Mixed Integer Linear Models for the Optimization of ... - Semantic Scholar

Mixed Integer Linear Models for the Optimization of Dynamical Transport Networks Bj¨orn Geißler∗

Oliver Kolb∗

Jens Lang∗

Alexander Martin‡

G¨ unter Leugering†

Antonio Morsi∗

December 21, 2010

Abstract We introduce a mixed integer linear modeling approach for the optimization of dynamic transport networks based on the piecewise linearization of nonlinear constraints and we show how to apply this method by two examples, transient gas and water supply network optimization. We state the mixed integer linear programs for both cases and provide numerical evidence for their suitability.

1

Introduction

In this article we consider the optimization of dynamical transport networks, where the arcs or connections of such a network correspond to pipes or certain components such as valves, compressors or pumps. For a network to be ∗ Department of Mathematics, Technische Universit¨ at Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany, ([email protected], [email protected], [email protected], [email protected]). † Department of Mathematics, Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg, Martensstr. 3, 91058 Erlangen, Germany([email protected]). ‡ Department of Mathematics, Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg, Am Weichselgarten 9, 91058 Erlangen, Germany ([email protected]).

1

a transport network, we assume some flow through the components of the network, which should be optimized under the consideration of several side constraints. In one of the simplest cases, this may lead to a minimum cost flow problem, where a cost is assigned for the use of each connection and the goal is to achieve a certain flow between distinct nodes of the network at lowest cost. Since this problem can be described in terms of a mathematical model involving only linear constraints and a linear objective function, it can be solved by using algorithms from linear optimization, which are proven to be fast and to guarantee to find the global optimum. The situation gets more complex in cases where the network contains some kinds of switching components, i.e., components which can be in different discrete states. In order to reflect discrete processes in a mathematical model, one typically introduces binary variables and additional linear constraints involving these variables. This leads to a mixed integer linear optimization problem, which in general is harder to solve than a linear program. Fortunately, there are still algorithms available which guarantee to solve these kinds of problems to global optimality. If we consider a situation where the transported entities underlie nonlinear dynamics, typically according to physical laws, it seems like the guarantee of global optimality, provided by continuous and mixed integer programming solvers, is no longer at hand. Moreover, classical algorithms for nonlinear optimization problems are not able to handle discrete variables and constraints in an efficient way. Our approach is based on piecewise linearization of nonlinearities and the modeling of piecewise linear functions in terms of linear constraints. This brings us into a situation where we can apply global optimization algorithms from mixed integer linear programming to the optimization of dynamical transport networks. Many engineering problems like supply chain network optimization, traffic problems, the problem of gas network optimization and the problem of water supply network optimization can be stated as an optimization problem on a dynamic transport network. In the succeeding sections, we will focus on the latter two. The remainder of this article is organized as follows: In Section 2, we introduce a network model suitable for the general case of dynamic transport networks. In Sections 3 and 4, we focus on the dynamics of gas and fresh water in pipeline networks, which are mainly described by a coupled set of hyperbolic partial differential equations. For these equations, we give a discretization scheme, suitable for our mixed integer linear model in Section 5. In Sections 6 and 7, we introduce our mixed integer models for the optimization of gas and water supply networks. Section 8 is dedicated to 2

the approximation of nonlinearities by piecewise linear functions and their modeling in terms of linear constraints. Finally, computational results are shown in Section 9 before we give a summary in Section 10.

2

Network Model

We model a dynamic transport network by means of a directed finite graph G = (V, A). The set A of arcs is partitioned into different sets for the various components in the network, e.g., a set AC of compressors, a set AV of valves, a set AP of pipes and a set AR of control valves in case of a gas network. In a water supply network the set of arcs A consists of a set of pipes AP , a set of tanks AT , a set of pumps APu and a set of valves AV . The set V of vertices or nodes consists of a set VP of intersection points of the segments (also called inner nodes) and a set of boundary nodes VB , which further divides into a set VS of sources, and a set VD of sinks. Sources are considered as gas or water delivering points and sinks reflect demands specified by the quantities flow and pressure. We denote by δν+ (δν− ) the set of all arcs a ∈ A outgoing (ingoing) from (to) the node ν ∈ V. To allow description of physical state variables at the ends of arcs, each node ν ∈ V is associated with a family of intermediate vertices {νa }, a ∈ δν+ ∪ δν− , see Fig. 1. We shall use a = νa wa to model a directed connection in the transport network. The task is to route the gas or water through the network to satisfy the consumers’ demands such that the costs for the control elements, that is fuel gas consumption of the compressors or power consumption of the pumps, is minimized.

3

Gas Flow in Pipes

The gas flow in a pipe is governed by the system of Euler equations supplemented by a suitable equation of state. In several situations, we can assume a nearly constant temperature T = T¯ of the gas - e.g., pipes in Germany are typically at least one meter beneath the ground. In such a situation, an isothermal flow is an appropriate model. Taking into account a non-ideal gas 3

m * b 

m

a

   m νb - νam ν νm c  HH c H HH j m

Figure 1: Network Model. Suppose three directed pipes a, b, c and one node ν representing an intersection are given as shown. Then, we have δν− = {a} for the set of ingoing arcs and δν+ = {b, c} for the set of outgoing arcs. The node ν is associated with the family of intermediate nodes {νa , νb , νc }, where values of physical state variables are described.

behavior, the Euler equations reduce to the continuity and the momentum equation, together with the equation of state. On each pipe a ∈ AP of the network, we have for t > t0 ∂t ρ + ∂x (ρv) = 0 , λ ∂t (ρv) + ∂x (ρv 2 ) + ∂x p = −gρ∂x h − ρ|v|v , 2D p ρ = , z(p)R0 T¯

