Comparison of Different Controllers for Equitable Water Supply in Water Networks

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1 Comparison of Different Controllers for Equitable Water Supply in Water Networks G R Anjana 1, M S Mohan Kumar 2 and Bharadwaj Amrutur 3 1 Department of Civil Engineering 2 Department of Civil Engg., IFCWS, ICWaR, Robert Bosch Centre for Cyber Physical Systems 3 Department of Electrical Communication Engineering and Robert Bosch Centre for Cyber Physical Systems, Indian Institute of Science, Bangalore, India [ 1 anjanagr@civil.iisc.ernet.in] ABSTRACT Equitable distribution of water among consumers is one of the most important agenda for many water authorities around the world. Many methods have been listed in literature for equitable distribution of water. In recent years, different automatic controller based approaches were also introduced for flow control in water systems. In this work, a comparative study is carried out among Dynamic Inversion based PID (DI-PID), Model Predictive Control (MPC) and EnKF based PID controllers for flow control application in real time for a water distribution network- Bangalore inflow water distribution system. It was observed that the DI-PID and EnKF based PID controllers are better than other controllers for equitable distribution of water in a mega city like Bangalore. Keywords: Water distribution systems; Equitable water supply; Controllers. 1 INTRODUCTION Equitable supply of water is becoming one of the major performance indices for water distribution systems around the world [1,2]. Factors like topography, water scarcity etc. lead to inequitable water supply among different DMAs of an area [3]. Application of controllers for equitable supply of water was studied earlier using DI-PID, for steady state simulation [4,5]. In this work, a comparison of different controllers for equitable water supply of water is being studied, for 24 hours, for Bangalore Inflow system. The valve settings required to maintain the equitable water supply is the control variable for this study. The controllers under study are Model Predictor Controller (MPC), Dynamic Inversion Proportional Integrator Derivative Controller (DI-PID) and Ensemble Kalman Filter PID (EnKF-PID). 2 MATERIALS AND METHODS: As explained earlier, three different types of controllers are studied for equitable water supply to different GLRs in Bangalore inflow system, for 24 hours. Amount of water to be supplied to the GLRs depend on number of consumers in different zones of Bangalore city. The study area, and the different controllers and their working will be explained briefly in the following sub-sections. 2.1 Study Area Bangalore, one of the major cities in India. Bangalore city receives water from river Cauvery. Water from Cauvery is taken to TK Halli Water Treatment Plant (WTP), and from there it is supplied to the city in four stages. [5]- Stage I, Stage II, Stage III, and Stage IV-Phase I. The inflow network consists of 55 Ground Level Reservoirs (GLRs) and 2 Direct Supply Points (DSP). Details of these

2 reservoirs (GLR) are shown in Fig. 1.Figure 1 shows the location of the different GLRs in the city. Bangalore city is divided into 6 zones (Fig.2), for easy operation and better supply to all consumers and each zone consists of a dedicated set of GLRs. Figure 2 shows the number of GLRs supplying water to each zone. Figure 1: Schematic of Bangalore Inflow Network 2.2 Problem Formulation Based on latest data of demand and supply received from Bangalore water supply and sewerage board, the amount water supplied to each zones of the city is calculated and is given in Table 1. The equitable requirement of water for each zone is calculated based on zone wise population, number of domestic connections, number of high rise connections, number of non-domestic connections etc. Preliminary analysis of field data proves that presently, water is being supplied inequitably to the different zones of the system. In order to calculate the equitable amount, the system parameters are being used as in Manohar et al., 2014 [5]. According to this study, the required amount of water to the zones and the current water supply are given in Table 1. The weighing factor adopted for each zones are also given in Table 1. Weighting factor is used to distribute the total Unaccounted For Water (UFW) among the different zones. The weighting factors were calculated based on the zone

