Implementación del biofertilizante a escala invernadero

Solving the above field-wide shale-well scheduling problem involves several sources of uncertainty. These uncertainties include parameters in the well and reservoir model (4.3)–(4.5), uncertainties in the spot price estimate ˆGspot_{, operational un-}

4.5. Receding horizon optimization

certainties and disturbances such as a sudden drop in the rate for some wells, unplanned shut-ins or required maintenance, and changes in the line pressures. Moreover, the overall system may change during the planning horizon _{K if new} wells are tied to the gathering system. Some control scheme is hence necessary to ensure reliability of the natural-gas supply to the NGPP.

To handle operational disturbances and uncertainties, as well as a varying gas spot price Gspot_{, we implement a receding horizon scheme (Rawlings and Mayne,}

2009; Mayne et al., 2000), also referred to as model predictive control (MPC). At the current control time tk, we compute optimal production settings on a prediction

horizon K, while we only implement the first optimal control input, i.e. pt,j1and

yd

j1 for all j ∈ J and d ∈ D. At the next time control time, tk+1, we reoptimize

the system on a receding horizon K + 1. Between the control inputs, we collect system information and measurements, hence introducing feedback in the system. At each iteration tk of the receding horizon scheme, we use the optimal solution of

the Lagrangian from the previous iteration, tk−1, as a starting point for the MILP

subproblems (4.19), together with (¯λ, ¯π) and γ from iteration tk−1 as initial guess

for the multipliers (λ1_{, π}1_{) and the initial penalty parameter in the bundle method}

(4.25).

Optimize well

Shale-gas field

Tuned proxy model

Implement first well schedule,

Receding horizon control with Given firm gas

demand dNGPP k

Measured rates, pressures and states of the wells (i.e. shut-ins) Update proxy (performed less Exported gas rates Spot price Source: Statoil.com estimate ˆGspot k

reoptimization of shut-in schedule

frequently) schedule Planned events yd j1, d∈D, and tubinghead pressures pt,j1

Fig. 4.7: Modules in the receding horizon scheme.

The implemented receding horizon scheme is illustrated in Fig. 4.7. At a given control time tk, we assume that a spot price estimate ˆGspottk for the prediction

horizon K is given as a system input, provided by a third-party. If the reoptimiza- tion of the well schedule is performed on a daily basis, then each evening after closing hours of the gas trading, the gas spot price for the day ahead delivery is collected (FERC, 2012), together with an updated gas-price estimate ˆGspottk+1. Note

that the gas price ˆGspot₁ for the first timestep in the optimization, i.e. for the first day ahead, will be the actual settled gas price. The updating of the spot price will hence correct the price estimate ˆGspot2 used in the previous computed well schedule

at iteration tk−1. This set-up of the gas spot price introduces a trade-off between

trusting the future price estimate ˆGspot_k compared to the known one day ahead price. To accommodate this trade-off when computing the optimal well schedule,

we introduce a discount factor, ˆ GspotDisc_k = Gˆ spot k (1 + ξ)(k×∆k). (4.30) The discount rate ξ in (4.30) represents risk more than depreciation of income: Assigning a high value to ξ would reduce the risk of losing profit if the price should decrease. In contrast, a low discount factor corresponds to trusting the spot price estimate ˆGspot_k , in which the gas production to the spot market may be choked to wait for an estimated higher future spot price.

For each well, we collect available measurements such as the gas rate and tub- inghead pressure. In particular, we collect the state of each well, i.e. if the well is producing or shut-in. Wells may experience sudden, unplanned shut-ins due to several reasons, including mismatch in actual and predicted rate (e.g. the gas rate drops below qgc) and failure of surface equipment. These events amount to opera-

tional uncertainties. Wells may further have to be shut-in for certain periods due to planned maintenance. Either of these events are handled through the feedback in the receding horizon scheme, by fixing binaries yd0

jk to 1 for the next k0timesteps

for those wells that must be shut-in.

To obtain closed-loop behavior in practice by a receding horizon control scheme, there are several other aspects that must be considered. Unless a measurement of the complete state vector is available, then some estimation technique, typically a Kalman filter or a moving horizon estimation technique (MHE) (Rao et al., 2001), must be applied to estimate the initial condition for iteration tk+1. For the shale-

well problem, this amounts to estimating minit

j . In the current implementation,

however, we omit for simplicity the state estimator, and hence apply the pseu- dopressure predicted by the proxy to update minit

jtk+1. Furthermore, an updating

strategy for the proxy model should be applied, using any collected measurements of the rates and pressures. This can be performed using either the simple updating strategy described in Knudsen et al. (2014a), or by some MHE based technique (Rao et al., 2001).

In conventional stabilizing tracking MPC (Mayne et al., 2000), the change in control input is normally penalized to avoid excessive actuation causing wear and tear of the equipment. The proposed receding horizon scheme, however, resembles economic MPC (EMPC) (Amrit et al., 2011), in which adding such a penalization term would deteriorate the economic interpretation of the objective (4.6). The constraints (4.17) added to limit the switching frequency of the wells, however, serve a similar purpose as a penalization term on the change in control input. To enforce these constraints also in the shift from one receding horizon iteration to the next, we implement a move-blocking strategy Cagienard et al. (2007). Before each reoptimization, we check if the well has changed its state, either from on to off or vice versa, and if so, we fix yd0

jk to its current value for the next τ1− 1 or τ2− 1

timesteps. The same type of variable fixing is applied to yd21

jk to avoid excessive

changes in the state of the second valve in Fig. 4.5. Note that by fixing either yd0

jk or yd21jk , we also fix the variables (yjkd1, yd2jk), (yd22jk , yjk¬d2), respectively, due to

the SOS1 constraints (4.32p) and (4.34f) associated with the reformulation of the disjunctions (4.10d).