The Potential of Fractional Order Distributed MPC Applied to Steam/Water Loop in Large Scale Ships

: The steam/water loop is a crucial part of a steam power plant. However, satisfying control performance is difﬁcult to obtain due to the frequent disturbance and load ﬂuctuation. A fractional order model predictive control was studied in this paper to improve the control performance of the steam/water loop. Firstly, the dynamic of the steam/water loop was introduced in large-scale ships. Then, the model predictive control with an extended prediction self adaptive controller framework was designed for the steam/water loop with a distributed scheme. Instead of an integer cost function, a fractional order cost function was applied in the model predictive control optimization step. The superiority of the fractional order model predictive control was validated with reference tracking and load ﬂuctuation experiments. methodology, R.D.K., C.-M.I. and S.Z.; software, S.Z. and R.C.; formal analysis, S.Z.; writing—original draft preparation, S.Z.; writing—review and editing, S. Z., R.C., R.D.K. and C.-M.I.; supervision, C.-M.I. and R.D.K.; funding acquisition, S.Z. and C.-M.I.


Introduction
A total of 80% trade volume is moved with shipping industry, which is becoming an important part of the world economy [1]. In order to obtain large carrying capacity and low operation cost, there is a large demand for large-scale cargo ships. To achieve energy saving and a stable energy supply, research is required to optimize the power plant in large-scale ships [2]. The steam/water loop in large scale ships has the function of supplying water and recovering the waste steam for the steam power plant. There are five output variables including the drum water level, exhaust manifold pressure, deaerator pressure, deaerator water level and condenser water level. The steam/water loop has the features of nonlinearity, multi variables and strong coupling system. During the operating of the ship, the frequent disturbance and load fluctuation make it difficult to obtain a satisfying control effect for the five sub-loops [3]. The challenges in the control of steam/water loop can be summarized as follows: • the steam/water loop is a system of nonlinearity, strong interactions and multivariable; • load and disturbances change frequently with large amplitude; • operating conditions changes frequently (there are ten levels of the sea state); • demands for mobility and rapidity are growing high.
where T s is the sampling time and Γ(·) is Euler's Gamma function.
In this paper, the original contribution lies in that the constant values of the weighting factors in classical MPC is replaced with time-varying weighting factors in fractional order MPC (FOMPC). The weight factors in the classical MPC have N p + N c elements for tracking error and control effort need to be defined, where N p is prediction horizon and N c is control horizon. The optimal problem for the weighting factors will be very complex. Hence, the weighting factors are mostly kept as constants in MPC. In the FOMPC, the weighting factors are obtained with the fractional orders, in which optimal different weighting factors can be tuned with two fractional orders, one for the tracking error and another for control effort. Hence, the FOMPC will result in a better system performance than the MPC method. In this work, fractional order MPC is proposed firstly for steam/water loop with GL definition. Then, the integer order cost function is replaced with a fractional order one in the MPC. In order to obtain a relatively good value set for the five sub-loops, different fractional orders are required for each sub-loop. The results show that the bigger fractional order leads to better control performance. Finally, reference tracking and load fluctuation experiments are conducted to verify the effectiveness of the FOMPC.
The paper is structured as follows: firstly, the steam/water loop is introduced in Section 2. Section 3 gives a brief introduction about Extended Prediction Self Adaptive Controller (EPSAC), and the fractional order cost function is designed. Section 4 shows the results of FOMPC compared with classical MPC. Conclusions are discussed in Section 5.