3 topography and distance from the source. In this study, we are trying to achieve equitable supply for 24 hours in the WDS. In order for that, a zone wise model was developed and a nominal demand pattern was adopted as shown in Figure 3. A zone usually has a dedicated set of GLRs (Fig.2). Once, the equitable water supply to each zone is calculated, the supply to the corresponding GLRs of that zone is calculated using a top to bottom approach. For achieving the same, around 118 valves need to be dynamically controlled for 24 hours. For achieving the target flows to each GLRs, we use controller methodology, as explained in the next section. Figure 2: Different Supply Zones for Bangalore City 2.3 Application of Controllers in WDS As explained earlier, in this study 3 different controllers are being used and its applicability and efficiency for equitable water supply is extensively studied. The controllers under consideration are: Model Predictive Controller (MPC), Dynamic Inversion based PID (DI-PID), and Ensemble Kalman Filter based PID (EnKF-PID) Model Predictive Controller (MPC) Model predictive control (MPC) is an advanced control methodology used in many process industries. As the name suggest, this controller tries to achieve the target state, by considering the future states of the system as well. The framework of the controller is that of an optimization problem, subjected to system constraints, and the idea is the minimize the objective/cost function. The main advantage of MPC is that it optimises the current timestep, while keeping future timeslots in account. This is achieved by optimizing a finite time-horizon, known as the Receding Horizon (T), but only implementing the control for the current timeslot. Earlier,

4 Figure 3: Zone wise Demand Pattern Table 1: Current and equitable supply required for each zone, and corresponding weighting factor Zones Current Supply (MLD) Equitable Supply (MLD Weighting factor North East West South South East Central Shankar et al. [6], have used MPC for optimal control in WDS with storage facilities. In this work, MPC is implemented to on an optimization problem as given below: Np i = = = Subject to Where, Q is the simulated flow and is the target flow, which is the required flow through the valve. T is the receding horizon. U is the fractional valve opening for valve i at time t. The main objective here is to reduce the error between the simulated and the target flows to the GLRs, given the valve setting. Here, the system flows, Q i, i: 1.Np, where, Np: number of targets, is the system states and the valve setting, U j, j : 1. Nv, where, Nv: Number of valves to be controlled. In this work, the objective function is minimized using Genetic Algorithm.

5 2.3.2 Dynamic Inversion based PID (DI-PID) Dynamic inversion uses the feedback signal from the system to cancel inherent system dynamics and achieve a desired system response, or the target [7]. The main idea of DI is to algebraically transform a nonlinear system into a fully or partly linear-equivalent system. A PID controller on the other hand attempts to achieve the target by correcting the error between a measured variable and a desired value and then outputting a corrective control action that can adjust the process accordingly [4]. DI-PID controllers were used earlier in literature to achieve equitable flow for Bangalore city [5]. According to DI-PID controller, the control variable, valve settings are achieved as follows: General form of a system dynamic equation can be expressed as: X = X + X u 3 where X : derivative of the state vector, u : control vector, Y : output vector, f : nonlinear state dynamic function, : nonlinear control distribution function. For the desired states, final form of DI control law is written as: u = 1 X [X s X ] Equation 2 forms the DI part of the control objective. The general PID control equation can be written as in equation (5), where valve setting is the control variable, u; and error is represented as e: [(Δu) is the incremental change in valve setting] Δ = p + + Where, K p, K i and K d are the proportional, integrational and differential gain. And according to Zeigler Nicholas tuning method, these gains are calculated according to equation (6): p = ; = an = K u : Ultimate gain of the system and P u is the oscillation amplitude. Final valve loss equation is expressed as (by combing the DI and PID components): = A l A A l. l.. + p + + where : present flow in pipe, : target flow in pipe, : head at the starting node of pipe, : head at the end node of pipe, A : area of selected pipe, l : length of selected pipe. For each

6 valve which are throttled one equation is formed as in equation (7) and they are solved for valve throttling. The complete derivation of equation is available in Manohar et al., 2014 [ 5] Ensemble Kalman Filter based PID (EnKF-PID) EnKF-PID controller is a combination of state estimation and control strategy. The state estimation module (EnKF) uses the measurements from the system to estimate the actual state of the system, i.e, flow in every link and pressure in every node at that time period. The actual state of the system at the time period helps in the control process, since it gives a better estimate of the starting system state, hence the control application will be more efficient. It also helps in overcoming the calibration errors associated with the system model. Once the system state is estimated, the PID controller is used to derive the control variable required to achieve the equitable flow, as in equation (5). The EnKF, state estimation module is explained using the following equations [7]: + =, Where x is the system state, and f represents the forward simulation model. w t is the system input. The measurements for the next time step will be predicted as y t+1. + = + The updated state at the time step t+1 is calculated as below, where represents the actual measurements from the field. And in equation (10), K denotes the Kalman gain which is calculated as shown in equation (11) + = Where, = H H H + (11), C is the ensemble covariance and H is the Hessian of the measurement transformation equation shown in equation (9). RESULTS AND DISCUSSIONS In this work, MPC, DI-PID and EnKF-PID is being used for achieving the required target flow. Each controller type is compared for its settling time, error reduction and overall system performance. From the results, it was found that for MPC controllers, it is difficult to bring the error to zero. It is because MPC controllers tries to minimize the error between the target and simulated values for the future time periods, i.e for the Receding Horizon. Hence, the control action taken is usually a conservative value of valve settings. For DI-PID controller, it gives the valve settings required for that time period, irrespective of the requirements of the next time period. Hence, in most of the cases, the errors did tend to zero valve within 10 minutes. For EnKF-PID, it is assumed that the water networks model used have a few errors, and hence, the system measurements are used to estimate the system state for that time period. This updated system state at time t, acts as the boundary condition for the t+1 th time step, hence, the valve settings calculated from this simulation will be more appropriate for field application. For each controller, for a particular time period, the errors are being compared. 4 different valves are used to draw a generalized controller efficiency

