A largescale parallelunit seawater reverse osmosis desalination plant contains many reverse osmosis (RO) units. If the operating conditions change, these RO units will not work at the optimal design points which are computed before the plant is built. The operational optimization problem (OOP) of the plant is to find out a scheduling of operation to minimize the total running cost when the change happens. In this paper, the OOP is modelled as a mixedinteger nonlinear programming problem. A twostage differential evolution algorithm is proposed to solve this OOP. Experimental results show that the proposed method is satisfactory in solution quality.
The shortage of fresh water has become a bottleneck of the economic development in many countries. Seawater desalination is an effective way to solve this problem. Reverse osmosis (RO) desalinating is one of the most popular ways to generate freshwater from seawater and has made a rapid progress over the past four decades [
Now, a kind of largescale parallelunit SWRO desalination plant, which is composed of multiple parallel RO units, has appeared. This kind of plant has huger capital cost and more complicated operation processes. So, before it is built, an optimal design is made to select the suitable devices and system performance to match the operating condition [
In this paper, a mathematical model of operational optimization problem (OOP) for a largescale parallelunit SWRO desalination plant, which includes objective function and constraint functions, is made. In order to solve this OOP, a twostage differential evolution (TSDE) algorithm is proposed. The simulating results show that the TSDE is excellent in searching ability than basic DE and genetic algorithm (GA).
The single SWRO desalination unit is a multistage process, which includes seawater intake, pretreatment, RO desalination, and posttreatment [
Schematic diagram of a single SWRO desalination unit.
In the intake stage, the raw seawater is pumped from a deep well, which is located close to the shoreline, to the flocculators to filter; in the pretreatment stage, the most suspended matters and colloids are filtered out of seawater by flocculators, mechanical filters, and precision safeguard filters successively to ensure that SDI (silt density index) is lower than 5 to meet the RO modules requirement; in the RO desalination stage, one part of the fed seawater is pressurized by the high pressure (HP) pump and the other by energy recovery device (ERD) and booster pump up to 40 ~ 50 bar and then is desalinized by RO modules; in the posttreatment stage, the produced freshwater flows into the product freshwater tanks (PFWT) to supply into the municipal water network; the brine is disposed reasonably.
The schematic diagram of a largescale parallelunit SWRO desalination plant is shown in Figure
Schematic diagram of a parallelunit SWRO desalination plant.
The OOP of a largescale SWRO desalination plant is to make an optimal scheduling plan to minimize the plant’s total running cost (TRC). This optimal schedule plan determines the on/off status and amount of generated freshwater of each RO unit, and the amount of supplied freshwater by each PFWT at each time period. When computing the TRC, the price factors, such as timeofuse electricity price, the operation cost, and the maintenance cost, are taken into consideration.
A lumped parameter model of this problem is built in this paper. The
The
The constraints of OPP include technical limitations and the design requirements, which are as follows.
The OOP of the largescale parallelunit SWRO desalination plant is a mixedinteger nonlinear programming problem (MINLP) over the time horizon. In the OOP,
Differential evolution algorithm was originally proposed by Storn and Price for solving continuous optimization in the mid1990s [
The original population is generated in the search space randomly, which contains
For each individual vector
The crossover operator generates an offspring
The selection operator is to generate next population
If the stopping criterion (maximum number of iterations) is satisfied, computation is terminated; otherwise, Steps
In the OOP, there are two types of constraints: boundary constraints and technical limited constraint functions. The following is the treatment strategies of them.
Sometimes
Penalty function is one of the most effective methods to solve the evolutionary constraint optimization problem [
In this paper, the MINLP with constraints (
In this paper, the
The OOP of a largescale parallelunit SWRP desalination plant has such features: the permeate rate of each RO unit is huge; that is, the amount of freshwater generated by each RO unit is usually more than 10000 tons a day; the permeate rate changes within an allowable range, but this range is much smaller than its amount. So, when a RO unit’s status changes, the
In this paper, a novel DE, TSDE, is proposed to solve the OOP. The TSDE is divided into two periods: Stage One and Stage Two. In Stage One, the permeate rate of each RO unit is supposed be a constant, which equals the median of its allowable range. A DE algorithm is used to compute the run/off status of each RO unit in this stage. When the DE is satisfying the stopping criterion, a preliminary scheduling will be worked out.
Then, the best 30 percent individuals in Stage One remain to Stage Two. The other 70 percent individuals in Stage Two are generated in the search space randomly (here, these two stages have the same population size
For numerical experimentation, a largescale parallelunit SWRO desalination plant in Liuheng Island, China, which has the capacity of 100,000 m^{3} freshwater a day, is considered over a 24hour time horizon. The basic parameters of this OOP are shown in Tables
The basic parameters of OOP in Liuheng SWRO desalination plant.
Parameters  Sign  Value 