(1) (2) (3)

where (ρ, v, p) is the state vector consisting of the density, the flow velocity and the pressure of the gas, respectively. The two terms on the right-hand side of (2) describe the influence of gravity and friction. Here, g is the acceleration constant, ∂x h is the slope of the pipe, λ is the pipe friction value, and D is the diameter of the pipe. Since in our practical computations all pipes are nearly horizontal, we will neglect the gravity term. The friction factor λ is implicitly given by the Prandtl-Colebrook law,   1 2.51 k √ = −2 log10 √ + , (4) λ Re λ 3.71 D with the Reynolds number Re = Dρ|v|/η, where η is the dynamic viscosity of the gas, and with the roughness k of the pipe. The compressibility factor 4

z(p) in (3) is approximated by z(p) = 1 + 0.257

p Tc p − 0.533 ¯ , pc T pc

(5)

where pc and Tc are the pseudo-critical pressure and temperature, respectively. This formula from the American Gas Association works quite well for pressures up to 70 bar. Due to the constant temperature T¯, we get z(p) = 1 + α p with α < 0, leading to 0 < z(p) < 1. Finally, R0 in (3) is the normalized gas constant. In practical gas network calculation, typically the following two state variables are considered: the pressure p and the gas flow q under norm conditions, where q = Aρv/ρ0 with the cross-sectional area A of the pipe and the norm density ρ0 . Replacing the density ρ and the velocity v in (1)-(2) and defining C0 = R0 ρ0 T¯/A, we get the following system of equations   p ∂t + C0 ∂ x q = 0 , (6) z(p)   A z(p) 2 C0 z(p) q + ∂x p = − λ(|q|) q|q| . (7) ∂t q + C0 ∂x p ρ0 2D p This system of partial differential equations has to be completed by initial, boundary and coupling conditions across the whole network. Suppose initial data p(x, t0 ) = p0 (x) and q(x, t0 ) = q 0 (x) are given. Admissible boundary values must be chosen in accordance to the characteristics [8]. Therefore, it is important to study the eigenvalue structure of the underlying flux function. Setting P = p/z(p) = p/(1 + αp) and reformulating (6)-(7) in terms of the vector u = (P, q) as ∂t u + ∂x F (u) = Q(u), a short calculation yields the eigenvalues of the Jacobian ∂u F (u). We obtain λ1/2 = v ± c(p), where c is the speedpof sound in the fluid defined by c2 = ∂ρ p. From (3), we have c(p) = z(p) R0 T¯. Gas flow networks are operated in the subsonic flow region, that is, |v| < c. Usually, we even observe |v|  c in practically relevant situations. We thus can conclude that although the characteristics are solution-dependent, they do not change their sign, that is, the directions information (like perturbations of a stationary state) travels are maintained and we have to pose exactly one condition at every boundary node of the network. In our applications, we make the gas flow q or the pressure p available at sources and sinks. 5

Analogously, for the inner nodes of the network, we have to pose exactly as many conditions as the number of arcs incident to the node. At each node ν ∈ V with ingoing arcs δν− , outgoing arcs δν+ and a family of intermediate vertices {νa }, a ∈ δν− ∪ δν+ , we enforce conservation of mass X X q(νa , t) = q(νb , t) (8) a∈δν−

b∈δν+

and consistency of the pressure p(νa , t) = p(ν, t)

for all a ∈ δν− ∪ δν+ .

(9)

Condition (8) is known as Kirchoff’s law and is often referred to as RankineHugoniot condition [8] at a node.

4

Water Flow in Pipes

In water supply networks, we are dealing with pressurized water networks. Due to the incompressibility of water, pressure p can equivalently be expressed as an elevation difference ∆h =

p , gρ

(10)

where g is the gravity constant and ρ is the constant water density. In water management, pressure is therefore often measured by the elevation above sea level, called the head h, which is the sum of the actual geodetic height and the elevation difference corresponding to the hydraulic pressure. Thus, a pressure of 5 bar at 100 m above sea level corresponds to a head h = 151 m. For this kind of network, the governing equations in all pipes a ∈ AP are the so-called water hammer equations [1]: ∂h c2 ∂q + ∂t gA ∂x ∂q ∂h + gA ∂t ∂x

= 0, = −λ

(11) q|q| , 2DA

(12)

where (h, q) is the state vector consisting of the piezometric head and the flow. Again, g is the gravitational constant, c is the speed of sound in the pipe, A and D are the cross-sectional area and the diameter of the pipe. 6

The term on the right-hand side of (12) models the influence of friction. As for the gas networks, the friction coefficient λ is implicitly given by (4). The flow characteristics of (11)-(12) are similar to the ones of (6)-(7). Here, the eigenvalues of the Jacobian ∂u F (u) are λ1/2 = ±c. Therefore, we have steady characteristic directions and we have to pose exactly one condition at every boundary of a pipe. Thus, at each node in the network, we have to pose as many conditions as the number of adjacent arcs at the node. At sources and sinks, we may prescribe the pressure head or the flow rate, and analogously to (8)-(9), we enforce conservation of mass and consistency of the pressure head as coupling conditions: X X q(νa , t) = q(νb , t) (13) a∈δν−

h(νa , t) = h(ν, t)

5

b∈δν+

for all a ∈ δν− ∪ δν+ .

(14)

Discretization

As we already mentioned, the information directions of the partial differential equations (6)-(7) and (11)-(12) are maintained. In this case, implicit box schemes are known to work very effectively. Box schemes, originally introduced by Wendroff [17], have been used for several years. They are conservative schemes, i.e., they guarantee exact conservation of the flux at the level of the box. Nonphysical oscillations, often caused by parasitic solution components in standard finite difference or finite volume approximations, are avoided. Box schemes are stable under mild conditions or even unconditionally stable and therefore allow large time steps, while they are as easy to program as explicit methods. The basic idea of box schemes is to locate the degrees of freedom at the center of the nodes instead at the center of the arcs as in finite volume schemes. We consider a sequence of discrete time points t0 < t1 < . . . < tN . Let unνa and unwa be grid functions of the state vectors at time tn and the ends of pipe a = νa wa . Then, for the equations (6)-(7) and (11)-(12) written

7

in the general form ∂t u + ∂x F (u) = Q(u), our box scheme on a reads n+1
un+1 un + unwa τn νa + uwa n+1 F (un+1 = νa − wa ) − F (uνa ) 2 2 ∆xa n+1 Q(un+1 νa ) + Q(uwa ) + τn . (15) 2

Here, ∆xa = |a| is the box length, and τn = tn+1 − tn is the time step size. The scheme is symmetric and first order. For scalar conservation laws, it is stable for τn ≥ c∆xa , where c > 0 depends on the flux function, and converges to the entropy solution [13]. Finally, to built up the fully coupled grid equations, the coupling conditions (8)-(9) or (13)-(14) are discretized at t = tn+1 .

