Application of «Gradient» algorithm to modeling thermal pipeline networks with pumping stations Текст научной статьи по специальности «Медицинские технологии»

Journal of Siberian Federal University. Engineering & Technologies 1 (2013 6) 28-35

УДК 697.34.001.24

Application of «Gradient» Algorithm to Modeling Thermal Pipeline Networks

with Pumping Stations

Alex Y. Lipovka* and Yuri L. Lipovka

Siberian Federal University, 79 Svobodny, Krasnoyarsk, 660041 Russia

Received 11.02.2013, received in revised form 18.02.2013, accepted 25.02.2013

The paper deals with the analysis of flow distribution in regulated pipeline systems. Proposed inclusion of nodal elevations and piezometric heads into «global gradient» system of equations allows estimation of hydraulic regimes of thermal networks with pump stations, flow- and pressure-control valves during the process of iterative approach. Application of proposed modification of the Todini's GGA method to calculation offlow distribution is shown on example of a small thermal network with pumps.

Keywords: water distribution network, flow equations, pumping station, heating networks, GGA.

Introduction

The equations describing the flow in complex looped heating networks are nonlinear, the problem of their effective solution still remains open, despite of the already proven methods of solution, the most famous of which are: the Hardy Cross method of single loop balancing, generalized methods of loop flows (Qequations) and nodal pressures (Hequations) by B. Xacn^eB A. MepeHKOB [1] and, of course, the «global gradient» algorithm (GGA) by E. Todini [2], which combines a fast convergence of the Qequations method with simplicity of Hequations.

Illustration of the sequence of iterations for the Qequations and GGA (QHequations) with two-stage algorithm solving first for the nodal pressures and then for flows sequentially in an iterative process provided by A. Simpson [3], in which a simple reference network with two tanks, two pipes and one node used to express equations and solutions based on these two methods.

Authors propose a lite modification of the method [2] and illustrate its application by calculating water distribution for a simple thermal network with various pumping stations.

Statement of the problem

The well-known one-dimensional mathematical model of steady isothermal flow distribution in hydraulic networks is described by three equations.

The first equation - the known law of hydraulic resistance of any link in the network

© Siberian Federal University. All rights reserved

* Corresponding author E-mail address: [email protected]

A/^+A/V^S^ " i = l.....n, (1)

where AH,- - value of pressure drop at the link , m; A//pm. - pretnure gain created by the pumping unit, m. For passive branches A//pm. = 0; QU - flow uh the link, m3/o; Sf - hydraulic resistance of link i, s2im5; n - n^nntj^r or links in network; r - exponent of the flow in the head loss equation. The recond - continuity on flow at each node;

^Ql=eQndi j = 1.....m, (2)

i

where ^ft Qi - algrb raic ssu-mm of flows in the links, having a common node j, m3/s; Qnd. - outflow from

i

thie node j, m3/s. Ii the node is a simple splitter (tees, crosses), than s?nd(. = 0; Uthe node?' is coesumer (water outflrw), Chen k„Cj. > 0; if"Slt^e node is source (water inflow), thine Qnd. < 0; m - the number of nodes with unknowo parameters.

And the Ihitd - three are equations that require the total pressure loss of zero withi= any loop in a nehwork, contisting of independenr loops

^TajFI = 0 c = l.....k. (3)

j

Solving system of equations (1)(3) givers us ^hne flow distribution in a nttwork theads at the nodes and flows in the linki). Thr difference in she use of a particular method is the time to achieve results with the required accuoacy and specified numb er of iseratione. For such co mpler hydraulic eetworks, like ehermal suppfying networks , it is very impsrtant, ctpecieily tf yon need to crlculate sccidnnts in real time.

Mathematical model

Formulation oS GGA iTods ^iL-lF'iiaiaitti Q-H equations) utet she generic term head H, without reference to the terrain, which requires additional computation to find piezometric heads

Sdldr-1 = 0. (4)

where H, H? - heads, respectively, at the beginning and end of the of the pipeline, m; n - the exponent in the head-loss equation fos the link.

We propose to replace heads in the original formula (4) with the dependency that takes into account efevations and piezometric herds at the nodes

SftlQI"-1 + fa + Zt) - № + Z,) he 0 (55)

wPtee Z,t Zj - elevations of nodeo, respectively, at the beginning and end of the pipe, mi P,-, P, -jfrieozzeom^treiii: heads, respectively, at nhie beginning and end of the eccounting ^«e^tiion of the pipeline, m. The enargy conservation law in case of the pumping station:

:(fto-a(§) tihfahZ^-CPihZi)

= o, i<5)

where <-- - the relative speed of the pump; h0, a, b- coefficients and the exponent of the approximation curve of pump c haracteristics.

