Self-Learning Salp Swarm Optimization Based PID Design of Doha RO Plant

: In this investigation, self-learning salp swarm optimization (SLSSO) based proportional-integral-derivative (PID) controllers are proposed for a Doha reverse osmosis desalination plant. Since the Doha reverse osmosis plant (DROP) is interacting with a two-input-two-output (TITO) system, a decoupler is designed to nullify the interaction dynamics. Once the decoupler is designed properly, two PID controllers are tuned for two non-interacting loops by minimizing the integral-square-error (ISE). The ISEs for two loops are obtained in terms of alpha and beta parameters to simplify the simulation. Thus designed ISEs are minimized using SLSSO algorithm. In order to show the effectiveness of the proposed algorithm, the controller tuning is also accomplished using some state-of-the-art algorithms. Further, statistical analysis is presented to prove the effectiveness of SLSSO. In addition, the time domain speciﬁcations are presented for different test cases. The step responses are also shown for ﬁxed and variable reference inputs for two loops. The quantitative and qualitative results presented show the effectiveness of SLSSO for the DROP system.


Introduction
The existence of civilization depends upon the availability of potable water. The studies are repeatedly indicating that the potable water is acutely depleted in the last few decades. The main reason for depletion of fresh water is overuse of ground water due to growing population and industrialization [1]. In order to meet the requirements due to increased population and industrialization, it is required to explore for sources of fresh water. The desalination of sea water is one of the possible solutions since the expense of sea water is abundant [2].
The process of removing salts from saline water is called desalination [3]. Desalination is achieved by (a) thermal distillation processes such as multiple effect evaporation, multi-stage flash (MSF), vapour compression, etc., and (b) membrane processes such as nano-filtration, electro-dialysis reversal, forward osmosis, reverse osmosis (RO), etc. Among these, RO has been proved to be economical [4]. RO process also produces high quality water with minimal energy consumption. Additionally, there is low initial investment in case of RO process.
RO is a filtration process which segregates fresh water from higher concentrated saline water. In RO process, a semi-permeable membrane is used which passes the fresh water through it and rejects the concentrated solution. The various plants which use the RO process are Doha RO system [5],

Reverse Osmosis System
The schematic block diagram of Doha reverse osmosis plant (DROP) system is provided in Figure 1 [4]. The RO desalination plant comprises of four stages, namely, pre-treatment, high pressure pump, RO membrane assembly and post-treatment.
describes the SLSSO algorithm. The results and comparisons for two loops are provided in section 6. Finally, the conclusions and key findings are discussed in section 7.

Reverse Osmosis System
The block diagram of Doha reverse osmosis plant (DROP) system is provided in Fig. 1 [4]. The RO desalination plant comprises of four stages, namely, pre-treatment, high pressure pump, RO membrane assembly and post-treatment. During pre-treatment stage, the saline water undergoes filtration to remove any physical or chemical impurities. This stage is important to ensure membrane longevity. The organic chemicals which can burn holes in the membrane and can cause irreparable damage to membrane During pre-treatment stage, the saline water undergoes filtration to remove any physical or chemical impurities. This stage is important to ensure membrane longevity. The organic chemicals which can burn holes in the membrane and can cause irreparable damage to membrane are trapped using activated cartridge filters. In addition, some inhibitors are added to decrease the pH of the feed-water which helps in reducing the scaling effect. Besides this, physical impurities like mud particles and microbial growth on membrane are also removed during this stage to increase the life of membrane.
High pressure pump is utilized to provide the necessary pressure required for reverse osmosis. For brackish water, it is in the range of 15-25 bars while it is 54-80 bars for sea water. The pressurized feed-water is then fed to the membrane assembly which is made up of two or more hollow or spiral wound composite polyamide membranes. This is the main part of the RO system. It does not allow impurities to pass through it. The membrane assembly must be able to withstand the high pressure. The rejected brine from membrane assembly is sent to the discharge channel. In post-treatment stage, calcium and bi-carbonate ions are removed from the water. In addition, the pH is raised slightly above 7 and water is prepared for distribution. The two controlled variables (flux and conductivity of output water) are manipulated using two manipulated variables (feed-water pressure and feed-water pH) in case of DROP model.

Control-Loops for DROP Model
The block diagram of DROP is given in Figure 2. The transfer function of interacting DROP model is given as where R 1 (s) and R 2 (s) are two manipulated variables which are feed-water pressure and feed-water pH, respectively, and Y 1 (s) and Y 2 (s) are two controlled variables which are flow and conductivity of output water, respectively. The transfer functions D 11 (s), D 12 (s), D 21 (s) and D 22 (s) are denoting the individual transfer functions in between two manipulated variables and two controlled variables. The transfer functions D 11 (s), D 12 (s), D 21 (s) and D 22 (s) are given as: D 11 (s) = k 11 (n 11 s + 1) s 2 + 2ζ 11 d 11 s + d 11 (2) It can easily be concluded from transfer functions (2)-(5) that both flow and conductivity are manipulated by feed-water pressure. However, the feed-water pH is affecting only conductivity. The values of different parameters of transfer functions (2)-(5), are given in Table 1 [4]. The interval transfer functions of (2)-(5) turn out to be (2)-(5) by considering 40% uncertainty.
It can easily be concluded from transfer functions (2)-(5) that the both flow and conductivity are manipulated by feed-water pressure but the conductivity is affected by feed-water pH only. The values of different parameters of transfer functions, (2)-(5), are given in Table 1 [4].