The size of the spatial steps in the discretization of the PDEs has been chosen equal to the length of the arcs/pipes since a finer discretization of each real pipe can be achieved by a representation with more than one arc. In fact, such coarse discretizations are usual in practical computations for daily operations. Often, quasi-stationary models are used here, which also consist of a two-point-discretization in space. Neither the selection of the spatial step size nor the temporal step size results from the mentioned stability condition. This only restricts the ratio dx/dt from above by a value approximately twice as large as the speed of sound in gas/water.

6

A Mixed Integer Model for the Transient Case of Gas Network Optimization

We base our mixed integer linear model on the network model introduced in Section 2. For each node ν ∈ V and time step t0 ≤ tn ≤ tN , we introduce pressure variables pnν , and for each arc a ∈ A, flow variables qan with constant lower max and q min , q max , respectively. and upper bounds pmin ν , pν a a For each pipe a = νa wa ∈ AP and time step t0 ≤ tn ≤ tN , we declare addin with respect to our discretization scheme. tional flow variables qνna and qw a Further, we introduce flow variables qνn for each boundary node ν ∈ VB . All 8

bounds are due to technical restrictions and known in advance. Besides the pipes, valves, compressors and control valves are the main components of a gas network. These are modeled by arcs of zero length. A valve is a control element, which can be opened or closed. A compressor compensates for the pressure loss in pipes due to friction and a control valve can be used to reduce pressure if necessary. We proceed with a detailed description of the modeling of each component. Pipes. Besides the flow bounds, we have to model the continuity (6) and momentum (7) equations. We apply our box scheme (15), introduced in Section 5, and obtain the discretized continuity and momentum equations Pνn+1 + Pwn+1 P n + Pwna q n+1 − qνn+1 a a a − νa + C0 wa = 0, 2τn 2τn La

(16)


2 !
n+1 2 n+1 n qνn+1 qw + qw qνn+1 qνna + qw C0 a a a a a + − − + 2τn 2τn La Pwn+1 Pνn+1 a a  n+1 |) q n+1 |q n+1 |  n+1 |) qνn+1 |qνn+1 | λ(|qw A pn+1 C0 λ(|qνn+1 wa wa wa − pνa a a a a + . =− n+1 ρ0 La 2D 2Pνn+1 2P wa a (17) Here, Pνn = P (pnν ) = pnν /z(pnν ) for all ν, the spatial step size is chosen equal to the length La of a pipe and τn = tn+1 − tn is the time step size. Since these equations involve nonlinear expressions, we have to construct a piecewise linear approximation in order to incorporate them into our mixed integer linear program. We refer to Section 8 for further details on this topic. Valves. As mentioned above, a valve is either closed or open and switching between these states is a discrete process, which is best described by introducing a binary variable sna ∈ {0, 1} for each valve a ∈ AV and time step t0 ≤ tn ≤ tN , where sna = 1 if and only if valve a is open in time step tn . The gas flow through a valve is zero if it is closed and ranges between its technical limits in case of an open valve. In order to model this, we introduce the following constraints: qamin sna ≤ qan ≤ qamax sna

∀a ∈ AV and t0 ≤ tn ≤ tN .

(18)

If a valve is open, the pressure values at its adjacent nodes must be equal and can be arbitrary if the valve is closed. To model this, we add Ma sna − pnνa + pnwa ˜ a sn + pn − pn M a νa wa 9

≤ Ma and ˜ a, ≤ M

(19) (20)

˜ a are appropriately chosen for all a ∈ AC and t0 ≤ tn ≤ tN , where Ma and M numbers according to the pressure bounds at the adjacent nodes. Compressors. Since the operation of a compressor is quite complex and often steered by empirical data, we use an idealized model of a compressor [11]. For each compressor a = νa wa ∈ AC and time step t0 ≤ tn ≤ tN , we introduce binary variables sna as in the case of a valve, indicating whether the compressor is running in time step tn (sna = 1) or is switched off (sna = 0). To model the flow bounds we add the constraints qamin sna ≤ qan ≤ qamax sna

∀a ∈ AC and t0 ≤ tn ≤ tN .

(21)

Each compressor has an accompanying bypass valve b(a) ∈ AV connected in parallel. Once the compressor is running, its bypass valve is closed. If the compressor is switched off, the valve is open. To model this, we state snb(a) + sna = 1 ∀a ∈ AC and t0 ≤ tn ≤ tN .

(22)

Additionally, we have to modify the constraints (19)-(20) slightly since the pressure difference between the adjacent nodes must not be arbitrary if the bypass valve is closed, i.e., the compressor is running. So, for bypass valves, we substitute (19) and (20) by ramax snb(a) − pnνb(a) + pnwb(a) pnνb(a)

−

pnwb(a)

≤ ramax and

(23)

≤ 0

(24)

for all a ∈ AC and t0 ≤ tn ≤ tN . Here, ramax denotes the maximum pressure increase achievable by compressor a. A running compressor consumes some amount of the transported gas. The fuel gas consumption fan of a compressor a in time step tn is given by the following formula !  n  γ−1 pwa γ n n n fa = da z(pνa )qa − 1 ≤ famax , (25) pnνa where da is a compressor specific constant and famax is the maximum fuel gas consumption of compressor a. It can be easily seen that the fuel gas consumption is a nonlinear, even nonconvex function. Therefore, the piecewise linear approximation of this function has to be done very carefully. How the linearized fuel gas consumption function can be incorporated in our mixed integer linear program is described in Section 8. Control Valves. A control valve a ∈ AR is similar to a valve and inherits all its properties, but a control valve never occurs as bypass valve of a 10

compressor. In addition, a control valve can be used to reduce pressure in a certain range [ramin , ramax ]. In order to model the pressure reduction, we introduce the following constraints for each control valve a ∈ AR and time step t0 ≤ tn ≤ tN : (Ma + ramin )sna − pnνa + pnwa ˜ a − ramax )sna + pnν − pnw (M a a

