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# Total gas recovery maximization

## Mathematical models

### Modeling and algorithmic considerations:<=
/h3>

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In the short term operation, the most important problem is related to th=
e **Total Gas Recovery Maximization**. In order to w=
ithdraw as much natural gas from a reservoir as possible, one option is to =
use waterflooding. This leads to the problem of finding an optimal wat=
er injection rate with respect to different objectives, such as the ma=
ximal ultimate recovery, or the total revenues. Indeed there are sever=
al objective functions due to different aspects of the problem.

Consider two wells drilled on the surface of the gas reservoir, one for =
gas recovery and one for water injection. Therefore, let r(t) denote t=
he withdrawal rate of gas which is bounded by the maximum rate of gas =
extraction rm(t). Through the water injection, well water is injected into =
the reservoir at the nonnegative rate s(t). This model assumes a const=
ant g which is the ratio of gas entrapped behind the injected water to=
the volume of water at any time. The model aims at maximizing the ultimate=
gas recovery and can be posed in a nonlinear form. Some author discuss oth=
er several other objective functions. For example, the objective funct=
ion to maximize the present worth value of the net revenues for intern=
al rate of return**.**

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# Combined Heat and Power Production M=
anagement

~~Introduction~~

## Mathematical models

## Optimization methods

## =
Data and Software

## References

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The future development of energy production and distribution relies on t=
he combined exploitation of different, renewable energy sources, each one w=
ith its own costs and profits, sometimes dependent on weather and/or daylig=
ht hours (e.g., thermal solar). In this contest Combined Heat and Power (he=
reafter CH&P) assumes a increasing relevance due to its high efficiency=
when compared to conventional condensing power plants.

We consider CH&=
amp;P Energy production optimization a tactical and operational issue, beca=
use the management cycle has a yearly horizon and subsequent short term pla=
nning, with the aim of maximization of EBITDA (Earnings Before Interest, Ta=
xes, Depreciation, and Amortization).

The Mixed Integer Linear Programming (MILP) problems arising in this bus=
iness are particular cases of the well-known Unit Commitment (UC) problem (=
see, e.g. \cite{padhy2004unit, wood2012power}), in which the goal is to det=
ermine how to dispatch the committed units in order to meet the demands and=
other constraints cost-efficiently. As presented in \cite{bettinelli2016},=
a plant manager has to serve one or more demands (heat demand mainly, but =
also chill demand and/or electricity demand if the plant is connected to a =
building or to an industrial facility) and these demands can be satisfied b=
y producing energy using different machines, depending on the plant configu=
ration and the various economic drivers of input and output commodities. Al=
so, if the electricity produced by the plant is sold to the National Electr=
icity Network, there is an additional variable (the amount of energy sold) =
and an additional constraint (the amount of electricity committed to the ma=
rket the day before for the following day).

Manual management relies on=
several elements to manage the typical business processes:

- budgeting, to define the overall energy production for the year to come= ;
- revised budgeting, to modify the budget objectives on the basis of the = results of (at least) the first semester;
- weekly and daily operations, most of the times based on the experience = of the plant manager (in respect of the previous years).

The goal of a plant manager is to satisfy the customers=E2=80=99 demands= by choosing the best energy production mix to maximize the EBITDA. Consequ= ently, the decision-making process must take into account the following fac= tors:

- costs, profits and fiscal advantages (if any) of each energy source;
- technical constraints of the plant itself and of the machines;
- regulatory constraints;
- ordinary and extraordinary maintenance requirements.

For each machines, the workload has to be optimized for each time window= . Possible non-linear efficiency curves of the machines are can be approxim= ated in non-convex piecewise linear functions and linearized in the model. = Time is discretized in a finite set of time intervals, having a duration de= fined by the user. Smaller time windows yield to more detailed production p= lans but also increase the size of the associated problems and the correspo= nding computational effort required for their resolution. Fiscal advantages= , as well as power production regulatory constraints are defined by the leg= islator over indicators that measure the performances of the plant over a s= olar year and play a fundamental role in economic terms, since strict respe= ct of certain energy/heat rations is the condition to leverage on incentive= s that are often fundamental to economic sustainability of the plant. This = explains why even for operational day-ahead optimization, an extended time = horizon including the entire month or, more often, the remaining part of th= e solar year should be considered. The presence of thermal energy collector= s, as well as plant-specific period constraints (maximum number of start-up= over period) are other examples that binds together what otherwise may be = considered as single time-windows problems.