7 (Valve ID: 89,91, 260 and 269). The settling time required for each controller is compared as in Fig. 4. In these 4 valves, the MPC value did settle to zero, because the flow requirement for the time step t and t+1 was the same (T=14 &15 hrs). In all these analysis, the system is assumed to be in a steady state for that time period (t=1). And the boundary conditions/initial conditions are updated for every hour. From Fig.4, it is clear that MPC controller fluctuations are more compared to the other two controllers. It is mainly because of the random nature of GA used for the MPC optimization problem. The fluctuations are still within permissible limits because the initial population guess made by GA was near to the valve controls derived for the water network for equitable supply, without downstream demand at the GLRs. These values were taken from Manohar et al., 2014 [5]. DI-PID controller and EnKF-PID controllers have similar behaviour, but since the EnKF-PID controller has the real time feedback from the field it is more suitable in situations where real time measurements are available. EnKF is able to estimate the actual system state even in the presence of measurement noise. Also, the real time field measurements used in this study contained noise, and EnKF is able to filter this noise and still give an better estimate of the system state. In these situations, it can be inferred that. the EnKF-PID performance is better than MPC and DI-PID controller. All the three controllers give almost similar values of valve setting, for this time period. 3 CONCLUSIONS In this study we have achieved equitable supply to different zones in Bangalore city for 24 hours using control strategy. Even though the controllers (MPC,DI-PID and EnKF-PID) gave similar setting for valves for most of the time slots, there performance varied on the basis of time of computation and settling time. It was observed that MPC took much more time compared to the other two control strategies, because of the optimization and GA application. The fluctuation in error for MPC for a particular time was also more that DI-PID and EnKF-PID. All the 3 controllers had settling times less than 10 minutes. Since, EnKF-PID brings the real time measurements back into the loop, it might be performing better than the other two for real world applications. Further studies are still required to be carried out in order to compare the controllers more extensively.

8 Figure 4: Error profiles for valves 89,91,260 and 269 References [1] S.Rode, Equitable distribution of drinking water supply in municipal corporations in Thane District, Manage. Res. Pract., vol. 1(1), pp.14 25, [2] N. Asingwire, D. Muhangi, and J. Odolon, Study of factors influencing equitable distribution of water supply and sanitation services in Uganda, 31st WEDC Int. Conf. on Maximizing the Benefits from Water and Environmental Sanitation, WEDC, U.K, [3] S.R. Basu, and H.A.C. Main, Calcutta s water supply: Demand, governance and environmental change. Appl. Geogr., vol. 21(1), pp.23 44, [4] M. Prasanna Kumar, and M.S. Mohan Kumar, Comparative study of three types of controllers for water distribution networks. J. Am. Water Works Assoc., vol. 101(1), pp.74 86,2009. [5] U. Manohar, and M. S. Mohan Kumar, "Modeling equitable distribution of water: Dynamic inversion-based controller approach." J. of Water Resour. Plann. and Manage., vol ,pp ,2013. [6] G. S. Sankar,, M.S.Kumar, S.S.Narasimhan, S.Narasimhan, and M.S. Bhallamudi, Optimal control of water distribution networks with storage facilities. Journal of Process Control, 32, , [7] Georgie, J., and Valasek, J. (2003). Evaluation of longitudinal desired dynamics for dynamic-inversion controlled generic reentry vehicles. J. Guid. Contr. Dynam., 26(5), [8] P.L Houtekamer, and H.L. Mitchell, Ensemble kalman filtering. Quarterly Journal of the Royal Meteorological Society, 131(613), , 2005