Appendix A. Mathematical Model of Flowing Tank Farm Simulation

This appendix summarizes the mathematical formulation of the flow control algorithm implemented in the Digital Twin model.

Notation

Symbol Description
\(P_i(t)\)Flow from process unit i
\(D_j(t)\)Auxiliary flow from upstream tank farms
\(V_{in}(t)\)Total inlet flow (t/h)
\(V_{out}(t)\)Outlet flow (t/h)
\(S(t)\)Tank inventory (t)
\(S_{min},S_{max}\)Inventory limits
\(S_{target}\)Target inventory
\(V_{min},V_{max}\)Pump capacity limits
\(dV\)Maximum outlet flow change per step
\(\Delta t\)Simulation time step

1. Total Inlet Flow

$$ V_{in}(t)= \sum_{i=1}^{N}P_i(t)+ \sum_{j=1}^{M}D_j(t) $$

2. Material Balance

$$ S(t+\Delta t)= S(t)+ \left( V_{in}(t)-V_{out}(t) \right)\Delta t $$

3. Pump Capacity

$$ V_{min} \le V_{out}(t) \le V_{max} $$

4. Inventory Limits

$$ S_{min} \le S(t) \le S_{max} $$

5. Flow Ramp Constraint

$$ V_{out}(t)-dV \le V_{out}(t+\Delta t) \le V_{out}(t)+dV $$ or equivalently $$ \left| V_{out}(t+\Delta t)-V_{out}(t) \right| \le dV $$

6. Maximum Feasible Outlet Flow

$$ V_{out}^{max} = \min \left( V_{max}, V_{ramp}^{max}, V_{inventory}^{max} \right) $$ where $$ V_{ramp}^{max} = V_{out}(t)+dV $$ $$ V_{inventory}^{max} = V_{in}(t)+ \frac{S(t)-S_{min}} {\Delta t} $$

7. Minimum Feasible Outlet Flow

$$ V_{out}^{min} = \max \left( V_{min}, V_{ramp}^{min}, V_{inventory}^{min} \right) $$ where $$ V_{ramp}^{min} = V_{out}(t)-dV $$ $$ V_{inventory}^{min} = V_{in}(t)+ \frac{S(t)-S_{max}} {\Delta t} $$

8. Preferred Outlet Flow

Ignoring operational constraints, the preferred outlet flow is $$ V_{out}^{*}(t)= V_{in}(t)+ \frac{ S(t)-S_{target} } {\Delta t} $$

9. Selected Outlet Flow

The controller projects the preferred outlet flow onto the feasible operating region. $$ V_{out}(t)= \min \left( V_{out}^{max}, \max \left( V_{out}^{min}, V_{out}^{*}(t) \right) \right) $$

10. Optimization Objective

The controller seeks an outlet flow satisfying $$ S(t)\rightarrow S_{target} $$ subject to $$ V_{min}\le V_{out}(t)\le V_{max} $$ $$ S_{min}\le S(t)\le S_{max} $$ $$ \left| V_{out}(t+\Delta t)-V_{out}(t) \right| \le dV $$

11. Control Algorithm

  1. Calculate total inlet flow.
  2. Determine the operating mode.
  3. Compute feasible outlet-flow limits.
  4. Calculate the preferred outlet flow.
  5. Apply pump, inventory and ramp-rate constraints.
  6. Update the material balance.
  7. Advance the simulation time.
  8. Repeat until the end of the simulation.