UC problems, even when CH&P is not considered, are complex in practi= ce. The solution methods used to solve UC problems span from Lagrangian rel= axation, as proposed in (Borghetti, 2003) and (Li, 2005) to genetic (Kazarl= is,1996) or Tabu search (Mantawy,1998) heuristics. Attention h= as been put as well on obtaining efficient MILP formulations (see e.g. (Car= rion, 2006) and (Ostrowski, 2012)). Interdependencies between power and hea= t productions make realistic CH&P power units even more difficult to be= optimized (Song, 1999). In some cases general purpose MILP techniques are = applied, such as the Branch and Bound (see (Illerhaus, 1999)).= Resolution methods may otherwise rely on time-based decomposition, as in (= Lahdelma, 2003), dynamic programming (Rong, 2009)= or again Lagrangian relaxation (Thorin, 2005). In the business case c= onsidered in (Bettinelli, 2016) a Matheuristics based on time and machines = decomposition have been implemented to reach near optimal solutions in a re= asonable amount of time. The key idea of the heuristic strategies consists = in the resolution of several easier sub-problems rather than directly addre= ssing the original large MILP formulation.

1. Borghetti, A., Frangioni, A., Lacalandra, F., Nucci, C.A.: Lagrangian=
heuristics based on disaggregated bundle methods for hydrothermal unit com=
mitment. Power Systems, IEEE Transactions on 18(1), 313=E2=80=93323 (2003)<=
br>2. Bettinelli A., Gordini A., Laghi A., Parriani T., Pozzi M., Vigo D.: =
Decision support systems for energy production optimization and network des=
ign in district heating applications. Technical Report OR-16-5, DEI =E2=80=
=93University of Bologna, under review by Integrated Series in Information =
Management

3. Carrion, M., Arroyo, J.M.: A computationally efficient mix=
ed-integer linear formulation for the thermal unit commitment problem. Powe=
r Systems, IEEE Transactions on 21(3), 1371=E2=80=931378 (2006)

4. Iller=
haus, S., Verstege, J.: Optimal operation of industrial chp-based power sys=
tems in liberalized energy markets. In: Electric Power Engineering, 1999. P=
owerTech Budapest 99. International Conference on, p. 210. IEEE (1999)

5=
. Kazarlis, S.A., Bakirtzis, A., Petridis, V.: A genetic algorithm solution=
to the unit commitment problem. Power Systems, IEEE Transactions on 11(1),=
83=E2=80=9392 (1996)

6. Lahdelma, R., Hakonen, H.: An efficient linear =
programming algorithm for combined heat and power production. European Jour=
nal of Operational Research 148(1), 141=E2=80=93151 (2003)

7. Li, T., Sh=
ahidehpour, M.: Price-based unit commitment: a case of lagrangian relaxatio=
n versus mixed integer programming. Power Systems, IEEE Transactions on 20(=
4), 2015=E2=80=932025 (2005)

8. Mantawy, A., Abdel-Magid, Y.L., Selim, S=
.Z.: Unit commitment by tabu search. In: Generation, Transmission and Distr=
ibution, IEE Proceedings-, vol. 145, pp. 56=E2=80=9364. IET (1998)

9. Os=
trowski, J., Anjos, M.F., Vannelli, A.: Tight mixed integer linear programm=
ing formulations for the unit commitment problem. IEEE Transactions on Powe=
r Systems 1(27), 39=E2=80=9346 (2012)

10. Padhy, N.P.: Unit commitment-a=
bibliographical survey. Power Systems, IEEE Transactions on 19(2), 1196=E2=
=80=931205 (2004)

11. Rong, A., Hakonen, H., Lahdelma, R.: A dynamic re=
grouping based sequential dynamic programming algorithm for unit commitment=
of combined heat and power systems. Energy Conversion and Management 50(4)=
, 1108=E2=80=931115 (2009)

12. Song, Y., Chou, C., Stonham, T.: Combined=
heat and power economic dispatch by improved ant colony search algorithm. =
Electric Power Systems Research 52(2), 115=E2=80=93121 (1999)

13. Thorin=
, E., Brand, H., Weber, C.: Long-term optimization of cogeneration systems =
in a competitive market environment. Applied Energy 81(2), 152=E2=80=93169 =
(2005)

14. Wood, A.J., Wollenberg, B.F.: Power generation, operation, an=
d control. John Wiley & Sons (2012)

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