≤ Ma and ˜ a. ≤ M

(26) (27)

If a is closed at time step tn (i.e. sna = 0) the pressure difference between the adjacent nodes may be arbitrary, while for an open valve, the pressure decrease is within [ramin , ramax ]. Nodes. Because each running compressor consumes some amount of gas, the mass conservation constraint (8) has to be modified slightly: X X qan + qνna − a∈(A\AP )∩δ + (ν)

X

qan −

a∈(A\AP )∩δ − (ν)

X a∈AP ∩δ − (ν)

qνna +

a∈AP ∩δ + (ν)

X

fan = 0.

(28)

a∈AC ∩δ − (ν)

Besides the pressure bounds, no further constraints have to be fulfilled in each node. Further transient conditions. In addition to the gas dynamics described so far, we have to consider further constraints in our transient model. First, there are restrictions on the minimum runtime and downtime of each compressor, including start-up and shut-down costs, occurring whenever a compressor is switched on or off, which have to be incorporated in our objective function. Second, we must include a terminal condition for the overall gas volume flow of a network. Minimum Runtime and Downtime of a Compressor. Consider a compressor a ∈ AC with minimum runtime Ra ∈ N and downtime ra ∈ N. The minimum runtime of a compressor is modeled by the inequalities sna − sn−1 ≤ sm a a ,

for n + 1 ≤ m ≤ min{n + Ra − 1, N }.

(29)

For modeling the minimum downtime, we introduce the following constraints: sn−1 − sna ≤ 1 − sm a a ,

for n + 1 ≤ m ≤ min{n + ra − 1, N }.

11

(30)

We remark that a complete linear description of the polytope defined by (29) and (30) as well as an appropriate separation algorithm can be found in [7]. Start-Up and Shut-Down Costs of a Compressor. Since switching a compressor on or off generates costs which have to be considered in our objective function, we have to model the switching processes of a compressor. To this end, we introduce binary variables sna,up , sna,down ∈ {0, 1} for each compressor a ∈ AC and time step t0 < tn ≤ tN , where sna,up = 1 if and only if compressor a is switched on in time step tn and sna,down = 1 if and only if compressor a is switched off in time step tn . The following conditions assure these properties: sna − sn−1 − sna,up + sna,down = 0 a sna,up

+

sna,down

≤1

for

1 ≤ m ≤ N,

(31)

for

1 ≤ m ≤ N.

(32)

A complete linear description of the polytope defined by (31) and (32) can be found in [11]. Terminal Condition. Since our optimization process is restricted to a finite time horizon, we have to ensure operational availability of the gas network at the end of the considered timespan. If we do not include a terminal condition, the optimization process tends to produce very low pressure and flow values in the last time step. This is due to the fact that pumping a gas network dry is cheaper than transporting gas from the sources to the sinks. To avoid this, we follow the approach of [3, 4] and set lower bounds to the total gas volume in the network at time tN . According to [15], the gas volume Van in a pipe a = νa wa ∈ AP at time step tn is given by Van =

La Da2 π ρnm La Da2 π z0 T0 n = p , 4 ρ0 4 zm T p0 m

(33)

where La is the length and Da is the diameter of pipe a. The mean compressibility factor zm is given by an appropriate constant value and for the mean pressure pnm we take pnm =

pnνa + pnwa . 2

(34)

We require the total gas volume at the end of the considered time horizon to be at least as large as at the beginning. This can be modeled by the

12

following linear inequality: X

X

Va0 ≤

a∈AP

VaN .

(35)

a∈AP

Please note that the optimization process tends to fulfill (35) at equality, since transporting gas through the network causes costs. Optimization Task. To satisfy the demands of all consumers while keeping the running costs at an acceptable level, all control elements have to work efficiently to transport the gas through the network. The cost of each compressor a = νa wa ∈ AC is modeled by its entire fuel gas consumption. The fuel gas consumption fa is proportional to the power of the compressor Ha (pνa , pwa , qa ), i.e., Ha = dh fa with a constant dh > 0. We replace pwa in (25) using the formula for Ha and take the set of all Ha as control variables. In real-world gas networks, it is necessary to bound the compressor power. Therefore, we have to choose the outgoing pressure pwa such that 0 ≤ Ha ≤ Hamax . Due to efficiency, it is also common to require Hamin ≤ Ha whenever the compressor is on. Our continuous optimization problem for a gas supply network now reads as follows: X Z tN min J(p, q) := fa (t) dt (36) Ha ∈Haad ,a∈AC

a∈AC

t0

subject to all continuous state equations describing the instationary behavior of (p, q). Here, Haad = {H : H = 0 or Hamin ≤ H ≤ Hamax } is the set of admissible controls, which guarantees the boundedness of the compressor power for all a ∈ AC . Observe that the objective function is neither convex nor concave. In order to get a fully discretized model, the objective function is approximated by the trapezoidal rule, using the discrete time points tn . This leads to the linear objective function −1 X NX τn n min (f + fan+1 ), 2 a

(37)

a∈AC n=0

with time step size τn = tn+1 − tn . Further, we have to consider switching n n costs of all compressors. Let Ca,up and Ca,down be the costs (as fuel gas consumption) for compressor a being switched on in time step tn or switched off, respectively. Since we want to minimize the switching costs as well as

13

the fuel gas consumption, our objective function finally reads −1 N X X X NX τn n n+1 n n min (f + fa ) + Ca,up sna,up + Ca,down sna,down . (38) 2 a a∈AC n=0

7

a∈AC n=1

A Mixed Integer Model for the Transient Case of Water Supply Network Optimization

