1st International WDSA / CCWI 2018 Joint Conference
Kingston, Ontario, Canada July 23-25, 2018
Pressure dependency of total demand in water distribution networks Tomasz Janus∗1 and Bogumil Ulanicki2 1
2
Independent Researcher Water Software Systems, De Montfort University, Leicester, United Kingdom *
[email protected]
ABSTRACT This paper investigates the relationship between inlet pressure and total flow in the network assuming that local demands alike leakage are pressure-driven and follow a well understood power law. Having a good understanding of this pressure-flow dependency on a global scale within a water distribution network or a district metering area is important for finding the system curve used in pump scheduling and pressure reducing valve design and tuning. The study shows that a power law cannot, in practice, describe the relationship between inlet pressure and total flow but can be used to approximate this relationship. However, both coefficients of the power law will depend on the topology of the network and will additionally vary in time due to spatial and temporal variations in nodal demands across the network. The global pressure dependency coefficient estimate was found to depend on the elevation of demand nodes with respect to the reference node. It decreases the more demand is placed at nodes located below the reference node and increases if the opposite is true. Keywords: pressure dependent demands, pressure reducing valve, leak law
1
INTRODUCTION
Pressure dependent outflow via an orifice, whether an intentionally opened tap or a rupture in a pipe forming a leak has been widely investigated and disseminated in scientific literature. Most of the research has been done on leakage modelling where pipe/joint opening deformation with pressure causes the pressure-flow characteristics to deviate from the well known Toricelli’s equation. To account for those effects two modifications to Toricelli’s equation were introduced. The first Q = k (H − z)α is the so called power law. The equation coincides with Toricelli’s for α = 0.5, √ k = c A 2g and when the piezometric head H − z is equal to the water level H above the orifice. The second modification Q = k1 (H − z)0.5 + k2 (H − z)1.5 is called the linear law as it captures the effects of a linear dependence of opening area vs. pressure. Although both equations proved in agreement with experimental data on individual leaks i.e. on local scale, only one paper has been published so far on the subject of global pressure dependency of total leakage and demand within a water distribution network (WDN) or a district metering area (DMA) [1]. The authors performed averaging calculations for a number of leaks modelled with either the power law or the linear law under randomly varying pressure heads H, flow coefficients k and pressure dependency coefficients α, and concluded that the global leak law exponent is likely to be larger than the corresponding mean local leak law exponent. All calculations were performed with respect to average pressure head in a district, what differentiates the work of [1] from this study which is focused on the relationship between total demand (and indirectly leakage) in a DMA/WDN and inlet pressure.
1st International WDSA / CCWI 2018 Joint Conference
Kingston, Ontario, Canada July 23-25, 2018
The relationship between demand and leakage (ie. total flow in the system) and inlet pressure forms the so-called system curve which is needed in pressure-management of WDNs. One application is in pump-scheduling where an optimum is sought with regards to pumping costs but also water lost in the system due to leakage. Another application which motivates this study is related to the design and tuning of electronically controlled pressure reducing valves (PRVs). As explained and demonstrated in [2, 3], PRVs can become unstable under low flow conditions due to inverse relationship between its static gain and valve opening causing the downstream pressure to become more sensitive to valve position adjustments at low openings. It was shown in [4] that this increase in static gain is reduced by the system curve and the extent of this reduction is proportional to the value of the global pressure dependency coefficient α. Finding a system curve is equivalent to finding global kˆ and α ˆ estimates Pn α ˆ αi ˆ such that k (H0 − z0 ) ≈ i=1 ki (Hi − zi ) where ki , αi , Hi and zi denote, respectively, the flow coefficients, pressure dependency coefficients, pressure heads, and elevations at each of the demand nodes and H0 and z0 are, respectively, the pressure head and the elevation at the reference node (usually at the inlet to the WDN or the DMA). The manuscript begins with the simulations on simple network elements, i.e. single pipe, two pipes in series, two pipes in parallel, and a loop. Next, the global power law coefficients are calculated on a simple mixed-type network with a loop and branches for a number of randomly selected operating conditions.
