 1993, 7, (4), pp. 163-169

3 McCANN, B.M.: ‘Measurement and control of power system heating load’. MSc Thesis, the Queen’s University of Belfast, 1993

4 HSU, Y.-Y. , and SU, C.-C.: ‘Dispatch of direct load control using dynamic programming’, IEEE Trans., 1991, PWRS-6, ( 3 ) , pp. 1056-1061

5 WOOD, A.J., and WOLLENBERG, B.F.: ‘Power generation, operation and control’ (Wiley, New York, 1996, 2nd edn.)

6 BOND, S.D., and FOX, B.: ‘Optimal thermal unit scheduling using improved dynamic programming algorithm’, IEE Proc. C, 1986, 133, ( I ) , pp. 1-5

7 HADLEY, G.: ‘Linear programming’ (Addison-Wesley, Reading, MA, 1962)

8 VILLASECA, F.E., and FARDANESH, B.: ‘Fast thermal gener- ation rescheduling’, IEEE Trans., 1987, PWRS-2, (l), pp. 65-71

9 RADI, K.M., and FOX, B.: ‘Power system economic loading with flexible emergency reserve provision’, IEE Pron. C, 1991,

10 BREWER, C., CHARLES, G.F., PARISH, C.C.M., PREWET- T, J.N., SALTHOUSE, D.C., and TERRY, B.J.: ‘Performance of a predictive automatic load-dispatching system’, Proc. IEE, 1968, 115, ( IO), pp. 1577-1586

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646 IEE Proc.-Gener. Tronsm. Disrrih.. Vol. 145, No. 6, November 1998