2. Steam System Description and Problem Statement
In an ethylene plant, the steam system generally provides power and heat to drive the compressors or pumps and heat the streams through the heat exchangers.
Figure 1 presents a schematic of the steam system, which is composed of multiple steam headers, including super-high-pressure steam (SPS), high-pressure steam (HPS), medium-pressure steam (MPS), and low-pressure steam (LPS). Furthermore, four devices, namely, boilers, steam turbines, heat exchangers, and letdown valves, are always present in a steam system.
As shown in
Figure 1, the SPS comes from the cracking furnace system and the start-up boiler system in a typical ethylene plant. The cracking furnace recycles heat from the high-temperature flue gas to produce SPS. The start-up boiler produces SPS by burning fuel gas to meet the process demands as a supplement. The SPS section supplies steam to the HPS section by extracting steam from the steam turbine and letdown valve. Section MPS is supplied by sections SPS and HPS. Section LPS is provided by sections HPS and MPS. Letdown valves L1, L2, and L3 are designed to regulate the pressure of the steam headers to guarantee the safety of the steam system. The process demands mainly include streams that are heated. Steam is converted into water, which can be used as boiler feed water after purification.
In this work, we aimed to minimize the operating cost of a steam system while satisfying the process demands, such as shaft power and heat. The following assumptions are presented to develop the steam system model:
- (1)
The costs of SPS from the cracking furnace and the start-up boiler are the same;
- (2)
the unit costs of SPS generated by the start-up boilers under different loads are the same;
- (3)
the costs of the driving pump by a steam turbine or an electrical motor are the same, and no switch cost is incurred; and
- (4)
the heat demands are constant during optimization.
For a fixed shaft work demand of the compressor, different combinations of inlet and extraction steam flow rates of a steam turbine could be provided. Hence, the flow rate of the extraction steam is selected as the optimization variable. The flow rates of the inlet steam for the letdown valve is also chosen as the optimization variable to regulate the steam header pressure. Depending on the cost deviations of steam and electricity for providing per unit shaft work, the steam turbine or electrical motor is selected to drive a pump. The use of steam turbine or motor is denoted by a binary variable. The model is also subject to mass and energy balance, process demand, and variable range constraints.
The literature review in
Section 1 indicate that the efficiency of steam turbines varies under different operating conditions. Therefore, the parameters of steam turbine models are considered as uncertain parameters. A DDRO paradigm is proposed for the optimization of the steam system by utilizing the hybrid model and historical data to eliminate uncertainties.
5. Case Study
To verify the capability of the proposed DDRO schema for steam system optimization under uncertainty, we present an actual case study from an ethylene plant. The steam system has four steam headers (SPS, HPS, MPS, and LPS), four extraction–exhausting steam turbines (EEST1–4), 29 back-pressure steam turbines (BST1–20), three letdown valves (L1–3), and more than 50 heat exchangers with fixed energy demands.
The initial conditions of the four EESTs, such as inlet and extraction steam flow rate and shaft power are summarized in
Table 1. The initial conditions of BSTs and electrical motors, such as inlet flow rates, rated power, and initial states, are summarized in
Figure 3. The process demands are shown in
Table 2. Furthermore, the prices of SPS, electricity, and BFW are 210 CNY/t, 1.25 CNY/kwh and 10 CNY/t, respectively.
Moreover, the parameters of letdown valve models are as follows: c1 = 1.075, c2 = 1.073 and c3 = 1.057. To compare the level of conservatism, we set as 0.04, 0.1, 0.2, 0.4, and 0.8 with corresponding budget of 4.01, 3.64, 3.32, 2.89, and 2.14, respectively. The models are coded in GAMS 24.0.2, and the DICOPT solver is used to solve the proposed DDROSS model.
The results of the deterministic model (DOSS) and the proposed DDROSS with different budgets are listed in
Table 3.
As shown in
Table 3, the optimal operating cost of the DOSS model is 102,313 CNY/h. The optimized operating cost of the DDROSS model with different budgets ranges from 5.34% to 8.19%, which is higher than that of the DOSS model. The DOSS model uses a set of fixed model parameters, and the DDROSS model is subject to an uncertain parameter set. Moreover, when the value of
increases, the size of the uncertainty set and the operating cost decrease. In this case study, the difference between
ε = 0.8 and
ε = 0.04 is approximately 2.63%, which is used to adjust the level of the robustness and the conservatism of the DDROSS model.
The changed binary variables of DDROSS model are listed in
Table 4.
Table 4 shows that more electrical motors are switched to run from standby, because the cost of electricity generation is cheaper than that of the additional steam generation. Some of the electrical motors are switched to stop from running to balance the pressure in the specific steam headers.
The extraction steam flow rates of EESTs under the initial condition, the DOSS model, and the DDROSS model with different budgets are presented in
Figure 4.
Figure 4 shows that the extraction steam flow rates of the DOSS model and the DDROSS model with different budgets decrease except for EEST2 in the DDROSS model, in which
e = 0.04, 0.1. The flow rates of extraction steam in the EESTs of DDROSS model with different budgets are greater than those of the DOSS model. The DOSS model has fixed parameters of EESTs, whereas the DDROSS model has uncertain parameters of EESTs, indicating that the latter is subject to more constraints during optimization. Depending on different efficiencies, the four EESTs have different variation trends to satisfy the shaft power and heat demands from the process.
Moreover, the steam flow rates of the letdown valves are almost zero after optimization mainly because the letdown valves are used to balance the pressure in the steam headers but do not provide power or heat to the compressor, pump, or heat exchanger. The well configuration of steam, electricity, and water can improve the efficiency and stability of the steam system.
To compare the performance of the proposed method and the data-driven adaptive robust optimization (DDARO) in research [
19] for steam system optimization under uncertainty, we present the results in
Table 5 using the same prices of the SPS, electricity, and BFW in [
19].
As shown in
Table 5, the numbers of continuous variables and constraints of DDROSS model are far less than those of data-driven adaptive robust counterpart for steam system (DDARCSS) model. This is because the dual transformation brings many times of auxiliary variables and constraints. The operating cost of DDARCSS model with
h = 0.05 and
α = 0.4 is less than that of DDROSS model with different parameters due to using adaptive mechanism. Therefore, the DDROSS model is suitable for large-scale optimization problem because its numbers of variables and constraints increase linearly. On the other hand, the DDARCSS model is suitable for the small-scale problem since its numbers of variables and constraints increase exponentially. This may help us choose different robust optimization frameworks for actual optimization problems under uncertainty.