water Article Numerical Modeling and Hydraulic Optimization of a Surge Tank Using Particle Swarm Optimization Khem Prasad Bhattarai , Jianxu Zhou * , Sunit Palikhe , Kamal Prasad Pandey and Naresh Suwal College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China; kpb63@outlook.com (K.P.B.); sunit.palikhe@hhu.edu.cn (S.P.); lotuskamal44@hotmail.com (K.P.P.); narcissussuwal@gmail.com (N.S.) * Correspondence: jianxuzhou@163.com Received: 25 February 2019; Accepted: 4 April 2019; Published: 6 April 2019 Abstract: In a pressurized water conveyance system, such as a hydropower system, during hydraulic transients, maximum and minimum pressures at various controlling sections are of prime concern for designing a safe and efficient surge tank. Similarly, quick damping of surge waves is also very helpful for the sound functioning of the hydro-mechanical system. Several parameters like diameter of the surge tank, diameter of the orifice, operating discharge, working head, etc., influence the maximum/minimum surge, damping of surge waves in the surge tank, and the difference of maximum pressure head at the bottom tunnel and maximum water level in the surge tank. These transient behaviors are highly conflicting in nature, especially for different diameters of orifices (DO ) and diameters of surge tanks (DS ). Hence, a proper optimization method is necessary to investigate the best values of DO and DS to enhance the safety and efficiency of the surge tank. In this paper, these variables are accurately determined through numerical analysis of the system by the Method of Characteristics (MOC). Furthermore, the influence on the transient behavior with changing DO and DS is investigated and finally, optimum values of DO and DS are determined using Particle Swarm Optimization (PSO) to minimize the effects of hydraulic transients on the system without compromising the stability and efficiency of the surge tank. The obtained results show significant improvements over the contemporary methods of finding DO and DS for surge tank design. Keywords: numerical analysis; method of characteristics; hydraulic transients; surge analysis; surge tank; particle swarm optimization 1. Introduction Hydraulic transients occur whenever there is a disturbance in the steady state condition of a system, causing variations in pressure and discharge in the system. The control of hydraulic transients in pressurized systems like water supply, wastewater, and hydropower systems, is a major concern for engineers regarding the safety and effective operation of the system throughout the life span of the project. Uncontrolled hydraulic transients in the system have led to several cases of accidents in hydropower stations in the past [1–3]. A number of methods and models have been developed for the analysis of hydraulic transients, like the graphical method, the algebraic method, the implicit method, the wave characteristic method, the Method of Characteristics (MOC), the Zielke model, the Brunone model, Surge2012, and much more commercial software [4–9]. Calamak et al. [10] developed a computer program using MOC, which accurately predicted pressures for various simulations in case studies having a Pelton and Francis turbine. Martin et al. [11] performed 3-D and 1-D simulations for various unsteady friction formulations and discovered that the convolution-based unsteady friction models, which consider a set of previous time steps in simulation, perform better than the instantaneous Water 2019, 11, 715; doi:10.3390/w11040715 www.mdpi.com/journal/water Water 2019, 11, 715 2 of 19 acceleration-based models. Among these methods, MOC is a very popular method, as it gives accurate results with a shorter computational time for analyzing the hydraulic transients in a pipeline system, and can be conveniently used in computer modeling [12,13]. In this study, MOC is used, considering the steady-state friction along the pipe, for numerical modeling and hydraulic transient analysis of the given generalized hydropower system, where the transient is generated by gradual closing and opening of the valve. Researchers have extensively studied hydraulic behaviors of the pressurized system to discover methods for improvements and successfully tackle hydraulic transient complications in the system. Yu et al. [14] performed 1-D analysis of a complex differential surge tank for a long diversion-type hydropower station using MOC and concluded that the high pressure difference produced between two sides of breast wall during transient conditions can be effectively reduced by setting appropriate pressure reduction orifices. Guo et al. [15] reviewed transient processes and safety measures for hydropower stations with long headrace tunnels, and revealed that the maximum pressure in hydropower stations with long headrace tunnels depends on the maximum water level in the surge tank, rather than the closing mechanism of the guide vane. Zhou et al. [16] introduced the addition of interconnecting holes in a differential surge tank, which significantly reduced the pressure on the breast wall and made gate maneuver much easier and more effective. Covas and Ramos [17] used inverse transient analysis for the identification of the location of a leak in a water pipeline system through a transient generated by fast change in flow conditions. Balacco et al. [18] carried out laboratory experiments on an undulating pipeline consisting of an orifice for air venting, creating a complex air-water environment. They analyzed the pressure surge build during rapid filling of the pipeline and found that the maximum pressure is reached during the mass