Design of decoupler
A decoupler is required to be designed to nullify the effect of interaction for interacting TITO system. Fig. 3 shows the decoupler along with the plant. This work employs simplified decoupling technique [17] to convert interacting TITO system into two non-interacting control

Design of Decoupler
A decoupler is required to be designed to nullify the effect of interaction for interacting TITO system. Figure 3 shows the decoupler along with the plant. This work employs simplified decoupling technique [17] to convert interacting TITO system into two non-interacting control loops. The transfer function of decoupler is obtained by Algorithms 2020, 13, 287 5 of 14 such that where C p and C d are, respectively, the transfer functions of plant and decoupler. From (10), it is obtained Hence, it is clear from (13)-(16) that by choosing x and y as given in (13) and (14), respectively, the interacting system is decoupled into two non-interacting loops with transfer functions, T 11 and T 22 as given in (15) and (16), respectively. Thus, two independent PID controllers, C PID1 and C PID2 , can be obtained for two non-interacting loops having transfer functions, namely T 11 and T 22 .

Decoupler Plant
In simplified decoupling, it is aimed such that where p C and d C are, respectively, the transfer functions of plant and decoupler. From (6), it is

Controller Structure
In literature, many structures of PID controller are existing. However, the simple and widely used parallel form given in (17) is used in this work.
The parameters, K p , K i and K d are, respectively, the proportional, integral and derivative gains. However, u(t) and e(t) denote the control effort generated by controller and error signal fed to controller, respectively. In Laplace domain, (17) becomes as given in (18).

The Determination of ISEs for DOHA Plant
The ISE is utilized for obtaining the controller parameters. The ISE is obtained in terms of alpha and beta parameters. The ISE for two independent loops, i.e., pressure-flux and pH-conductivity loops, is given below.
(a) ISE for pressure-flux loop: (b) ISE for pH-conductivity loop: The alpha and beta parameters are obtained utilizing alpha and beta tables. The alpha and beta tables are derived from respective error in Laplace domain. Suppose the error in Laplace domain is given as The alpha and beta tables are constructed as given in Tables 2 and 3, respectively [4]. The first two rows of Tables 2 and 3 are obtained from (21). The rest of the elements are obtained using elements of these two rows.

Salp Swarm Optimization
Salp swarm optimization (SSO) [25] is a nature inspired swarm-based meta-heuristic optimization algorithm. It is proposed recently by Mirjalili et al. [26] in 2017. This algorithm is derived from the swarming behavior of salps. Salps are found in deep ocean and are very similar to jelly fishes. The salps' population can be categorized into (i) leaders and (ii) followers. The leaders lead the whole population while followers follow the instructions of leaders, directly or indirectly. The salps form chain to move. The salps at front of chain are known as leaders and remaining salps are known as followers. The better half of the population is treated as leader salps. However, remaining salps are considered as followers.
At first, the candidate solutions for leaders are updated. Since, followers follow the leaders, the candidate solutions for followers are updated using the solutions obtained for leaders. Suppose, X i,j, i = 1, 2, · · · , M and j = 1, 2, · · · , N denote initial candidate solutions for whole population where M and N represent population size and number of decision variables, respectively. The candidate solutions for leaders and followers are updated as follows.

Update of Candidate Solutions for Leaders
The candidate solutions for leaders are updated as where X new i,j is updated candidate solutions for X i,j , T j is the position of food source, X − j and X + j represent the minimum and maximum values of decision variables, k 2 and k 3 are random numbers distributed uniformly in the range [0,1], and k 1 is modified during course of iterations as where T and t, respectively, are maximum number of iterations and current iteration.

Update of Candidate Solutions for Followers
The candidate solutions for followers are updated using the solutions of leaders. The mathematical expression used for updating the candidate solutions for followers is where X new i,j denotes updated candidate solution for follower X i,j . After updating the whole population as suggested in (22)   After updating the whole population as suggested in (17) and (19), candidate solutions violating the minimum and maximum values of decision variables are reinitialized at respective minimum and maximum values of decision variables. The pseudo code of SSO is presented in Table 1.

