Modelling of demand and losses in real-life water distribution systems

Urban Water 2 (2000) 131±139

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Modelling of demand and losses in real-life water distribution systems Dusan Obradovic * University of Belgrade, 11000 Belgrade, Yugoslavia Received 16 December 1999; received in revised form 24 August 2000; accepted 11 September 2000

Abstract Despite the rapid development of modelling methods in the water industry, some problems of water distribution network modelling are still unresolved, such as how to represent adequately real demand and losses. Both are dependent upon service pressure, yet the relationship is not fully known. The paper discusses various proposals and shows how the modelling procedure could be improved ± if that relationship were known. A case from practice is included for illustration. It describes how a real emergency was analysed on the model with the help of the local sta€ and using past experience. A few suggestions for further work are included at the end of the paper. Ó 2000 Published by Elsevier Science Ltd. Keywords: Water demand and losses; Pressure-related demand; Leakage; Modelling emergencies; Water distribution systems

1. The problem Mathematical modelling has become an indispensable tool for the design of water supply and distribution systems, especially since powerful PCs have become accessible to all. One may argue that modelling is, and always was, an integral part of the design process, which modern computers have made only easier. The next logical step was to apply this powerful tool for the operational management of real water distribution systems. Despite many successes in practice (Boulos, Heath, Freiberg, Ro, & Meyer, 1996; Burnell, Race, & Evans, 1993; Cesario, 1995; Davies, 1993; Dzadey & Price, 1996; Johnson, Casey, & Turner, 1993; Joyner & Wright, 1996; Nerenberg & Butterworth, 1996; Nguyen, 1994; Rance, Bounds, Tennant, & Ulanicki, 1993; Ray, 1996; Schulte & Malm, 1993; Warren & Langley, 1996; Williams, 1996), a number of diculties still remain to be solved. The foremost one ± in the author's opinion ± is how to model the demand and the losses at the level of individual nodes. In any real-life system this information is never fully available, even if the total demand and the total loss are known rather well. It is true that data on demand can be found in the literature (Bailey, Jolly, & Lacey, 1986; Edwards & Martin, 1995; Obradovic & Lonsdale, 1998), but only

*

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1462-0758/00/$ - see front matter Ó 2000 Published by Elsevier Science Ltd. PII: S 1 4 6 2 - 0 7 5 8 ( 0 0 ) 0 0 0 5 1 - 0

for the regular situations. Information about the possible in¯uence of local pressure upon demand is sadly lacking. Note that the designer's position is easier: for him the demand is a given or selected value, pressures should be within the given range, and losses might be allocated in a rather arbitrary way ± perhaps just by increasing the demand proportionally to include losses. These assumptions are not valid for the operational management of real-life systems. If one compares daily diagrams for total demand of the whole system with corresponding data captured at the level of (relatively small) demand management areas, one will discover that the ®rst has much smaller amplitude in comparison with the latter: the minimum night ¯ow (MNF) is relatively higher and the morning/evening peaks are less prominent, see Fig. 1 (Obradovic & Lonsdale, 1998). The question arises: why are these diagrams not similar? The answer is obvious: the losses must also be taken into account. As a rule, they are higher during the night ± when pressures in the network are high ± and smaller during the daytime, for the same reason (see pressure changes in Fig. 1(b)). A relationship between leakage and pressure was proposed by Ridley (1980) and has been widely accepted in practice (Bessey, 1985). The conclusion is not only that losses must be taken into account, but also that they must be modelled separately from the pure demand. One option is to introduce separate daily diagrams for losses as in the example in Fig. 2, where one such proposal is given alongside a typical domestic diagram for comparison. Of course,

132

D. Obradovic / Urban Water 2 (2000) 131±139

Q tot

Q

Data

Telemetry

MNF

0

24 h

Time

a) Total Demand

Distribution System Watermeter

p

Q p Data

Logger

Q dem

Notebook

MNF

A Demand Management Area 0

24 h

Time

b) One DMA

Fig. 1. Daily diagrams at di€erent levels.

such diagrams are highly hypothetical, but still are very valuable in the process of model calibration. Germanopoulos and Jowitt (1989) have tried a different approach: the losses are allocated to individual pipes and computed as Sij ˆ c1 Lij …Pijav †1:18 ;

…1†

where Sij is the leakage out¯ow from the pipe connecting nodes i and j, c1 is a constant depending on the state of network, Lij is the length of the same pipe, and Pijav is the average pressure along the pipe. The demand patterns are presumably pressure-invariant. A similar approach was used by Salgado, Rojo, and Zepeda (1993). The data for spatial allocation of demand and losses might be collected from various sources: the telemetry,

control watermeters, water bills, GIS, etc. Successful model calibration was reported by several authors, but again mostly for the regular circumstances, i.e., the same as those prevailing at the time when original data has been captured. However, it is questionable whether the same diagrams should be used in a di€erent situation, like emergencies, when certain parts of the network are subjected to much higher pressures, while others are depressurised completely. Some relationship between demand and losses on one hand, and service pressures, on the other, is clearly needed. If it were known, and in an algebraic form, then further development of present methods would be possible as will be demonstrated below.