Matrix form of the system of equations of the modified «globalgradient» method:

G -Ai [Q

-A1

0

A0(P0 + Z0) + AZ + H0pm Qo

= 0,

(7)

where GO - square diagonal matrix of coefficients; Qo, Q - vector of given outflows in nodes and the unknown flows in pipe s; P0, P - vectors of given and unknown piezometric heads; Z0; Z - vectors of elevations, respectively, for nodes with given and unkvown piezometric hecds; aA0, A - incidence matrix, respectively, for nodes with given and unknown piezometric herds; H0pm - vector of products h0o)2 of coefficient from pump curve and its relative speed.

This nonlinear system of equations, solved by Newton's method has the form:

j(e(fc),nm)

se(fc+1)

5a(fc+1)

= -f( ^a«).

(8)

where J(Q(k), PV) - Jacobian matrix; f(Q(k), P— - viscfor o- deviations; (k), (k+1) - number of iteration; 8Q(+1) - unknown vector of ifcrements of volume flowsi m3/s; 5P(r+1:) - unknown vector of the increments of piezometric heads, m.

Sy ttem of linnar equations (8) can bt written as

G'(t) -Al

-Al 0 ,

After teansformations, we obtain the solution: a (k+1) iferafion vectoo of the piezometric heads, obtained solely on the I)asis of previous (k) iteration of flows

Q(0+l) _ ft

.^Ccn) - 1-rfc).

G<r)f m - Aii-rn + z) - n2r-e -f z0) - a0pm . -n?frk) - ffo .

(9)

pfc+i) = {a7;g'-1A1}-1{a1;(G'-1(GQ(1) o E) o q(g)) o qo}

(10)

and (k+1) iteratif nvectoe of flows, which depends on the previous (k) iteration of flows, and (k+1) iteration of piezometric head.

Q(1+1) = G'-1(A1P(1+1) + G + (G' o G)Q(1)), where IE - auxiliary vector of known variables

E = AtZ + A2(Pe + Z o)+H°-m

(11)

(12)

Creating and updating square diagonal matrices G, G' h G'"1 during the solution is -very simple. For pipeline cosresponding entries G(itt h G'(i,i) are given as:

G(i,s)=rtlQ ¡1 G '(s,s) = 2G(s,t)

(13)

(14)

Coetesponding entries G(i,iS h G'(i,i) for pumps: G(i,i) = ao)2-G\Q\b~1 G'(t=) = ¿G(t, f)

(15)

The inverse matrix GM(/',/') is easily calculated by the formula

^0.0=^ (17)

An example of using the modified algorithm

Let us analyze the convergence of the proposed modified algorithm GGA on example of a simple thermal pipeline network with pumping stations (Fig. 1).

The results of calculation of flow distribution: piezometric heads at the nodes, flows in the pipes and operating points in pumping systems are given in Tables 1-3.

Relative deviations at each step of the iterative process for the flows are shown in Fig. 2, for piezometric heads - in Fig. 3, for head gains in pumps - in Fig. 4.

Diagram of infinite norm of vector of increments for piezometric heads is shown in Fig. 5.

Analysis of the calculations shows that piezometric heads are close to solution even after the first iteration, and after the fourth iteration the unknown parameters of the flow distribution are almost stabilized.

Fig. 1. The scheme of thermal network: Pm1, Pm2, Pm3 - circulating, pressure rising, water supplying pumping stations; 1-5 - numbers of network nodes; (1)-(8) - numbers of network pipes

Table 1. Piezometric heads in the network nodes at each iteration, P m

Number of iteration Nodes

2 3 4 5

1 45,275464 62,135573 38,83471 15,321225

2 44,994661 61,982059 38,969458 15,041589

35 44,994661 61,983504 38,97235 15,041677

4 44,994661 61,983502 38,972342 15,041680

5 44,994661 61,983502 38,972342 15,041680

Table 2. Volume flows in pipes of the network at each iteration, Q, m3/s

Number of Plots

iteration 1 2 3 4 5 6 7 8

1 0,002778 0,002778 0,015686 0,015686 0,015686 0,015686 0,012908 0,012908

2 0,002778 0,002778 0,014905 0,014905 0,014905 0,014905 0,012127 0,012127

3 0,002778 0,002778 0,014881 0,014881 0,014881 0,014881 0,012103 0,012103

4 0,002778 0,002778 0,014881 0,014881 0,014881 0,014881 0,012103 0,012103

5 0,002778 0,002778 0,014881 0,014881 0,014881 0,014881 0,012103 0,012103

Table 3. Volumetric flows through pump, Q, m3/s and piezometric heads at the suction and discharge nozzles, Pu Pout, m at each iteration