The Description for Steam/Water Loop
The steam/water loop in the large scale ships plays the role of supplying water to the boiler and recycling waste steam from turbine. The system is composed of a boiler drum, exhaust manifold, deaerator and condenser. The system operating principle can be found in Figure 1, in which there are two main operating loops. One is indicated with the red line for the steam loop, while the green line for the water loop. The description for the system are as follows: firstly, the condensed water from condenser goes to the deaerator for preheating and deoxygenation. Secondly, the feedwater is pumped to the drum. Due to the high density of the water than the steam, the water goes to the mud-drum. In the risers, the water absorbs the heat and becomes the mixture of steam and water. The steam gets separated in the drum and goes to the turbine after being heated in the economizer. The waste steam from the turbine and other auxiliary machines goes to the exhaust manifold. Most of the waste steam goes to the condenser, and the left part of waste steam goes to the deaerator for deoxygenation. Finally, the condensate water goes to the deaerator to work again. In the steam/water loop, the input variables u = [u 1 , u 2 , . . . , u 5 ] are valves for the five sub-loops, and the output variables y = [y 1 , y 2 , . . . , y 5 ] are the drum water level (y 1 ), exhaust manifold pressure (y 2 ), deaerator pressure (y 3 ), deaerator water level (y 4 ), and condenser water level (y 5 ), respectively [3]. The transfer functions and constraints of the steam/water loop are shown as follows: where , and other transfer The operating point and ranges of variables are shown in Table 1.

Brief Introduction for EPSAC
In this part, a brief introduction about EPSAC is given, and the detailed theory is introduced in [26]. For a discrete system: where t is the discrete time index; y(t) and x(t) are the system output and model output, respectively; and w(t) indicates the disturbance. The model output x(t) can be obtained according to the system model and past model outputs and inputs: The following two parts compose the future input in EPSAC: where the u base (t + k|t) indicates the basic future control actions and δu(t + k|t) is the optimized future control actions. With these two parts of future input, the predicted future system output can be divided with the following two parts: where y base (t + k|t) is produced with the basic future control action u base (t + k|t) and y opt (t + k|t) is obtained with optimized future control action δu(t + k|t). According to the discrete input scenario, the y opt (t + k|t) can be obtained with: where the h i and g i are the impulse response and step response coefficients of the system; N c is the control horizon and N p the prediction horizon. The system output can be re-written as: where N 1 is the time delay of the system. The disturbance term w(t) in Equation (5) is taken as a filtered white noise [26], which is given by: where q −1 is the backward shift operator; The cost function gives: where p k and q k are nonnegative weighting factors, which are kept as constants.
If there are constraints in the optimization problem, it can be solved with quadratic programming. Without constraints, the optimal input sequence for δu(t + k|t) gives: For the fractional order MPC, the cost function can be expressed with where [N 1 , N p ] and [1, N c ] are the integration intervals, and γ I N p N 1 and λ I N c 1 are the symbols for fractional order integral with γ and λ the fractional orders. The γ I N p N 1 and λ I N c 1 can be rewritten as: According to [27], the Equation (15) can be approximated by: The m i with fractional order α gives: where n is the number of the m i .
From Equation (18) to Equation (21), the weight matrix can be easily tuned with fractional order γ and λ in the cost function J FOMPC . And the weighting factors in the FOMPC are time-varying along the prediction horizon and control horizon. If there are constraints, the optimization problem can be solved with quadratic programming. Without constraints, the optimal input sequence for δu(t + k|t) in FOMPC can be obtained as:

Applied the EPSAC to the MIMO System with Distributed Scheme
Due to the flexibility and robustness of the distributed control, the EPSAC is applied to the steam/water loop in distributed scheme, in which each sub-loop works as independent modules with communication network. The application of distributed fractional order MPC on the steam/water loop is similar to the method introduced in [3], and the integer cost function is replaced with fractional order cost function. Only a brief introduction is given here about the distributed MPC.
In the distributed MPC, the inference from other sub-loops should be considered to optimize the future error and control effort for sub-loop i, hence, the item GU in Equations (13) and (17) should be replaced with 5 ∑ j=1 G ij U j , and G ij is the transfer function of i th output from j th input. A pseudo-code for the distributed MPC is provided in Algorithm 1.