Self-Learning Slap Swarm Optimization
In this phase, a self-learning rule exploit the space in close proximity of the individual position to reach its global optima [37]. The rule provides an opportunity to each learner to enhance the individual knowledge from self-surrounding space. This phase can be mathematically expressed as: Where,  determines the individual self-learning capability and termed as self-learning factor,  

Results and Comparisons
Update candidate solutions using (22) Update candidate solutions using (24) Modify 1 k using (23)

Self-Learning Salp Swarm Optimization
A self-learning rule exploits the space in close proximity of the individual position to reach its global optimum [23]. The rule provides an opportunity to each learner to enhance the individual knowledge from self-surrounding space. This phase can be mathematically expressed as: where, X new i,j (k) is the new solution vector in this phase, λ is the self-learning factor which determines the self-learning capability of each individual and r ∈ [0, 1]. The value of λ is considered 3 in this work. Update the solution vector, X i,j (k), using greedy selection. Thus, the updated X i,j (k) after this phase takes part in the next iteration.

Results and Comparisons
Two case studies are performed to highlight the findings and contributions of this article. One proportional-integral-derivative (PID) controller is designed for pressure-flux loop and other PID controller is tuned for pH-conductivity loop. In both the case studies, ISE of respective loop is minimized using SLSSO algorithm [23]. For fair presentation of efficacy of SLSSO algorithm, other algorithms namely, SSO, ABC, LJ, NMS, SCA and PSO are applied to tune the PID controller.

Case Study I
The results presented in this case study are obtained for pressure-flux loop by minimizing the performance index given in (19). Table 4 presents PID gains i.e., K p , K i , and K d obtained using SLSSO. Same table also tabulates the PID gains obtained using ABC, LJ, NMS, PSO, SCA and SSO. The mean and minimum values of ISE along with standard deviation are also presented in this table. The time domain specifications of step response obtained due to all algorithms (ABC, LJ, NMS, PSO, SCA, SSO and SLSSO) are tabulated in Table 5. The Figure 5 shows the step response obtained due to ABC, LJ, NMS, PSO, SCA, SSO and SLSSO. In addition to this, the response of pressure-flux loop for variable input due to SLSSO algorithm is plotted in Figure 6. The Figure 7 is showing the response with disturbance.
The mean value of ISE is minimum in case of SLSSO algorithm as shown in Table 4. The same is true for minimum value of ISE and standard deviation. The time domain specifications tabulated in Table 5 also supports the response obtained for pressure-flux loop. The step response of Algorithms 2020, 13, 287 9 of 14 SLSSO algorithm is also reaching steady state as shown in Figure 5. For variable input and input having disturbance also, the SLSSO tuned controller is providing stable step response as shown in Figures 6 and 7. Hence, it can be inferred that SLSSO algorithm is a better choice for tuning the PID gains for pressure-flux loop of DROP system.

Case Study II
This case study presents the analysis of results obtained for pH-conductivity loop. In this case, the performance index given in (20) is minimized using SLSSO algorithm. The PID gains (K p , K i , and K d ) obtained using SLSSO are presented in Table 6. The Table 6 also provides gains obtained due to ABC, LJ, NMS, PSO, SCA and SSO. The mean and minimum values of ISE along with standard deviation are also shown in Table 6. The time domain specifications (rise time and settling time) of step response are given in Table 7 which are obtained due to ABC, LJ, NMS, PSO, SCA, SSO and SLSSO. The Figure 8 plots the step response of pH-conductivity loop due to ABC, LJ, NMS, PSO, SCA, SSO and SLSSO. Moreover, the response of pH-conductivity loop for variable step input and step input having disturbance due to SLSSO algorithm is presented in Figures 9 and 10, respectively.
In this case also, the mean value of ISE is the minimum for SLSSO algorithm as shown in Table 6. The same conclusion can be derived for minimum value of ISE and standard deviation. Table 6, Figures 8-10 also prove that the SLSSO algorithm is an excellent alternative for tuning of PID controller for DROP system.

Amplitude
Step response

Conclusions
This contribution presented design of proportional-integral-derivative (PID) controller for Doha reverse osmosis plant (DROP). The DROP system is interacting in nature and has two inputs and two outputs resulting in TITO system. Since, this TITO system is interacting, the interaction effect should be eliminated before designing the PID controllers for two manipulated variables. In this article, simplified decoupling technique is adopted to design decoupler for interacting TITO DROP system. After designing the decopuler, two PID controllers for pressure flux and pH conductivity loops are designed to manipulate the pressure and pH in order to generate specified flux and conductivity. For designing the PID controllers, integral-square-error (ISE) of the respective loop is minimized using lelf-learning salp swarm optimization (SLSSO) algorithm [23]. It was proved that SLSSO algorithm is an excellent choice for tuning the PID gains for two loops of the DORP system.
As far as the further extension of this work is considered, it is aimed to obtain the interval model in discrete domain. The reduced-order model for discrete domain will be designed in further research. It would also be interesting to investigate the controller design using intelligent optimization algorithms like fireworks algorithm [12], bacterial foraging optimization [13], elephant herding optimization [10], whale optimization algorithm [2,14], monarch butterfly optimization [15], brain storm optimization [16], etc.
Funding: This research was funded by SERB, grant number ECR/2017/000212.

Conflicts of Interest:
The authors declare no conflict of interest.