2
SIMULATION RESULTS
The following sections present the results obtained via calculations performed on four basic building blocks of a network: single pipe (a), two pipes in series (b), two pipes in parallel (c), and single loop (d) - see Fig. 1. Since the paper is focused on pressure dependency of total flow which, during daytime, is dominated by demands not leakage we will assume from now on that at each node αi = 0.5 as in the Toricelli’s equation. This assumption will limit the number of special cases in our analysis whilst not being too restrictive as we consider demand to be a combination of a large number of outflows from an orifice for which Toricelli’s law is assumed to be valid. All simulation studies were performed in MATLABr on steady-state hydraulic models with headlosses calculated with Darcy-Weisbach formula. The equations were solved in MATLABr with fsolve whilst parameter estimation was performed with the non-linear least squares problem solver lsqnonlin.
Figure 1: Diagrams of basic building blocks of a water distribution network considered in the simulation studies.
1st International WDSA / CCWI 2018 Joint Conference
2.1
Kingston, Ontario, Canada July 23-25, 2018
Single pipe
The simplest case of a network is a single pipe with one demand node - see Fig. 1(a). The simulations are performed with and without friction. In both scenarios the system is calculated for different elevation differences ∆z = z0 − z1 and, in case of non-zero friction, also for different demands, i.e. k coefficients. It is not necessary to simulate the frictionless network for different k’s since the power law equation is linear with respect to k. Hence, we know that for a frictionless pipe kˆ is proportional to k and α ˆ is independent of k. The calculations are performed for different values of Hout ranging between 100 and 200 m producing a range of flow and pressure pairs used to fit the coefficiets kˆ and α ˆ ˆ of the global power law equation. The estimated parameters k and α ˆ together with 95%-ile confidence intervals are shown for the frictionless and rough pipework in Tables 1 and 2, respectively. The results in Table 1 indicate that both kˆ and α ˆ depend on the elevation difference between the two nodes with ˆ k increasing and α ˆ decreasing with ∆z. In other words, α ˆ < α if node 0 is above node 1 and α ˆ>α if node 0 is below node 1. The same relationship qualitatively applies to the system with head-losses, however now the results also depend on k which affects the value of kˆ for a given ∆z but, interestingly, not the value of α ˆ - see Table 2. Comparison of the results in Tables 1 and 2 for the same values of k and ∆z shows that the estimated values of k for the system with friction are lower compared to the frictionless pipe since pressure at the demand node is lower. Interestingly, introduction of friction does not affect the value of α ˆ , however only when α = 0.5. When α 6= 0.5, as in the case presented in Table 3 where α = 1 the value of α ˆ depends on k. The authors believe that α = 0.5 is a special case in which the interaction between the exponent of 0.5 in the Toricelli’s equation cancels out the quadratic term describing headlosses due to friction vs. flow. Another interesting finding is that the value of α at the demand node does not seem to have any significant effect on α ˆ which for α = 1 and ∆z = 0 is still close to 0.5. Both relationships still remain to be analytically investigated. In both cases, i.e. with and without friction, the quality of the fit of the global power law decreases with ∆z, which is indicated by larger relative confidence intervals of the fitted parameters at higher elevation differences. Table 1: Global kˆ and α ˆ coefficients for a single frictionless pipe system at various elevation differences ∆z for k = 0.2 m3−α s−1 and α = 0.5.