2. The node (H) equations method A part of a distribution network, around node i, is …i† shown in Fig. 3. It feeds the local consumption, qdem ,

Demand

q dem

q

q(i)

loss

q(i) Losses

Node "i" 0

loss

dem

6

12

18

Time (h) Fig. 2. Daily diagrams of domestic demand and relevant losses.

24

Qi j

Node "j" Fig. 3. Part of distribution network around the node i.

D. Obradovic / Urban Water 2 (2000) 131±139 …i† qloss .

and some water is lost there, The computation begins by allocating arbitrary heads to all nodes in the network. The ¯ow through all pipes can be computed afterwards; then the balance for each node will show whether the estimate was correct or not. The ¯ow through one pipe ± for example, from node i to node j ± can be computed as ÿ
u …2† Qij ˆ Cij jHi ÿ Hj j sgn Hi ÿ Hj ; where Hi and Hj are relevant pressure heads in nodes i and j, Cij the unit capacity of pipe ( ˆ ¯ow which corresponds to 1 m of pressure head di€erence), and the exponent u is

taining the linear terms only, the following result is obtained: Ni X odi odi hi ‡ hj ˆ ÿd…0† i ; ohi oh j jˆ1

ˆ

ˆ 0:54 for Hazen±Williams formula:

jˆ1

…i†

ˆu

…3†

By computing ¯ows through all Ni pipes connected to the node i and deducting local demand and losses, one ®nally arrives at: Ni X

…i†

Qij ÿ qdem ÿ qloss ˆ di …Hi ; H1 ; H2 ; . . . HNi †:

…4†

Here di represents the magnitude of error ± it should be equal to zero. The value of di is clearly a nonlinear function of the assumed values of head in all neighbouring nodes. By varying these heads, one should ®nd the solution, which is indicated by all di values close to zero. This can be achieved by di€erent methods; the Newton±Raphson method is widely used. The problem is to ®nd corrections of head, denoted by hi …0†

hi ˆ Hi ÿ Hi

…5†

in such a way that the nonlinear function di is reduced to zero. By expanding function i in series and req

Ni X oQij …i†0 …i†0 ÿ qdem ÿ qloss ohi jˆ1 Ni X

o ˆ ohi

Function ``signum'' is de®ned as for x > 0; for x ˆ 0; for x < 0:

ÿ

u

Cij jHi ÿ Hj j sgn Hi ÿ Hj

jˆ1

Ni X

uÿ1

Cij jHi ÿ Hj j

jˆ1

…i†0

odi odi ˆ ˆ ÿu Cik jHi ÿ Hk juÿ1 ˆ Bik : oHk ohk

0

s

‰Bim Šhm ˆ ÿd…0† i :

…7†

…i†0

…8†

…9†

The further procedure is well known (see, for instance, Obradovic & Lonsdale, 1998) and need not be repeated here in great detail. Brie¯y, by solving the system of Eqs. (9) one can get the unknown corrections hm , ®nd new values of pressure head Hm and compute all ¯ows in the system by using Eq. (4). If the error di is below an acceptable level in all nodes the solution is found; otherwise the procedure should be repeated again. Losses

Demand

1

0

q

k F= q a) Procedure III

…i†0

One equation of the type of Eq. (8) has to be written for each node in network. After rearrangement, this results in a system of linear equations

p p

…i†0

ÿ qdem ÿ qloss

…i†0

…i†0

1

p

!

ÿ qdem ÿ qloss ˆ Bii :

Demand

0




Note the ®rst derivatives of qdem and qloss ± they are functions only of local head, Hi , and not of the head in any of neighbouring nodes. For the other correction, hk , the derivative is

kF kL

dem

…6†

where hi is the unknown correction of head in the node i and hj is the same for the neighbouring node j. The partial derivative for hi correction is equal to ! Ni odi odi o X …i† …i† ˆ ˆ Qij ÿ qdem ÿ qloss oHi ohi ohi jˆ1

u ˆ 1=…r ‡ 1† ˆ 0:50 for Darcy±Weissbach formula;

‡1; sgn …x† ˆ 0; ÿ1;

133

dem 0 dem

p/p q

kL= q

0

loss 0 loss

b) WESNET

Fig. 4. Demand±pressure relationship.

