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Optimal inflow control in transport systems with uncertain demands ‐ A comparison of undersupply penalties

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We address the challenging task of setting up an optimal production plan taking into account uncertain demand. The transport process is represented by the linear advection equation and the uncertain demand stream is captured by an Ornstein‐Uhlenbeck process (OUP). With a model predictive control approach, we determine the optimal inflow. We use two types of undersupply penalties and compare the average undersupply as well as the number of undersupply cases in a numerical simulation.
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Received: 16 May 2019 Accepted: 12 June 2019
DOI: 10.1002/pamm.201900252
Optimal inflow control in transport systems with uncertain demands -
A comparison of undersupply penalties
Kerstin Lux1,and Simone Göttlich1,∗∗
1University of Mannheim, 68131 Mannheim, Germany
We address the challenging task of setting up an optimal production plan taking into account uncertain demand. The transport
process is represented by the linear advection equation and the uncertain demand stream is captured by an Ornstein-Uhlenbeck
process (OUP). With a model predictive control approach, we determine the optimal inflow. We use two types of undersupply
penalties and compare the average undersupply as well as the number of undersupply cases in a numerical simulation.
© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
1 Stochastic optimal control problem
Accounting for uncertainty is crucial in the modeling of many real-world processes. One example is the increasing interest in
solutions accounting for large fluctuations in renewable energy generation (see e.g. [1]). Another source of uncertainty in the
production context arises in terms of the unknown demand for goods (see e.g. [2]). A reliable supply is crucial to avoid high
costs of short-term external purchase and for the company’s reputation. We focus on the optimal inflow control in transport
systems avoiding undersupply. We use the stochastic optimal control framework set up in [3].
1.1 Supply system and demand dynamics
We consider a supply system where goods are transported from left (x= 0) to right (x= 1) with a constant velocity λ, i.e.
goods feeded into the system need 1
/λtime units to pass from left to right. Mathematically, this transportation process is
described by the linear advection equation with initial and boundary condition given by
zt+λzx= 0, x (0,1), t [0, T ], z(x, 0) = 0, z(0, t) = u(t), t [0, T 1
/λ]., (1)
where uL2([0, T 1
/λ]) is the inflow control. We set u(0) = 0. The output of the system is y(t) = z(1, t). We aim at
determing the inflow control u(t)such that the resulting supply y(t)optimally matches the uncertain demand (Yt)t[0,T ].
The demand dynamics are described by an Ornstein-Uhlenbeck process (OUP) being the unique strong solution of
dYt=κ(µ(t)Yt)dt +σdWt, Y0=y0.(2)
Wtis a one-dimensional Brownian motion, σ > 0, κ > 0are constants, and y0is the initial demand. The OUP is a popular
stochastic process in demand modeling (see e.g. [4] for electricty demand modeling). Its mean reverting property allows the
interpretation as random fluctuations around a given mean demand level µ(t)possibly obtained from historical demand data.
For further details and an extension of it including the possibility of jumps, please see [3].
The resulting stochastic optimal control (SOC) problem can be stated as
min
uL2([0,T 1
/λ]) ZT
1
/λ
OF (Ys, t0, yt0, y(s))ds subject to (1) and (2).(3)
Here, we assume u(t)to be Fˆ
ti-measurable for t[ˆ
ti,ˆ
ti+1], where ˆ
ti=i·tup,i∈ {0,1,··· ,T1
/λ
/tup}, and
tup [0, T 1
/λ]is the update frequency. Then, 0 = ˆ
t0<ˆ
t1<···<ˆ
tnT1
/λspecifies the grid of update times, where
the current demand at the market is observed. Accordingly, we subdivide our optimization horizon [0, T ]into subintervals
[ˆ
ti,ˆ
ti+1]and solve the SOC problem of type (3) thereon with updated state of the supply system and updated initial demand
(see control method CM2 in [3]).
1.2 Choice of cost function
Our focus is the choice of the cost function for the optimization. We will compare different types of undersupply penalties in
terms of different cost functions OF (Ys, t0, yt0, y(s)). The first one is taken from [5]:
OFpenC(Ys, t0, yt0, y(s)) = E(Ysy(s))2|Yt0=yt0
|{z }
I
+α·E(Ysy(s))2|Ys> y(s)Yt0=yt0
|{z }
II
.(4)
Corresponding author: e-mail: klux@mail.uni- mannheim.de, phone: +49 621 181 2472
∗∗ e-mail: goettlich@uni-mannheim.de
This is an open access article under the terms of the Creative Commons Attribution License 4.0, which permits use,
distribution and reproduction in any medium, provided the original work is properly cited.