Using the terminology of Section 2, let V be the set of vertices or nodes and let A be the set of arcs in our water supply network. Each arc a = νa wa ∈ A is defined by a starting node ν ∈ V and a finishing node w ∈ V. In our hydraulic model, we use head variables hnν for each node ν ∈ V and flow variables qan for each arc a ∈ A and time step tn ∈ {t0 , . . . , tN }. Head and flow variables are bounded from below and above by some time-independent max , q min and q max . In our model, a backward technical constants hmin ν , hν a a flow is indicated by a negative flow on a directed arc. The water supply into our network is represented by non-negative flow variables qνn for each source ν ∈ VS and time step tn ∈ {t0 , . . . , tN }. In the same manner, water withdrawal at each sink ν ∈ VD is described by non-positive flow variables qνn for each time step tn = t0 , . . . , tN . Since we want to model dynamics, we cannot assume to have a constant flow within a pipe and therefore, we bring n representing the flow at the beginning in two additional variables qνna and qw a and the end of a pipe a = νa wa ∈ AP at time step tn ∈ {t0 , . . . , tN }. Except for pipes, all other arcs are modeled to have no length, and so, we introduce only one flow variable qan per arc a ∈ A \ AP at time tn . Pipes. As we have seen, pipe dynamics are described by the water hammer equations (11) and (12). Applying the introduced implicit box scheme (15) to these partial differential equations yields the discretized equations n+1 n+1 − q n+1 hn+1 hn + hnwa c2 qw νa + hwa νa − νa + · a = 0, 2τn 2τn gA La

(39)

n n+1 qνn+1 + qw q n + qw hn+1 − hn+1 νa a a a − νa + gA wa = 2τn 2τn La  n+1 |)q n+1 |q n+1 |  λ(|qνn+1 |)qνn+1 |qνn+1 | λ(|qw 1 wa wa a a a a − + . (40) 2DA 2 2

14

Here, τn = tn+1 − tn is the time step size. In our model, the spatial step size equals the length La of pipe a. Fortunately, the discretized continuity equation (39) is linear, so we can include it directly into our mixed integer linear program. The momentum equation (40) however contains nonlinear expressions. Therefore, we have to approximate the nonlinear terms by piecewise linear ones to incorporate them into our mixed integer model. We refer to Section 8 for further details on how we build these piecewise linear approximations and how we include them into our model. Tanks. In water supply networks, tanks are used for intermediate storage of water. Roughly speaking, tanks are filled during periods of low demand and emptied during the peak demand periods, although in general the situation is not as simple as that. Mainly for optimal controls this behavior might differ. In our model, the node w, corresponding to the head node wa of a tank a = νa wa ∈ AT is of degree one. This node w (and of course wa ) can be seen as physical point inside the tank, whereas ν and νa represent the physical situation in front of it. At first glance this model might be a little bit confusing, but it has the main advantage that we can represent tanks as arcs in our graph. Using the assumption that tanks have constant crosssectional area Aa , we can derive the following relationship for the change of a tank’s filling level over time: d 1 hw (t) = qa (t) dt a Aa

(41)

A positive flow qa (t) can be imagined as tank filling at time t, in contrast a negative flow represents an outflow. To integrate this ordinary differential equation into our model, we apply the implicit Euler discretization scheme and get the equations n hn+1 1 n+1 wa − hwa = q (42) τn Aa a for each time step. Furthermore, the discharge law, i.e., p qa = Ca sgn (hνa − hwa ) |hνa − hwa | or equivalently (43) qa |qa | = Ca2 (hνa − hwa )

(44)

must hold. Here, Ca is a tank specific discharge coefficient. Since this law must hold at each point in time tn ∈ {t1 , . . . , tN }, we get the system of equations qan |qan | = Ca2 (hnνa − hnwa ). (45) 15

Finally, we piecewise linearize the nonlinear terms qan |qan | and add these approximated version of equations (45) to our mixed integer linear program. Pumps. In pressurized networks, water is distributed by the fact that water flows from points of high pressure to points of lower pressure. Hence, one must increase the pressure at certain parts of the network, e.g., when extracting ground-water or when water is transported uphill. Pumps are used to increase the pressure inside a water supply network. First of all, we introduce a binary variable sna , which indicates if pump a ∈ APu is active or shut down at time tn . Every active pump a = νa wa ∈ APu increases the pressure by some controlled non-negative amount ∆ha = hwa − hνa . Generally, in water networks there are two basic types of pumps. Pumps can have fixed speed. Then, we can describe its pressure gain by the characteristic pump curve (46) ∆ha (t) = αa − βa · qa (t)γa . Here αa > 0 is the maximum possible pressure increase of the pump. The efficiency parameters βa > 0 and γa ≥ 1 are also pump-specific. An inactive/closed pump of either type must have a flow rate qa = 0 and its pressure differential ∆ha can be arbitrarily. In the case of fixed speed pumps, this is modeled by the two constraints M a (1 − sna ) ≤ ∆hna − (αa − βa · (qan )γa ) ≤ M a (1 − sna ),

(47)

max for appropriately chosen constants M a and M a , e.g., M a = hmin wa −hνa −αa max min and M a = hwa −hνa −αa . Again, in the case of γa 6= 1, we get a nonlinear term, which we approximate by a piecewise linearization. Obviously, the discretized version of equation (46) holds, if the pump is running, i.e., sna = 1 and the pressure difference ∆hna of the pump is arbitrary, if the pump is shut down, i.e., sna = 0 and qan = 0. Now, we have to include additional constraints linking the values of qan and sna .

sna ≤ qan , qan

≤

qamax sna

(48) + .