<4-1 o Pumps

¡3 § Pm 1 Pm 2 Pm 3

z .1 Q, m3/s Pin, m Pout, m Q, m3/s Pin, m Pout, m Q, m3/s Pin, m Pout, m

1 0,015686 45,198490 62,212547 0,015686 57,058599 63,911684 0,002777 19,068071 59,207392

2 0,014905 44,924029 62,052690 0,014905 56,911428 64,040088 0,002777 18,997330 58,997330

3 0,014881 44,923980 62,054185 0,014881 56,912823 64,043028 0,002777 18,997330 58,997330

4 0,014881 44,923977 62,054185 0,014881 56,912817 64,043025 0,002777 18,997330 58,997330

5 0,014881 44,923977 62,054185 0,014881 56,912817 64,043025 0,002777 18,997330 58,997330

£ 1.01

4 5

Number of iteration

(1), (2) (3) — (6) (7), (8)

Fig. 2. Relative deviations of flowsin pipes (1) - (8i

1,02

"<3

Oh 1

0,995

3 4

Number of iteration

Fig. 3. Relative deviations of piezometric heads in nodes 15

1,01 1,005 1

0,995

a -d

Pi

0

0,985 0,98 0,975 0,97 0,965 0,96 50,955

Pm1 Pm2 Ppi3

3 4 5

Number of iteration

Fig. 4. Relative deviations of head gains developed by the pumps Pm1-Pm3

Number of iteration

Fig. 5 Infinite norm of piezometric heads inc rements

и 50 й

30

S20

u Л

H 10

53

38 45

32

19 26

9

5 5 5 5 5

GGA -L-C

1 2 3 4 5 6 7

Number of circuits with pumping units

Fig. 6 Comp6rison of the c onvergcoce of the calculation: Hardy Cross m ethod (L-C) and modified global g radient method (GGA)

5

0

Comparison of the convergence of the classical Hardy Cross method and the modified method of «global gradient» on a series of networks with pumping units in each independent loop was carried out, the results of which are shown in Fig. 6.

Conclusion

The modified algorithm, involving the introduction of piezometric heads and elevations directly into the equations of "gradient" algorithm, inherits high convergence of original GGA and provides ability control flow distribution and evaluate the reliability and sustainability of newly designed or existing thermal networks during the process of iterative calculation.

Proposed modification greatly increases the possibility of modeling pumping stations with both parallel and series layout of pumps with different characteristics, including frequency controllers. It also makes possible for fast tracking of operational point reposition for single pumps or whole pump stations due to changes in resistance characteristics of the thermal network, which may go beyond the optimum pump performance, and hence reduce its efficiency.

Research of high interest in this area is related to regulated pipeline systems with various flow and pressure control valves, which will be investigated further using authors' computer program.

References

[1] Меренков А.П., Хасилев В.Я. Теория гидравлических цепей. М.: Наука, 1985. 279 с.

[2] Todini E., Pilati S. // Computer applications in water supply, B. Coulbeck and C. H. Orr, eds., Wiley, London, 1988. P. 1-20.

[3] Simpson A. R. // Water Distribution System Analysis 2010 - WDSA2010, Tucson, AZ, USA, Sept. 1215, 2010.

Использование «градиентного» алгоритма применительно к моделированию тепловых сетей с насосными подстанциями

А.Ю. Липовка, Ю.Л. Липовка

Сибирский федеральный университет, Россия 660041, Красноярск, пр. Свободный, 79

В статье рассмотрены вопросы анализа потокораспределения в регулируемых трубопроводных системах. Предложено в систему уравнений метода «глобального градиента» ввести отметки рельефа местности и пьезометрические напоры, что позволяет в процессе итеративного приближения оценивать гидравлические режимы тепловых сетей с насосными подстанциями, регуляторами расхода и давления. На примере небольшой тепловой сети показана схема применения предлагаемой модификации метода Тодини к расчету потокораспределения.

Ключевые слова: потокораспределение, уравнения потока, насосные подстанции, тепловые сети.