Algorithm 1
The iterative DiMPC 1: Sub-loop i receives an optimal local control action δU i at the iterative time as iter = 0 according to the EPSAC, and the local control action δU i can be marked as δU iter i , where δU i indicates the vector of the optimizing future control actions with length of N ci ; 2: The δU iter j (j ∈ N i , N i = {j ∈ N : G ij = 0}) will be communicated with the loop i, and the δU iter+1 i will be calculated again with the δU iter j from other loops; 3: If one of the termination conditions is reached, the δU iter+1 i will be adopted to the system. Otherwise, the iter will be set as iter = iter + 1, and return to Step 2. The termination condition is shown as follows: where ε is a positive constant and iter indicates the upper bound of the number of iteration times. 4: Calculate the optimal control effort as U t = U base + δU iter , and the control effort will be applied to the system; 5: Set t = t + 1, return to Step 1.

Results and Discussion
In this paper, reference tracking and load fluctuation experiments are conducted to verify the effectiveness of the FOMPC. The parameter configuration is listed in Table 2.

The Effect of Different Fractional Orders on the Overall System Performance
The choice of fractional orders of γ and λ have important influence on the system. To explore the effect of different fractional orders, the fractional orders listed in Table 3 are applied for each sub-loop. Table 3. Different fractional orders for the five sub-loops.

Loops Fractional Orders
Drum water level control loop [6. Intuitively from the results shown in Figure 2, all the outputs of the five sub-loops indicate that the larger fractional order leads to better responses both in response time and overshoot in a certain range. Also, the overshoot reduces with large fractional order value.

Reference Tracking Performance
According to the results from previous section, The fractional orders for the five sub-loops are chosen as 6.2, 0.62, 2.2, 1.2 and 1.2, respectively. The following indexes are applied to evaluate the performance of the FOMPC and MPC, including the Integrated Absolute Relative Error (I ARE), Integral Secondary control output (ISU), Ratio of Integrated Absolute Relative Error (RI ARE), Ratio of Integral Secondary control output (RISU) and combined index (J).
where u ssi is the steady state value of i th input; C 1 ,C 2 are the two compared controllers; the weighting factors w 1 and w 2 in equation (27) are chosen as w 1 = w 2 = 0.5.
In the Reference tracking experiment, step signals are introduced to the system at different time. The setpoints for the five loops are shown in Table 4. The simulation results about reference tracking performance are shown in Figure 3. And the performance indexes are shown in Tables 5 and 6.   Table 6. Indexes for RI ARE and RISU in the reference tracking experiment (MPC is the C 2 and FOMPC the C 1 according to Equations (25) and (26)). The combined index J for the reference tracking case is 1.1711. Both from the Figure 3 and performance indexes, it can be concluded that the FOMPC outperforms the MPC in reference tracking experiment. The response time and overshoot are both smaller in FOMPC than those in the traditional MPC. From Table 5, the ISUs for sub-loop 5 are similar, however, the I AREs are much different from each other. This is caused by the interaction between water levels of deaerator and condenser water level. The condenser has a smaller volume than the deaerator, hence, the water level in condenser changes a lot along with the deaerator water level.

Load Disturbance Rejection Performance
In the simulation of the load disturbance rejection experiment, 20% load increase from time period 100 s to 150 s and 20% load decrease from time period 850s to 900s are introduced in the steam/water loop. And the simulation results for load disturbance rejection experiment are shown in Figure 4 and Tables 7 and 8.     Table 8. Indexes for RI ARE and RISU in the load fluctuation experiment (MPC is the C 2 and FOMPC the C 1 according to Equations (25) and (26)). The combined index J for the load fluctuation case is 2.1549. Compared with MPC, the FOMPC method reduces the tracking error by 168%, and reduces the amount of control effort changes by 62.7%. The FOMPC has a significant improvement than the MPC method. With less control effort, the FOMPC can achieve better load disturbance rejection performance than the MPC.

Conclusions
By replacing the integer cost function with a fractional order one, the fractional order MPC based on EPSAC framework is applied to the steam/water loop in large scale ships. The comparison between FOMPC and MPC shows the FOMPC has better performance both in the reference tracking and load disturbance rejection. Different fractional orders are applied to the five sub-loops in steam/water loop, and it is concluded that within a certain range of the fractional order, the larger the order leads to better performance.