Initial train optimization

In this section, the train optimization was performed by giving the inputs from table 6.3 according to the formulation described in chapter 6.

In order to perform an optimization in gPROMS, an initial guess of the decision variables has to be provided so that the program can initialize with those values. Besides that, the lower and upper bounds of those variables have to be provided so that the searching area of optimization becomes smaller, making it easier to find the optimal solution. Also, good bounds and initial guesses are half step to avoid possible discontinuities, local minimums and unrealistic solutions that the solver can deliver during the iteration process.

For this reason, it is important to have a good idea of what are the limits and typical values for the decision variables in this system: number of compressors and pressure ratio.

In the report followed by this study, the average pressure ratio of the compressors was around 2.67, so that value was assumed as the initial guess for the optimization. As for the bounds, based on what is commonly used, a wide range between 1.2 and 4 was used in order to allow all feasible possibilities.

As a first attempt, concerning a good understanding of the system and due to the complexity of the problem, the number of compressors was fixed and not included as a decision variable. Instead, many Non-Linear Programming (NLP) problems were solved by optimizing the pressure ratio and fixing the number of compressors into different values. This way, the problem becomes much simpler, it is certain that the optimal solution is found and it is also possible to observe the effect of both decision variables, separately, in the objective function.

Using this strategic approach, the number of Degress of Freedom (DOF) determined in section 6.4 changes from 8 to 1 because, having the number of compressors fixed, the only decision variable left is the pressure ratio. This way, the problem becomes very simple and quick to solve. In fact, it is pos- sible that, given the number of possibilities and environments where the CO2train can be designed, the best way of finding the optimal solution could be running many of these ”simple” optimizations in parallel for different number of compressors, with the advantage that the user has access to all the

integer solutions of the problem.

Sometimes, the best economical solution to the design conditions is not the best one to build, since there is also the need for the train to have some flexibility in terms of load and discharge pressure. So it could be important for the user to test the best two or three solutions from the program in off-design conditions and compare their performance in terms of OPEX and also check the existence of surge or choke in the centrifugal compressors.

According to figure 5.2, although there are only three compressors before the dehydrator, two of them have intercooling between stages. So, since in this superstructure the design of each compres- sor is performed in a section basis, that can lead to more than three compressor sections due to the fact that there’s no intercooling between stages.

Having this in mind, the point optimization was performed for a range of number of compressors between three and the maximum allowed by the superstructure (eight compressors) in order to have the wider range possible excluding infeasible trains with just one or two compressors.

Both constraints, maximum discharge temperature of each compressor and range of final dis- charge pressure, formulated in section 6.1 were taken into acount in the optimization and the results were the following:

Figure 7.1:Optimal total cost in function of number of compressors (a), Pressure ratio of each compressor from an eight compressors compression train (b), Minimum total cost and main components in function of number of compressors (c).

By analyzing figures 7.1(a) and 7.1(b), it is possible to see that the optimal solution for this problem, the one that minimizes total cost, corresponds to a compression train with 4 compressors and a pressure ratio of 2.3. Also, as expected, the pressure ratio decreases with the number of compressors, since the compression work is being divided by more compressors.

OPEX has a much bigger contribution to the total cost than CAPEX, representing around 85%

of total cost. This relationship remains almost constant while changing the number of compressors.

Although OPEX has a much bigger weight in the objective function than CAPEX, the variation of OPEX with the number of compressors has the same order of magnitude than CAPEX. So, both components are important and the optimal solution results in a balance between them.

Figure 7.2:Optimal value of CAPEX in function of number of compressors.

CAPEX increases with the number of compressors, since the bigger the train, the more compres- sors and coolers are needed, leading to more spare parts, control loops and a bigger project.

By analyzing the contribution of each component of CAPEX, the electric drive is the most important one (representing in average 33% of CAPEX), followed by the compressor and project cost (with an average weight of 28% and 16% respectively) and the rest of the components represent about 20 to 24% of total cost.

It is also possible to see that, although the electric drive has the biggest weight in CAPEX, its variation with the number of compressors is much smaller than for the compressors, coolers and project costs. In fact, the components with larger variation with the number of compressors are the compressor cost and the project cost (both increase around 10000$ for each compressor and cooler added).

Figure 7.3:Optimal value of OPEX in function of number of compressors.

According to figure 7.3, the electricity cost (C elec) clearly has the biggest weight in OPEX, with an average weight of 95%, while the cooling water (C cw), maintenance (C main) and interests (C int) have a combined weight of 5%. Again, in spite the fact that the major contribution to OPEX comes from the electricity cost, the variation of the component costs around the optimal solution (4, 5 and 6 compressors) is similar between electricity and cooling water.It is possible to observe that the number of compressors that minimize electricity cost is five but the actual number that minimizes OPEX is four due to the effect of the other components.

Observing the variation of OPEX with the number of compressors, there is a clear minimum lo- cated in 4 compressors. At a first sight, it seems to be against thermodynamics, since the more compression steps followed by intercooling the system has, the closer the compression work gets to the ideal isothermal work, reducing the electricity cost has it was referred in section 2.3.2.

In order to determine the effect that’s changing the expected behavior of the system, the electricity cost and polytropic efficiency of the compressors are show in the next figure:

Figure 7.4: Optimal electricity cost in function of number of compressors (a), Polytropic efficiency of each com- pressor from the optimal train with eight compressors (b).

As it was mentioned before and looking at figure 7.4(a), it is possible to see that the minimum value of OPEX comes from the minimum of the electricity cost effected by the other components. The

reason of this minimum is due to two different and opposite effects: the approximation to the ideal isothermal work, described previously, and the relationship between the polytropic efficiency of the compressor and its diameter.

As the CO2becomes compressed, its volume flowrate decreases, making the compressor diame- ter decrease as well so the diameter of the compressor decreases along the train. Since the efficiency penalty decreases with the diameter (see section A.1.5), the last compressors of the train are the ones with the lowest efficiency, illustrated in figure 7.4(b). Having these two influences, the final number of compressors from the optimization results in a balance between the approximation to the isothermal ideal work and the decreasing of efficiency.

In spite of having the problem solved by solving many small problems for fixed values of number of compressors, as a proof of concept, the problem was also solved with the original formulation from chapter 6. In other words, the problem was solved as a MINLP optimization problem by optimizing both number of compressors and pressure ratio.

Table 7.1:Critical output results from initial point optimization.

Output variable Variable Ns Fixed Ns Deviation (%)

Pressure ratio 2.3073 2.3068 0.02

Number of compressors 4 4 -

Total cost (M$/yr) 6.515 6.513 0.03

CAPEX (M$/yr) 0.856 0.855 0.12

OPEX (M$/yr) 5.659 5.658 0.02

Average MINLP iterations 2 0 -

Average NLP iterations 21 10 -

Average total CPU time (s) 12.1 7.5 -

The results from table 7.1 were obtained using 3 as an initial guess for the number of compressors and using 3 and 5 as lower and upper bounds, knowing that the solution is already inside that range.

As for the pressure ratio, the initial guess and bounds were the same used in the previous optimization (initial guess of 2.67, lower bound of 1.2 and upper bound of 4).

Even thought the complexity of the problem is higher than the previous one, since the number of DOF increased from 1 to 4, the optimal solution can be considered the same, since the deviation is smaller then 0.5% for all critical variables of the problem. Moreover, both problems are still relatively simple, since the number of DOF is reasonably small, so it’s normal that the difference between both methods is not a problem in terms of optimality and optimization time.

Note that, during this optimization, the compressor duty (around 9200 Hp) and cooler area (around 220 m²) were inside the range of applicability of the cost functions from section 6.2.2.