The number of RO units 

8 
The number of PFWTs 

4 
The number of RO units which feed freshwater to each PFWT 

2 

2  

2  

2  
The upper limit of the permeate rate of RO Unit 

460 

460  

570  

570  

570  

800  

570  

570  
The lower limit of the permeate rate of RO Unit 

380 

380  

470  

470  

470  

655  

470  

470  
The initial value of PFWT (m^{3}) 

340 

340  

340  

340  
The maximum capacity of PFWT 

1680 

1680  

2280  

1680  
The minimum capacity of PFWT 

320 

320  

320  

320  
The maintenance cost coefficient 

11.5 
The repair and replacement fees 

155 
The correlation coefficient of energy consumption and generated freshwater 

2.86 
The user’s demands for freshwater
Time 
Demands for flash water (m^{3})  Time 
Demands for flash water (m^{3}) 

1  1660  13  2525 
2  1520  14  2430 
3  1345  15  2300 
4  1370  16  2490 
5  1865  17  2765 
6  2235  18  3030 
7  2700  19  3095 
8  2830  20  2970 
9  2770  21  2655 
10  2555  22  2350 
11  2705  23  2085 
12  2640  24  1870 
The timeofuse electricity price
Time 
1–8  9–11  1213  14–19  2021  22  2324 

Electricity price 
0.27  0.69  0.27  0.69  0.89  0.69  0.27 
Here, the length of individual
The CDC of this plant is considered as a constant and ignored when calculating the TRC.
The DE’s search ability for different problems depends on its parameters [
Firstly, the effects of two important parameters, the maximum number of iterations
The different pairs
Analysis of NP and
NP 

Best solution ( 
Mean value ( 
Std. dev. 

200  250  89758  90376.7  351.6491 
250  200  89662  90471.4  556.8511 
334  150  89116  89666.7  410.5998 
500  100  88822  89926.1  423.0399 
625  80  88725  89508.5  451.2566 





It is worth noting that if
The scale factor
Usually the
Analysis of
Analysis of
In Figure
As mentioned in the above sections, the TSDE algorithm is divided into two stages. The two maximum iteration numbers of these two stages,
Analysis of


Best solution ( 
Mean value ( 
Std. dev. 






500  500  88464  88878.9  383.1357 
700  300  88556  88850  166.2274 
1000  0  88275  89052  489.2724 
The pseudocode of TSDE is shown in Algorithm
Line % Pseudocode of TSDE
(1)
(2)
(3) % Stage One
(4)
randomly in the range of
initialized as constants, which equal the average capacity of each RO unit)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32) % Stage Two
(33) Initialize population
(34)
(35)
(36)
(37)
(38) Randomly selected three integers
(
(39)
(40)
(41) Randomly selected index rnbr(
(42) CR = rand * 0.2 + 0.7
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
(57)
(58)
(59)
(60)
(61)
In this paper, genetic algorithm (GA) and basic DE are chosen to compare the search ability with STDE for this problem. The main parameters of GA are set as follows: the maximum number of iterations
Each of algorithm runs 10 times, and the statistical results are listed in Table
Comparison of the best solutions of STDE with basic DE and GA (
Algorithm  Best solution  Worst solution  Mean value  Std. dev. 

GA  89680  91102  90556  622.9885 
Basic DE  88429  89266  88955.4  376.5983 





Finally, we compare the optimal operation with manual operation in the OOP. The scheduling strategy of manual operation is that all RO units are running until reaching the high amount limits of the PFWTs from time 1 to time 8 every day, when the timeofuse electricity price is the lowest. When the freshwater amount of each PFWT is at high limit, the corresponding RO units stop. At other time periods, the running status of each RO unit is only determined by the user’s freshwater demand, and the timeofuse electricity price will no longer be considered. The optimal operation obeys the scheduling plan computed by TSDE.
Figure
The comparison of capacity of each RO unit.
Figure
The comparison of capacity of each RO unit.
From Figure
Both of these two schedules open the RO units to generate freshwater from time 1 to time 8, and the sum of freshwater reaches its peak at time 8;
the manual operation does not take the timeofuse electricity price as the cost factor of TRC after time 8, so even at the highest electricity price period, such as time 15, the sum of freshwater of PFWTs is still increasing. That is, the RO units are still opening at these time periods, so that the TRC under manual operation cannot get its optimal value;
the optimal operation takes full use of its advantage in global optimization, so at times 8, 13, and 18, the sum of generated freshwater reaches its local peaks before the timeofuse electricity price gets higher. Therefore, when the electricity price is higher, the storage of freshwater is used to satisfy user’s demand, and the RO units will not be opened unless the storage of freshwater reaches its lower limit. In this way, the TRC of this plant under optimal operation is 5% lower than manual operation (Figure
The comparison of TRC.
The OOP of largescale parallelunit SWRO desalination plant is modeled as a MINLP, in which binaryvalued vectors indicate the on/off statsues of RO units and realvalued vectors indicate the amounts of freshwater generated by RO units and the amounts of supplied freshwater by PFWFs. The objective function of the OOP is the total running cost of the desalination plant, and the constraint functions include the technical limitations, the design requirements of each RO unit, and the freshwater demand of user. A novel DE, twostage DE, is presented to solve this OOP, and the effects of its main parameters on the search ability are analyzed in this paper. Applying this TSDE to a 100,000 ton SWRO desalination plant in Liuheng Island, China, it is observed that the proposed TSDE can successfully improve the utilization rate of RO units to reduce the TRC.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by National Key Technology Research and Development Program of the Ministry of Science and Technology of China (2009BAB47B06) and the grants of National Natural Science Foundation of China (no. 61374142).