(49)

We can chose the parameter  > 0 to be the minimal relevant non-zero flow. By constraint (48), a flow of less than  implies the pump to be inactive (sna = 0) and a positive flow of more than  forces sna = 1 by (49). Other pumps can operate at variable speed. The technical model of these pumps involves the non-dimensional relative speed ωa (t) of a pump a ∈ APu 16

at time t. Then, the pressure increase ∆ha of a variable speed pump depends on two variables, the flow rate qa (t) and the relative speed ωa (t):     qa (t) γa 2 . (50) ∆ha (t) = ωa (t) αa − βa · ωa (t) As we can see, fixed speed pumps are a special case of variable speed pumps with constant relative speed ωa (t) ≡ 1. Indeed, we model variable speed pumps in exactly the same manner as fixed speed pumps, which reads as    n γa  qa n n n 2 M a (1 − sa ) ≤ ∆ha − (ωa ) αa − βa · ≤ M a (1 − sna ). (51) ωan Again, we have to replace the nonlinear terms by piecewise linear ones in order to include inequalities (51) in our mixed integer model. Valves. Our water supply network model uses three different kinds of valves. The simplest type is a check valve (CV). A check valve is used to avoid backward flow, which can be modeled by qan ≥ 0 for all a ∈ ACV and all time steps. All other valves have in common that they are controllable. For that purpose, we introduce binary variables sna ∈ {0, 1} for each valve a ∈ AV \ ACV and time step tn ∈ {t0 , . . . , tN }. The first class of controllable valves is called gate valves (GV). The flow through a gate valve is zero if it is closed and untouched otherwise. This condition is modeled by introducing qamin sna ≤ M a (1 −

sna )

≤

≤ qamax sna

qan hnwa

−

hnνa

≤ M a (1 −

and sna ),

(52) (53)

for each gate valve a = νa wa ∈ AGV . Constraint (52) forces the flow rate qan to be zero, when the valve is closed (sna = 0). For an opened valve, it follows from inequalities (53) that the head differential is zero. For appropriately max and M = chosen constants M a and M a , for example M a = hmin a wa − hνa max min hwa − hνa , the pressure differential remains unaffected when the valve is closed. Another kind of valve, the flow control valve (FCV) is used to restrict the flow rate to be at most qamax via (1 − sna ) + qan ≤ qamax qamax sna

≤

qan .

and

(54) (55)

For an  indicating the smallest possible amount of positive flow, inequality (54) forces the flow control valve to be in an active state (sna = 1) if qan = qamax and becomes redundant for any 0 ≤ qan < qamax . By inequality (55), the valve 17

is set to an open state (sna = 0) if qan < qamax and becomes redundant in the case qan = qamax . Note that these valves’ states, in contrast to all others, depend on the flow only. Thus, the binary variables sna may be viewed as state variables and not as control variables. In our model, an inactive flow control valve a = νa wa ∈ AF CV causes no pressure loss, whereas this condition does not hold for active valves. This is modeled by 0 ≤ hnνa − hnwa ≤ Ma sna ,

(56)

min for an appropriately large constant number Ma , e.g., Ma = hmax νa − hwa . So far, the presented valves only control the flow rate. Another class of valves is used to control the pressure – particularly pressure breaker valves (PBV). An active pressure breaker valve a = νa wa ∈ AP BV (sna = 1) provides a fixed pressure differential hdif a at time tn , i.e., n n n hdif a sa ≤ hνa − hwa

qamin sna

≤

qan

≤ hdif a ≤

qamax sna .

and

(57) (58)

Optimization Task. Similar to gas networks, in the case of water distribution networks, our objective is the fulfillment of consumer demands (w.r.t flow and head) with minimal costs. Here, the costs are mainly due to electric power consumption of the pumps. The power consumption Pa of a single pump a ∈ APu of either class (fixed or variable speed) can be described by an equation of the form   qa (t) 3 ¯ Pa (t) = ωa (t) α ¯ a − βa · , (59) ωa (t) which depends on the relative speed ωa (t) and the throughput qa (t). The discretized version then reads as   qan n n 3 ¯ Pa = (ωa ) α ¯ a − βa · n . (60) ωa Now, we have to consider that a pump has relative speed equal to zero (and hence no power consumption) if it is shut down. Therefore, we add the condition ωan ≤ sna wamax . (61) This inequality forces ωan to be zero for inactive pumps and leads to ωan ≤ wamax for a working pump. Again, we have to build piecewise linear approximations of the nonlinearities in equation (60). We remark that for fixed speed 18

pumps, i.e. ωan ∈ {0, 1}, equation (60) is even linear. Finally, the minimum power consumption over the optimization time horizon [t0 , tN ] is given by X Z tN min Pa (t) dt, (62) a∈APu

t0

subject to all continuous state conditions describing the instationary behavior of (h, q). For our time-discretized model, we use −1 X NX
τn min Pan + Pan+1 , 2

(63)

a∈APu n=0

with time step size τn = tn+1 − tn , by applying the trapezoidal rule to (62). Please note that additional requirements like minimum running and shutdown times or start-up and shut-down costs of pumps can be included in exactly the same manner as we have already seen for compressors in gas network optimization in Section 6.

8

Piecewise Linearization

Approximation of Nonlinear Functions. In Sections 6 and 7, we introduced the discretized continuity and momentum equations (16), (17), (40) for gas and water supply networks as well as the constraints describing the fuel gas consumption (25) of a compressor and the pressure increase (51) and power consumption (60) of a pump. All these equations involve nonlinear terms. In order to reflect the effects of these equations in our linear model, we build piecewise linearizations of all nonlinear terms. In case of univariate nonlinear expressions, like in the discretized continuity equation of the gas model, this leads us to an univariate piecewise linear function, while approximating the nonlinearities in the discretized momentum equation in the case of a gas network is done by constructing a two-dimensional triangulation. For the trivariate nonseparable fuel gas consumption equation, we have to use triangulations in terms of tetrahedra. Since the number of simplices crucially affects the number of variables in our mixed integer linear program, we use an adaptive approximation scheme, which generates many simplices in regions of high curvature and only few where the approximated function is more planar. 19