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https://doi.org/10.1002/pamm.201900252 © 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
2 of 2 Section 15: Uncertainty quantification
Iis of tracking type and II represents a multiple αof the expected quadratic undersupply given the information of being in
an undersupply case. The multiple αgives the possibilty to control the intensity of penalization. For the effect of different
intensities on the optimal supply, we refer the reader to [5]. We formulate an alternative penalization of undersupply as
follows:
OFpenI(Ys, t0, yt0, y(s)) = E(Ysy(s))2|Yt0=yt0
|{z }
I
+α·E(Ysy(s))2Ys>y(s)|Yt0=yt0
|{z }
III
.(5)
In contrast to (4), it might be of greater practical relevance for a producer to also account for the probability that an undersupply
occurs. Note that the difference between the two objective functions is determined by the latter probability P(Ys> y(s)):
(OFpenC OFpenI )(Ys, t0, yt0, y(s)) =α·E(Ysy(s))2|Ys> y(s)Yt0=yt0·(1 P(Ys> y(s))) 0.
We always have OFpenC(Ys, t0, yt0, y(s)) OFpenI(Ys, t0, yt0, y (s)) with equality if P(Ys> y(s)) = 1, i.e. undersupply is
penalized less for OFpenI. That goes along with two intuitive hypothesis numerically analyzed in the Section 2:
(H1) The average undersupply E(y(s)Ys)Ys>y(s)|Yt0=yt0for OFpenI is higher than that for OFpenC .
(H2) The number of undersupply cases is expected to be higher for OFpenI.
2 Numerical results
With only a slight modification of the reformulation of equation (5) in [5], we again have a deterministic reformulation of
the SOC problem (3). We use the numerical procedure set up in [5] adapted to the alternative cost function OFpenI to test
our hypothesis (H1) and (H2) numerically. We take 103Monte Carlo repetitions for the following parameter setting: T= 1,
λ= 4,µ(t) = 2 + 3 ·sin(2πt),κ= 3,σ= 2,y0= 1.
0.25 0.375 0.5 0.625 0.75 0.875 1
Time
0
50
100
150
200
250
300
350
(a) Number of undersupply cases
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time
-0.2
-0.18
-0.16
-0.14
-0.12
-0.1
-0.08
(b) Average undersupply
Fig. 1: Comparison of OFpenI with O FpenC
In Figure 1, we see that the number of undersupply cases is higher for OFpenI . The average undersupply (negative value)
resulting from OFpenI lies clearly below the one based on the optimization with respect to OFpenC . Both the binary measure
of undersupply as well as the quantitative height of the average undersupply show the higher undersupply penalization by
OFpenC, and numerically confirm our hypotheses (H1) and (H2). However, the increasing or decreasing tendency is the same
for both measures indicating that the use of either one is reasonable.
Acknowledgements The authors are grateful for the support of the German Research Foundation (DFG) within the project “Novel models
and control for networked problems: from discrete event to continuous dynamics” (GO1920/4-1).
References
[1] D. Bienstock, M. Chertkov, and S. Harnett, Chance-constrained optimal power flow: risk-aware network control under uncertainty,
SIAM Rev. 56(3), 461–495 (2014).
[2] J. L. Higle and K.G. Kempf, Production planning under supply and demand uncertainty: a stochastic programming approach, in:
Stochastic programming , Internat. Ser. Oper. Res. Management Sci., Vol. 150 (Springer, New York, 2011), pp. 297–315.
[3] S. Göttlich, R. Korn, and K. Lux, Optimal control of electricity input given an uncertain demand, arXiv preprint: 1810.05480, 2018.
[4] M. T. Barlow A diffusion model for electricity prices, Math. Finance 12(4), 287–298 (2002).
[5] S. Göttlich, R. Korn, and K. Lux, Optimal inflow control penalizing undersupply in transport systems with uncertain demands, arXiv
preprint: 1901.09653, 2019.
© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim www.gamm-proceedings.com
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