kˆ α ˆ
∆z = −40m
∆z = −20m
∆z = 0m
∆z = +20m
∆z = +40m
0.021 ± 0.011 0.923 ± 0.113
0.091 ± 0.010 0.645 ± 0.024
0.200 0.500
0.335 ± 0.014 0.409 ± 0.009
0.483 ± 0.028 0.346 ± 0.013
Table 2: Global kˆ and α ˆ coefficients for a single pipe system at various elevation differences ∆z and demand coefficients k for α = 0.5 and pipe resistance R = 32.32 s2 m−5 . ∆z = −20m k = 0.2 k = 0.4 kˆ α ˆ
0.060 ± 0.007 0.645 ± 0.024
0.077 ± 0.009 0.645 ± 0.024
∆z = 0m k = 0.2 k = 0.4 0.132 0.500
0.169 0.500
∆z = +20m k = 0.2 k = 0.4 0.22 ± 0.009 0.409 ± 0.009
0.282 ± 0.012 0.409 ± 0.009
1st International WDSA / CCWI 2018 Joint Conference
Kingston, Ontario, Canada July 23-25, 2018
Table 3: Global kˆ and α ˆ coefficients for a single pipe system at various elevation differences ∆z and demand coefficients k for α = 1.0 and pipe resistance R = 32.32 s2 m−5 . ∆z = −20m k = 0.2 k = 0.4 kˆ α ˆ
2.2
0.065 ± 0.009 0.679 ± 0.028
0.075 ± 0.009 0.657 ± 0.025
∆z = 0m k = 0.2 k = 0.4 0.151 ± 0.001 0.523 ± 0.001
∆z = +20m k = 0.2 k = 0.4
0.167 0.508 ± 0.001
0.261 ± 0.010 0.426 ± 0.009
0.283 ± 0.012 0.415 ± 0.009
Two pipes in series
The same simulation procedure as above was also performed for a system of two pipes in series with an intermediate demand node at the junction - see Fig. 1 (b). This study shows what effect the interaction between demands (k coefficients) at both demand nodes has on the estimated parameters of the global power law in two rough pipes connected in series. The system was simulated for different elevations and flow coefficients k1 and k2 at demand nodes 1 and 2. The pipes are assumed to have lengths L1 = 2000 m and L2 = 1000 m, and diameters D1 = 0.8 m and D2 = 0.5 m which translates to resistances of R1 = 12.72 s2 m−5 and R2 = 78.02 s2 m−5 . The results of the simulations and subsequent parameter estimations are presented in Table 4. Variations in k1 and k2 do not affect α ˆ when z1 = z2 , i.e. ∆z1 = ∆z2 but do affect α ˆ if ∆z1 6= ∆z2 . The values of kˆ and α ˆ depend on the values of k coefficients in individual nodes relative to their elevations. α ˆ increases and kˆ decreases if higher k values are associated with nodes for which ∆z < 0 whilst the opposite is true for ∆z > 0. ¯ 1 and Q ¯ 2 which denote averages from The demand flows from nodes 1 and 2 are represented with Q all nodal outflows as the upstream head is increased in a step-wise fashion. We can also observe that P P kˆ > ki if α ˆ < α and kˆ < ki if α ˆ ≥ α. Table 4: Results of simulations and subsequent parameter estimation of the system of two pipes in series - see Fig. 1 (b) No.
∆z1 m
∆z2 m
k1 m2.5 s−1
k2 m2.5 s−1
¯1 Q m3 s−1
¯2 Q m3 s−1
kˆ m3−α s−1
α ˆ –
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 0 +20 +20 +20 −20 −20 −20 −20 −20 −20 +20 +20 +20
0 0 0 +20 +20 +20 −20 −20 −20 +20 +20 +20 −20 −20 −20
0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2
0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1
1.464 1.599 1.937 1.611 1.759 2.131 1.299 1.418 1.718 1.429 1.582 1.822 1.473 1.584 2.005
0.627 0.793 0.528 0.690 0.873 0.581 0.556 0.704 0.468 0.718 0.913 0.651 0.524 0.658 0.373
0.212 0.242 0.250 0.355 ± 0.015 0.406 ± 0.017 0.418 ± 0.017 0.096 ± 0.011 0.110 ± 0.012 0.113 ± 0.013 0.236 ± 0.001 0.296 ± 0.002 0.239 ± 0.005 0.152 ± 0.009 0.154 ± 0.011 0.178 ± 0.016
0.500 0.500 0.500 0.409 ± 0.009 0.409 ± 0.009 0.409 ± 0.009 0.645 ± 0.024 0.645 ± 0.024 0.645 ± 0.024 0.482 ± 0.001 0.466 ± 0.002 0.511 ± 0.005 0.563 ± 0.012 0.585 ± 0.015 0.565 ± 0.019
1st International WDSA / CCWI 2018 Joint Conference
2.3
Kingston, Ontario, Canada July 23-25, 2018