Modeling Piecewise Linear Functions. In order to describe a method for the incorporation of a piecewise linear function into a mixed integer linear program, we consider some continuous nonlinear function φ : Rm → R, x 7→ φ(x). We further assume that this function has been approximated by a piecewise linear function φ as described in the last paragraph. Therefore, we can regard φ as a triangulation T = (∆, P), where Rm+1 ⊃ P = {(v1 , φ(v1 ))T , . . . , (vk , φ(vk ))T } is the set of vertices and ∆ = {S 1 , . . . , S l } is the set of m-simplices of the triangulation. We further assume T to be connected in such a way that for each subset ∆1 ⊂ ∆ and ∆2 = ∆\∆1 , there are m-simplices S 1 ∈ ∆1 and S 2 ∈ ∆2 such that S = S 1 ∩ S 2 is a (m − 1)simplex and a face of both S 1 and S 2 . For each simplex S i , we index its 0 vertices such that vij denotes the jth vertex of simplex S i and vim = vi+1 holds for i = 1, . . . , l − 1. This induces a complete linear ordering of the simplices (cf. Figure 2). How to obtain such an ordering is shown in [5]. Starting with this setting, we use the well known incremental method (sometimes also called δ-method) [9, 18] to describe φ in terms of linear constraints. Therefore, we introduce nonnegative variables δij and binary variables yi for i = 1, . . . , l and j = 1, . . . , m. Additionally we add the following constraints to our MILP: x = v10 +

l X m X

(vij − vi0 )δij ,

(64)

i=1 j=1

φ(x) ≈ φ(x) = φ(v10 ) +

l X m X

(φ(vij ) − φ(vi0 ))δij ,

(65)

i=1 j=1 m X

δij

j=1 m X j δi+1 j=1

≤ 1, for i = 1, . . . , l,

(66)

≤ yi , for i = 1, . . . , l − 1,

(67)

yi ≤ δim , for i = 1, . . . , l − 1,

(68)

δij

≥ 0, for i = 1, . . . , l and j = 1, . . . , m,

(69)

yi

∈ {0, 1}, for i = 1, . . . , l − 1.

(70)

The incremental method is superior to the standard textbook approach [12], which uses convex combinations of variables associated to the vertices to describe a point on the triangulation since the polytope defined by (66) - (69) together with the nonnegativity constraints yi ≥ 0 for i = 1, . . . , l is integral [14, 18]. A third way to incorporate piecewise linear functions 20

into a mixed integer linear program is to use branching on special ordered sets (SOS) [12, 16], which was successfully implemented for the case of gas network optimization in [10, 11], but is up to now not suitable in conjunction with irregular grids.

(a) A triangulation where the vertices are (b) The dashed arrows show a path indexed within each triangle in such a way through the triangulation, which is defined 0 holds for i = 1, . . . , 5. that vi2 = vi+1 by the vectors vi2 − vi0 for i = 1, . . . , 6.

Figure 2: A triangulation with vertex indices meeting the requirements of the incremental method.

9

Computational Results

In this section, we would like to present some numerical results from gas network optimization as well as from the optimization of water supply networks. To solve the underlying mixed integer linear problem, we used CPLEX 11.1 [6] as branch and cut solver. Afterwards, we verified the computed results of the linearized model by simulation. To this end, we applied our simulation tool [2], which solves the nonlinear model equations. All computations are carried out with a time limit of one hour on a computer with 8 Dual-Core AMD Opteron 8220 processors and 128 GB of main memory. Gas network optimization. We consider the gas network given in Figure 3 consisting of four pipes, two compressor stations with accompanying bypass valves and one control valve. Starting from a stationary state, we optimized

21

compressor C1

compressor C2

source

sink S2

control valve

sink S1

Figure 3: A gas network consisting of four pipes, two compressors, one control valve, one source and two sinks. time Compressor C1 Compressor C2

0 12,000 kW 12,000 kW

1 15,755 kW 10,448 kW

2 15,064 kW 11,590 kW

3 14,683 kW 12,340 kW

4 10,496 kW 15,302 kW

Table 1: Best feasible control for the network given in Figure 3.

the network operation over a time span of 4 hours. The source delivers gas with a constant pressure of 65 bar. Sink S1 has a constant flow demand of 3 3 500, 000 mh , while the demand at S2 increases linearly from 1, 500, 000 mh 3 to 1, 600, 000 mh . Moreover, we have minimum pressure constraints at both sinks. The pressure at S1 must not drop below 51.5 bar and the pressure at S2 has to be at least 47 bar. The corresponding MIP model consists of 2596 constraints, 3104 continuous and 656 binary variables. After 32s CPLEX has proven optimality of the solution given in Table 1. This solution results in a fuel gas consumption of 30, 746 m3 . We remark that the maximum pressure deviation from the simulated nonlinear model to our optimal control solution is below 0.15 bar, which is within an acceptable range from a practical point of view. As a second example, we consider a substantially larger network consisting of two sources, four sinks, thirteen pipes and three compressors (cf. Fig. 4). Again, we optimize the network over a timespan of 4 hours. The demands 3 3 at all but the second sink increase linearly from 360, 000 mh to 422, 350 mh . 3 At the second sink, the flow is hold constant at 185, 000 mh . The pressure at the first source has to stay constantly at 70bar. The second source feeds 3 725, 000 mh of gas into the network in each of the four hours. The resulting mixed integer linear program has 17159 constraints and 19792 variables from which 5317 are binary. The best solution found after 3600s

22

time Compressor C1 Compressor C2 Compressor C3

0 1997.02 kW 1993.88 kW 1997.11 kW

1 1170.87 kW 2047.01 kW 1328.55 kW

2 1418.12 kW 1701.50 kW 2043.69 kW

3 1418.92 kW 1340.26 kW 2476.76 kW

4 2915.20 kW 2018.11 kW 2779.84 kW

Table 2: Best feasible control for the network given in Figure 4.

is given in Table 2. The corresponding objective value is 7, 388m3 . The relative gap amounts to 16.59%. Verifying our solution by simulation shows that the maximum pressure deviation is below 0.13 bar, which again is within the acceptable range. Although it would be desirable to discover good feasible solutions and especially a proof of optimality faster, our approach seems to be promising since state of the art global solvers for nonconvex MINLP try even harder on similar problems [2]. sink A02

sink A05

source compressor Comp01

compressor Comp02

compressor Comp03

source

sink A06

sink A04

Figure 4: A gas network consisting of thirteen pipes, three compressors, two sources and four sinks.

