Hydraulics linearization

[Sourabhhkt]

Fixes/Addresses/Summary/Motivation:

Co-optimize method had two non-linearities: Hazen-William Rule and Pump Cost rule. These non-linearities result in a very slow computation by non-linear solvers. We make piecewise linearization of the term flow raised to the power 1.85 in the RHS of Hazen-Williams equation. In the LHS of the same equation, we model the product of pressure change and diameter selection binaries with a set of linear constraints. Finally, for the variable (electricity) component of the pump cost is modeled by compartmentalization of the volumetric flow through the pump. Thus, the formulation is completely linearized.

Changes proposed in this PR:

• the term in RHS of Hazen-William equation is modeled using a piecewise linear function.
  Brief idea: Consider a non-linear function of one variable with any point on the curve represented by (x, f(x)). We can fit a piecewise linear function as an approximation to f(x). Say the new point (x, y) is an approximation of the original point as shown in the figure below. Then, it is possible to model this (x, y) as a system of linear constraints. This can be done by introducing logarithmic number of binary variables as the number of sections of piecewise linear graph. Refer section 2 of https://link.springer.com/chapter/10.1007/978-1-4614-1927-3_10
  

• For LHS of the Hazen-William equation, we have a multiplication of non-negative continuous variable(pressure drop) and binary variable(diameter choice). When we have such a product, it can be modeled using linear constraints. For example, consider a term P = Hz where H is a non-negative continuous variable with upper bound M and z is a binary variable. Then we can model P using
  a) P>=0 b) P<=H c) P<=Mz d) P>= H - M(1-z)

• In Pump cost rule, the variable cost is modeled using comparmentalization of flow. For example, say the section which the flow is in is 20000bbl/day to 30000bbl/day. Then the variable cost would be modeled as vρgΔH*20000

• The documentation of hydraulics is updated to reflect the proposed changes in the formulation.

• Node pressure rule for co_optimize method is corrected in the documentation. Earlier it had flipped sign of pump head and valve release.

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[ElmiraShamlou]

Please consider exploring alternative methods to linearize the pump cost function in the co-optimize approach rather than replacing the pump head with constant values. Co-optimize is designed to incorporate pump cost within the objective function, accounting for elevation changes and frictional losses. Using a constant pump head overlooks the critical impact of elevation changes and frictional losses in the network on the objective cost. Without variable pump heads, hydraulic post-processing might suffice instead of co-optimize.

Additionally, it's important to note that we cannot use a constant number from a small case study and generalize it to all other case studies.

[Sourabhhkt]

Thanks for the comment and feedback! The pricing for the variable component of pump now has the pump head variable. The linearization is now done based on the flow. Say grid size for flow is 10000 bbl/day. Then for 0<flow<10000, the flow is assumed to be constant 0, for flow 10000<flow<20000, the flow is assumed to be constant 10000, and so on.

[codecov]

Codecov Report

Attention: Patch coverage is 95.94595% with 6 lines in your changes are missing coverage. Please review.

Project coverage is 88.64%. Comparing base (889ed4d) to head (3950d9f).

Files	Patch %	Lines
...anagement/strategic_produced_water_optimization.py	94.05%	6 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #285      +/-   ##
==========================================
+ Coverage   88.37%   88.64%   +0.26%     
==========================================
  Files          23       23              
  Lines        5059     5203     +144     
==========================================
+ Hits         4471     4612     +141     
- Misses        588      591       +3     

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[tarnold17]

Nice work - looks good for the most part to me, but a bit of cleanup is needed.

Also, would it be possible to develop a Jupyter notebook tutorial showing/explaining how to use the Hydraulics.co_optimize_linearized method? That would be a nice touch if wouldn't take too long to put together.

[tarnold17]

Great work!

[ElmiraShamlou]

Approved. Great work!