Two pipes in parallel
Similar conclusions to the setup with two pipes in series can be drawn for two pipes in parallel see Fig. 1 (c). Here the pipes have the following parameters: lengths L1 = 1000 m, L2 = 2000 m, diameters D1 = D2 = 0.55 m, and resistances R1 = 46.63 s2 m−5 and R2 = 93.87 s2 m−5 . We can see that α ˆ values in Tables 4, 5 and, in fact, also Table 6 are the same for simulation runs 1-9, showing that when all demand nodes share the same elevation, the value of the estimated global pressure dependency coefficient is solely a function of ∆z and not of network topology or pipe characteristics. We can also see that in case when elevations of the demand nodes differ from each other and the inlet node z0 , α ˆ values tend to be larger when higher k’s are associated with nodes having negative ∆z values and smaller when higher k’s are attributed to the nodes with positive ∆z values. Table 5: Results of simulations and subsequent parameter estimation of the system of two pipes in parallel - see Fig. 1 (c).
2.4
No.
∆z1 m
∆z2 m
k1 m2.5 s−1
k2 m2.5 s−1
¯1 Q m3 s−1
¯2 Q m3 s−1
kˆ m3−α s−1
α ˆ –
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 0 +20 +20 +20 −20 −20 −20 −20 −20 −20 +20 +20 +20
0 0 0 +20 +20 +20 −20 −20 −20 +20 +20 +20 −20 −20 −20
0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2
0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1
0.815 0.815 1.166 0.896 0.896 1.283 0.723 0.723 1.034 0.723 0.723 1.034 0.896 0.896 1.283
0.709 0.905 0.709 0.780 0.996 0.780 0.628 0.803 0.628 0.780 0.996 0.780 0.628 0.803 0.628
0.154 0.174 0.190 0.258 ± 0.011 0.292 ± 0.012 0.318 ± 0.013 0.070 ± 0.008 0.079 ± 0.009 0.086 ± 0.010 0.137 ± 0.003 0.168 ± 0.002 0.150 ± 0.006 0.150 ± 0.002 0.157 ± 0.003 0.206
0.500 0.500 0.500 0.409 ± 0.009 0.409 ± 0.009 0.409 ± 0.009 0.645 ± 0.024 0.645 ± 0.024 0.645 ± 0.024 0.522 ± 0.005 0.508 ± 0.003 0.544 ± 0.008 0.506 ± 0.003 0.520 ± 0.005 0.486
Loop
The loop is constructed from the two pipes featured in the previous subsection with a third pipe linking nodes 1 and 2 - see Fig. 1 (d). The parameters of the third pipe are as follows: L3 = 1500 m, D3 = 0.55 m, R3 = 69.95 s2 m−5 . The difference between this configuration and two pipes in parallel lies in the fact that under some conditions, part of Q1 can pass through node 2 whilst under different conditions part of Q2 can pass through node 1, resulting in flow reversal in pipe 1 − 2, as depicted with symbol sgn(Q1,2 ) in the last column of Table 6. sgn(Q1,2 ) = +1 denotes flow direction from node 1 to node 2 whilst negative sign of the flow indicates flow from node 2 to node 1. When the results in Tables 5 and 6 are compared same α ˆ values are noticed for the simulation runs 1 − 9 since, as already mentioned, α ˆ depends only on the difference in elevations between demand nodes and the inlet node, not on demand variations or friction when the demand nodes have equal elevations. However, when we look at the experiments 10-15 we notice that the values of α ˆ for the looped network tend to be more dependent on the elevations of demand nodes 1 and 2 than in the case of two pipes in parallel. α ˆ estimates for two pipes in parallel are within 0.508 − 0.544 for the
1st International WDSA / CCWI 2018 Joint Conference
Kingston, Ontario, Canada July 23-25, 2018