Water supply network optimization. We consider a water supply network which is used to transport drinking water from one valley to another across a hill of height 100 m. The network consists of one source and one sink, two pipes, a fixed speed pump and an intermediate tank, which is located on top of the hill (cf. Figure 5). The intermediate tank has a cross-sectional 23

tank

11 00 00 11 00 11

source

pump

1 0 0 1

11 00 00 11 00 11

sink 11 00 00 11 00 11

Figure 5: A water network consisting of two pipes, one pump, one tank, one source and one sink. time Pump power [MW] Tank level [m] time Pump power [MW] Tank level [m] time Pump power [MW] Tank level [m]

0 10.0 9 6.17 11.2 18 11.5

1 9.9 10 6.17 11.3 19 9.8

2 9.6 11 6.17 11.2 20 8.7

3 9.0 12 6.17 11.0 21 8.0

4 8.8 13 6.16 10.9 22 7.7

5 6.1 14 6.17 11.0 23 7.6

6 6.14 8.2 15 6.18 11.5 24 7.6

7 6.15 9.8 16 6.20 12.4

8 6.16 10.7 17 6.24 13.8

Table 3: Best feasible control found by branch and cut for the network given in Figure 5.

area of 1573 m2 and a height of 20 m. Its initial filling level is 10 m. For safety, we state that the filling level has to be always at least 6 m. From the source, water can be taken at a constant pressure level of 10 m. The pump is located next to the source and is used to transport the water uphill. It is able to increase pressure by at most 200 m and runs with a maximum
power π of 10 MW. The flow demand of the sink is given by q(t) = 2 sin 24 t , where 3 q is measured in ms and t in hours. We optimize the network over a period of one day, which results in an overall power consumption of 74.07 MWh. The resulting control and corresponding tank level is shown in Table 3. The mixed integer linear problem has 4241 rows and 3146 columns. Thereof 522 variables are binary. The instance was solved within 128s. Our second example on water supply network optimization is taken from a real-life application. The network consists of one source, four sinks, 20 pipes, three pumps and two intermediate tanks (cf. Figure 6). The situation is similar to our first example. Supplied water is pumped uphill and is then either stored in intermediate tanks on top of a hill or it is transmitted directly

24

pump P3

pump P2 supplier

RSA

RSB

RSC

RSE

pump P1

Figure 6: A water network consisting of 20 pipes, three pumps, two tanks, one source and four sinks.

to the consumers. From the source, water can be taken at a constant pressure level of 130 m. Next to the source, three parallel pumps can operate at fixed speed to increase the pressure. Each pump can increase the pressure by at most 720 m. The maximum power of a pump is restricted to 10 MW. The intermediate tanks are located at an elevation of 495 m and each of them has a cross-sectional area of 1573 m2 and a height of 14.5 m. Both initial tank levels are set to 63%. The consumers are located downhill at a height of 400 m, 300 m, 200 m and 100 m, respectively. The consumers’ demand for sinks i ∈ {RSA, RSB, RSC, RSD} is given by  Qi 4 t
,0 ≤ t ≤ 4 , qi (t) = (71) 2π Qi 0.9 + 0.1 cos( 20 (t − 4) , 4 ≤ t ≤ 24 3

with Q1 = 1, Q2 = 0.8, Q3 = 1 and Q4 = 1.2. Here qi is measured in ms and t in hours. The given scenario is optimized over a horizon of one day with a time step size of one hour. The constructed mixed integer linear program consists of 25000 constraints and 25077 variables, whereof 10839 are binary. An optimal solution was found after 743s and has an overall power consumption of 145.51 MW. The resulting pump controls and corresponding tank levels of the optimal solution are shown in Table 4. The solution is verified by a simulation.

25

time Pump P1 power [MW] Pump P2 power [MW] Pump P3 power [MW] Tank T1 level [m] Tank T2 level [m] time Pump P1 power [MW] Pump P2 power [MW] Pump P3 power [MW] Tank T1 level [m] Tank T2 level [m] time Pump P1 power [MW] Pump P2 power [MW] Pump P3 power [MW] Tank T1 level [m] Tank T2 level [m] time Pump P1 power [MW] Pump P2 power [MW] Pump P3 power [MW] Tank T1 level [m] Tank T2 level [m]

0 8.8 8.8 7 4.0 4.0 4.0 4.3 4.3 14 3.4 8.5 8.5 21 3.4 11.8 11.8

1 8.1 8.1 8 4.0 4.0 4.0 6.4 6.4 15 3.7 3.7 9.2 9.2 22 4.1 4.1 4.1 12.5 12.5

2 6.6 6.6 9 3.3 6.9 6.9 16 3.7 3.7 10.1 10.1 23 3.4 12.6 12.6

3 4.0 4.0 10 4.1 4.1 4.1 8.1 8.1 17 9.0 9.0 24 4.1 4.1 4.1 13.3 13.3

4 3.6 3.6 3.4 3.4 11 3.4 8.8 8.8 18 4.1 4.1 4.1 9.8 9.8

5 4.0 4.0 4.0 4.8 4.8 12 6.7 6.7 19 3.4 9.0 9.0

6 4.0 4.0 13 4.1 4.1 4.1 7.6 7.6 20 4.1 4.1 4.1 11.3 11.3

Table 4: Best feasible control found by branch and cut for the network given in Figure 6.

26

10

Conclusion

In this article, we have presented a network model which can be used to describe a general dynamic transport network. We have shown how to use this network model to formulate optimization problems on dynamic transport networks as mixed integer linear programs by means of two examples, namely gas and water supply network optimization. Such kinds of problems may involve nonlinear constraints, which cannot be incorporated into linear programs directly. To this end, we applied a method to approximate such constraints appropriately such that we are able to include them into our model. Finally, we presented some computational results showing that our approach is suitable for global optimization of dynamic transport networks. Future steps focus on extending the presented methods in order to apply them to larger networks and a higher number of time steps.

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