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D. Obradovic / Urban Water 2 (2000) 131±139

Source "A" Res-B

Res-C

Res-D

Res-E

LocDem-A PRV-B

Zone II/C

Zone II/D Res-A

PST-1

Zone II/E PST-2

LocDem-B PRV-A

Booster

SucT-1 PRV-C

SucT-2 Entry

I Pressure Zone 1

2

3

4

5

Wells

Source "B" Fig. 5. Water supply system of the town.

3. Pressure-related demand It has already been stressed that available data on demands and losses are captured under normal circumstances. If the same data are included in the model as demand, and the results ± heads and ¯ows ± are close to the observed values, then everything would be ®ne and acceptable. In the opposite case, the model has to be calibrated by changing some parameters until a satisfactory agreement is reached. Note that both ¯ows and pressures have to match the real value as closely as possible. But the model has to be applied to other situations as well, some being very di€erent from normal state. It is quite possible that in emergencies water will be pumped ± at very high pressures ± to the areas where it is needed. It is equally possible that in the times of shortages, local pressures could fall close to zero and air would be sucked into pipes. Obviously the hypothesis of pressure-invariant demands (and/or losses) will not hold in all such circumstances. This diculty can only be solved by introducing a relationship between demand and local pressure; as reliable data are extremely rare, assumptions must be made. For instance, in the program ``Procedure III'' (Jones, 1984) the relationship between demand and service pressure is linear (case ``a'' in Fig. 4); losses are presumably incorporated into the demand. In one other case (program WESNET; Williams, 1993a,b) the assumption is that the demand is proportional to the square root of pressure, while the losses rise faster than that (case ``b'' in Fig. 4).

These relationships could be used in Eqs. (4)±(7). Of course the assumptions are too simpli®ed, but might serve well until better data are made available through practice. Note that these relationships could be easily introduced in the model described above. The question is whether they describe the reality suciently closely? The answer can be found only by testing the hypothesis against real-life cases. One such case is included here.

Fig. 6. Total demand as on 23 September 1988.

D. Obradovic / Urban Water 2 (2000) 131±139

135

Fig. 7. Results of simulation for a normal day.

4. The ``Oldtown'' case ± how to deal with an emergency This is an ancient town in Herzegovina (Bosnia and Herzegovina), built along the old trade route coming inland from the Adriatic coast. After the Second World War it expanded rapidly, as several large factories were erected there, with new schools, trade organisations and even a University. The demand for water consequently rose sharply, and the existing source could not cope. New sources were found in the vicinity and the whole system was reconstructed and expanded. Fig. 5 shows a

simpli®ed schematic of this water supply system (Obradovic, 1998). The old source (denoted by ``A'' in Fig. 5) is rather far from the town; water ¯ows by gravity to the reservoir ``Res-A''. This water is of a very good quality, but the quantity can satisfy only about a third of the maximum demand ± the rest must be taken from elsewhere. The new source (``B'', Fig. 5) was found along the river, further upstream. Water is captured in several wells and pumped to the reservoir ``Res-B'' from where it ¯ows by gravity through an asbestos cement (A/C) pipe trunk

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D. Obradovic / Urban Water 2 (2000) 131±139

Fig. 8. Water consumption and local pressure at a large factory.

(800 mm). The main pressure zone is therefore supplied from two sides; any surplus is collected in suction tanks of two pumping stations and transferred to the higher zones (Fig. 5). Note also a booster station at the far end. These pumping stations and reservoirs are relatively new and are well maintained, but the network is generally very old ± some pipes are made of wood (!) and were laid a century ago. The replacement program is very modest because of shortage of funds; consequently the losses were high, reaching 50% of total production despite the e€orts of local sta€. This is plainly visible in Fig. 6 where a daily diagram of total demand is shown ± the minimum night ¯ows are too high and the maximum peak too low in comparison to the daily average, giving it a ``¯at'' appearance, typical for all high-loss systems. Despite high losses, the system operates rather well. The available storage space is low, equal to about 4 h of maximum demand, but the sta€ are well trained and alert. Water from the source A is used to the full, but the reservoir Res-A is protected from emptying by a levelsustaining valve (``PRV-A''). Well pumps feeding the reservoir Res-B are all levelcontrolled. The out¯ow from this reservoir is controlled by valve ``PRV-B''; further downstream is another pressure-reducing valve, ``PRV-C'', which could be used in an emergency. The supply is e€ectively controlled by two control valves PRV-A and PRV-B which are ®nely balanced. Three pumping stations (``PST-1'', ``PST-2'' and ``Booster'') are all level-controlled in order to keep the corresponding reservoirs reasonably full. A model of this system was created in 1988 and several normal and emergency situations were analysed with the sta€. Fig. 7 shows the main results for a normal day, when the demand was close to its maximum. Program WESNET (Williams, 1993a,b) was used for simulation runs. The demand and losses were treated separately, both being pressure-related (see Fig. 4(b)). It was assumed that all pumping stations, res-