experiments 10-12 and 0.486 − 0.520 for the experiments 13-15, which scatter less around the local α value of 0.5 compared to 0.435 − 0.456 and 0.603 − 0.657 respectively for the loop configuration. Additionally, flow direction in the third pipe does not seem to have any significantly effects on α ˆ. This requires further theoretical explanation. Table 6: Results of simulations and subsequent parameter estimation of the system of three pipes forming a loop - see Fig. 1 (d). No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2.5
∆z1 m
∆z2 m
0 0 0 +20 +20 +20 −20 −20 −20 −20 −20 −20 +20 +20 +20
0 0 0 +20 +20 +20 −20 −20 −20 +20 +20 +20 −20 −20 −20
m
k1 2.5 −1 s
0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2
m
k2 2.5 −1 s
0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1
¯ m s
Q1 3 −1
0.898 1.054 1.115 0.988 1.160 1.227 0.796 0.935 0.989 0.874 1.037 1.054 0.894 1.034 1.163
¯ m s
Q2 3 −1
0.642 0.812 0.777 0.706 0.893 0.855 0.569 0.720 0.689 0.649 0.862 0.747 0.630 0.753 0.770
kˆ m
3−α −1
s
0.167 0.228 0.178 0.282 ± 0.012 0.381 ± 0.016 0.297 ± 0.012 0.077 ± 0.009 0.103 ± 0.012 0.081 ± 0.009 0.219 ± 0.002 0.333 ± 0.007 0.231 ± 0.002 0.089 ± 0.008 0.098 ± 0.015 0.102 ± 0.009
α ˆ –
sgn(Q1,2 ) –
0.500 0.500 0.500 0.409 ± 0.009 0.409 ± 0.009 0.409 ± 0.009 0.645 ± 0.024 0.645 ± 0.024 0.645 ± 0.024 0.456 ± 0.002 0.434 ± 0.005 0.456 ± 0.001 0.618 ± 0.020 0.657 ± 0.032 0.603 ± 0.019
+1 +1 −1 +1 +1 −1 +1 +1 −1 +1 +1 +1 −1 +1 −1
Network
Finally, the global pressure dependency coefficient was determined for a network composed of a combination of the previously considered individual network elements, i.e. serial and parallel pipe connections and a loop. The network schematic is shown in Fig. 2. Two simulation scenarios were performed: (a) where ki coefficients in all nodes were varied randomly and independently via random sampling from a uniform probability distribution between 0.001 and 0.009 m2.5 s−1 and (b) an average flow coefficient k¯ was randomly drawn via inverse transform sampling from a discrete probability distribution obtained from a typical diurnal flow profile with a mean of 0.0045 m2.5 s−1 after which each individual flow coefficient ki for nodes 1-6 was randomly chosen such that the relative differences between k¯ and ki fell into a uniform probability distribution with lower and upper bounds of ¯ +0.1 ki, ¯ h−0.2 k, ¯ +0.2 ki, ¯ and h−0.4 k, ¯ +0.4 ki ¯ respectively. Each simulation scenario was h−0.1 k, executed 2000 times, each time with a different combination of ki coefficients. In each simulation run the pressure head in node d was varied between 100 m and 200 m in 10 m steps, thus generating 11 steady-state solutions for each demand pattern. These 11 flow vs. head data pairs were then used to identify the global power law parameter estimates kˆ and α ˆ . As in previous studies all αi were assumed to be equal 0.5. The simulation run (a) where ki were randomly sampled was performed for three combinations of nodal demands at nodes 1-6 (see Fig. 2) in order to investigate how demand node elevations affect the global pressure dependency coefficient estimate α ˆ . The average demand node elevation was, respectively, 10m lower, equal and 10m higher from the reference node elevation z0 . The simulation run (b) where ki were allowed to vary only up to some maximum fraction of the
1st International WDSA / CCWI 2018 Joint Conference
Kingston, Ontario, Canada July 23-25, 2018
average network flow coefficient k¯ were performed at different levels of variability of individual demand flow coefficients ki to see how spatial variability of demands across a network affects the width of the frequency distribution of the global α ˆ estimates. The obtained histograms of α ˆ estimates for the simulation scenarios (a) and (b) are presented in Fig. 3a and Fig. 3b respectively.