Fig. 9. The total demand during three-day emergency.

ervoirs and pipes are in good condition and operate as planned. Fig. 8 shows the changes of demand and service pressure at a large factory; note the high night ¯ows (due to regular use and leakage) and one broad peak in the ®rst shift. These results were accepted by the sta€ as very close to their own experience (unfortunately no operational data were available apart from water levels in reservoirs). The model was accepted and later used for planning and design purposes. The sta€ worried about the vulnerability of the A/C pipe between Res-B and the town. It is laid alongside the main road with heavy trac and could be easily damaged. In such cases about 60% of total production would be lost, at least until the crack is found and repaired. By past experience, this takes at least 19 h ± much longer than the available storage would last. A strategy was developed to deal with this emergency. The pressure at the entry point is carefully monitored; any sign of a sudden drop is an indication that the trunk is damaged. Then the state of emergency is declared and further steps are taken: · close down that end of the A/C trunk; · declare the emergency state; · reduce the opening of valve PRV-A, so that pressures in the main zone drop to about 1±1.5 bar only compared to the more usual 3±5 bar; · stop pumps in PST-1, PST-2 and Booster stations. This policy will leave all high zones without water, the upper stores in high buildings in the town centre will be supplied inadequately, but at least the main pipes will remain full and the people can satisfy their basic needs. As soon as the damage is repaired, and water starts to ¯ow in from the wells, the situation will

D. Obradovic / Urban Water 2 (2000) 131±139

137

Fig. 10. The system has survived this three-day emergency.

be restored by slowly opening PRV-A and other valves. This plan was tried couple of times, always with good results. This event was analysed by using the model. The demand and losses were assumed to follow variations of local pressure in the manner shown above in Fig. 4(b). The simulation had to cover three full days, according to the following scenario: · system has started to operate as usual;

· the break has occurred in the early morning hours, in the shape of a huge out¯ow somewhere along the A/C trunk (between PRV-C and the entry point); · valve ``Entry'' was closed shortly afterwards; · valve PRV-B changed its mode of operation: instead of sustaining water level in Res-A it has kept pressures in the network as low as 1.5 bar; · the damage was repaired the next day around noon;

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D. Obradovic / Urban Water 2 (2000) 131±139

· the valve Entry was then opened slowly to let water from source B back to the town; · the situation was gradually restored back to normal. Note that no particular action is necessary in regard of three pumping stations in the town ± the pumps will be automatically stopped when water level in their suction tanks drops too low. In the same way, these pumps will be automatically restarted when the storage is again suciently high. Fig. 9 shows the changes of total demand during this emergency; note the sudden peak caused by the pipe burst, the much lower value during the repairs, and the slow build-up afterwards. The main results of simulation are visible in Fig. 10. Both Res-A in the town and Res-B at the source B are kept at a high level as a precaution. Only one well was operating, supplying local needs. Note the smooth recovery of the system, albeit rather slow ± at the end of the third day the situation was not yet completely stabilised. The consumers in the main zone had to be content with reduced quantities of water (Fig. 11). But the factory did get some water, and pressures were always between 2.5 and 3 bar, permitting the safe operation of various processes in the factory. On the negative side, many customers remained without water for some 20 h ± they had to fetch necessary water from their luckier neighbours. The sta€ has enthusiastically participated in this exercise; by their judgment the simulation has described the last incident quite realistically. Needless to say, if demands and losses were pressure-invariant, the results of simulation would have been meaningless ± negative pressure heads would appear in many places. The plans were made to organise monitoring of ¯ows, pressures and water levels at selected places, then to calibrate the model and verify these ®ndings, but events outside our control have prevented further work.

Fig. 11. This emergency did not force the factory to close down.

5. Conclusion The case just described has only highlighted the need for further study of factors in¯uencing water demand and losses; service pressure being only one of them. Obviously much, much more data are needed from the ®eld and of the better quality. Apart from using the usual sources of information (telemetry, water bills, control watermeters), new investigations should be organised with the speci®c purpose to analyse the in¯uence of service pressure on the demand and losses, alongside the usual parameters such as ``Consumption per head'', etc. (see Bailey et al., 1986, or Edwards & Martin, 1995). If properly organised, this e€ort would give a valuable insight in this poorly understood problem. The measurements must cover various categories of users, in all seasons and across a broad range of service pressures. The results should be published freely, thus permitting the accumulation of knowledge in the whole water industry. Perhaps a ``Code-of-practice'' should be compiled in order to standardise measurement procedures, data capturing, archiving and analysis.

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