Figure 2: Diagram of the water distribution network used in the Monte Carlo simulation study In both simulation scenarios α ˆ follows Gaussian distribution in which the mean depends on the difference between the average elevation of the demand nodes z¯ (here 1-6) and the elevation of the reference node for which the global power law is determined z0 (node 0). In the default configuration where z¯ = z0 = 50m the mean of the α estimate µαˆ ≈ 0.5. In case z¯ = 40m < z0 and z¯ = 60m > z0 µαˆ ≈ 0.45 and µαˆ ≈ 0.55 respectively - see Fig. 3a. It is also interesting to note that for µαˆ ≈ 0.45 the distribution has a smaller variance than in the other two cases, however the reason for this is currently not known and needs to be theoretically investigated.
(a)
(b)
Figure 3: Flow vs. inlet pressure characteristics of a single pipe system for (a) different elevation differences ∆z at zero friction losses (b) different elevation differences and demands with non-zero friction in the pipe. The variability of α ˆ during normal operation of a WDN was analysed in the second simulation scenario in which a restriction had been put on the maximum difference between the local coefficients
1st International WDSA / CCWI 2018 Joint Conference
Kingston, Ontario, Canada July 23-25, 2018
¯ The results are plotted in Fig. 3b and indicate that µαˆ is ki and the mean network flow coefficient k. characteristic of the network’s elevation profile while the width of the distribution of α ˆ is determined by how much local coefficients ki in individual demand nodes vary between each other. It can be ¯ σαˆ = 0 for all demands i.e. that the observed global pressure dependency shown that if ∀i, ki = k, coefficient estimate depends only on the elevations of the demand nodes with respect to the reference node for which the global power law is determined.
3
CONCLUSIONS
During normal operation of a network the global α estimate α ˆ varies in time as the demands change independently at different parts of the network. Its probability distribution function forms a bellcurve with the mean µαˆ being characteristic of the elevation profile of the demand nodes with respect to the reference node whilst the standard deviation σαˆ depends on the differences (in time) between local ki coefficients with respect to each other. In case the average elevation of the demand nodes is lower from the elevation of the reference node α ˆ < 0.5 while when the demand nodes are above the reference node α ˆ > 0.5. For any network, when all ki coefficients are equal and vary proportionally α ˆ will be constant for all flows whilst when ki values differ from node to node and change independently α ˆ will take on different values depending on how much demand is focused in the nodes above and below the reference node. If higher ki values are attributed to the nodes with positive ∆z values then α ˆ < 0.5 whilst for negative ∆z values then α ˆ > 0.5. In the special case when all demand nodes are at equal heights, α ˆ = const irrespectively of the allocations of the demands between individual demand nodes. Additionally, it also seems, that in this special case the value of α ˆ is independent of the topology of the network, i.e. the incidence matrix which most likely stems from the observation that pressure variations due to headlosses do not seem to affect α ˆ values when local α coefficients are equal 0.5. This finding needs to be investigated further via theoretical analysis.
References [1] M. Ferrante, S. Meniconi, and B. Brunone, “Local and global leak laws - the relationship between pressure and leakage for a single leak and for a district with leaks,” Water Resources Management, vol. 28, no. 11, pp. 3761 – 3782, 2014. [2] B. Ulanicki and P. Skworcow, “Why PRVs tend to oscillate at low flows,” Procedia Engineering, vol. 89, pp. 378–385, 2014. [3] T. Janus and B. Ulanicki, “Hydraulic modelling for pressure reducing valve controller design addressing disturbance rejection and stability properties,” Procedia Engineering, vol. 186, pp. 635–642, 2017. [4] T. Janus and B. Ulanicki, “Improving Stability of Electronically Controlled Pressure Reducing Valves Through Gain Compensation,” Journal of Hydraulic Engineering, 2018, Accepted for Publication.