oscillation phase. A plethora of research could be found on the sensitivity analysis of parameters like diameter of the orifice (DO ), diameter of the surge tank (DS ), working head, operating discharge, valve closure mechanism, pipe material and diameter, etc., for various transient effects [19–23] Hydraulic transients in pressurized systems are complex phenomena which are interdependent of several parameters, and also highly site-specific. Thus, designing an effective and well-optimized hydraulic system is always a challenging work. Several attempts of optimizing the orifice in a surge tank have been made using 1-D and 3-D simulations [24–27]. Similarly, different optimizing tools have been successfully employed to obtain the most advantageous design of hydraulic systems. Afshar et al. [28] applied the NSGA-II optimization algorithm for developing an optimized closing-rule curve for valves to control the effects of a water hammer in pressurized pipelines. Kim et al. [29] implemented the genetic algorithm (GA) with the impulse response method, and then optimized the location and dimensions of a surge tank considering the safety and cost of the system. Fathi-Moghadam et al. [30] employed GA to optimize the diameters of a headrace tunnel, penstocks, and surge tank considering the benefit-cost ratio as the objective function. Their model successfully optimized the diameters in a system consisting of branching tunnels and penstocks with a surge tank. Jung et al. [31] integrated both GA and PSO simultaneously with hydraulic transient analysis to discover the best location and combination of transient protection devices in a pipe network. Similarly, Moghaddas et al. [32] compared central force optimization (CFO) and GA for optimizing the pipeline protection cost of a water transfer project considering hydraulic transients induced during pump maneuver. They optimized the volume of the air chamber along with the location and type of air-inlet valves with minimum cost as the objective function. Generally, researchers have performed the optimization of a surge tank considering the cost of the project as the objective function and including some hydraulic transient characteristics, while optimization dedicated to reducing the severe effects of the extreme transient conditions is very limited. This study solely emphasizes the hydraulic transient characteristics of the system and uses PSO for optimization of the diameter of the surge tank (DS ) and diameter of the orifice (DO ). Generally, researchers have used GA for optimization, while PSO has been proved to be more convenient and effective. GA is more suitable for discrete optimization and has been more rigorously studied and applied in diverse fields due to its much earlier introduction (in 1989) than other Water 2019, 11, 715 3 of 19 modern optimization techniques, while PSO is a much more recent technique (in 1995) and suitable for problems and has a higher converging speed towards the solution3[33–36]. of 19 Hassan et al. [37] compared PSO and GA and discovered that the computational effort and efficiency solution is for significantly less athan that of GA to reach the same high-quality solution. Similarly, required PSO to reach high-quality solution is significantly less than that of GA to reach the Kecskes et al. [38] and Duan etSimilarly, al. [39] also compared PSOand andDuan GA on their models and same high-quality solution. Kecskes et al. [38] et al. [39]respective also compared PSO and discovered that PSO was better and has a high convergence ratio compared to GA. On the other hand, GA on their respective models and discovered that PSO was better and has a high convergence ratio every engineering has their own unique characteristics which demand in compared to GA.problem On the other hand, every engineering problem has their owncertain uniquealterations characteristics general optimization procedures. In this study, we have attempted to optimize the surge tank based which demand certain alterations in general optimization procedures. In this study, we have attempted onto itsoptimize hydraulicthe transient characteristics with a highlytransient non-linear solution andwith constrained surge tank based on its hydraulic characteristics a highlyvariables. non-linear In solution these conditions, PSO is much easierIntothese formulate and apply formuch optimization it acquires the and constrained variables. conditions, PSO is easier to and formulate and apply global optimization value quickly and conveniently with fairly agreeable results. for optimization and it acquires the global optimization value quickly and conveniently with fairly In this study, agreeable results.a generalized system of a high-head hydropower station is selected for analysis during In extreme casesa generalized of hydraulicsystem transients. The system consists ofstation a constant-head this study, of a high-head hydropower is selected upstream for analysis reservoir, headrace tunnel, orifice surge tank, penstock pipe, and turbine at the end of the penstock. during extreme cases of hydraulic transients. The system consists of a constant-head upstream reservoir, Because thistunnel, study mainly focuses on the surge pipe, analysis hydraulic of the surge tank, headrace orifice surge tank, penstock andand turbine at the optimization end of the penstock. Because this thestudy downstream turbine can be simplified into a valve boundary, as shown in Figure 1. First, 1-D mainly focuses on the surge analysis and hydraulic optimization of the surge tank, the downstream numerical modeling and analysis the boundary, hydraulic transients, occurring to 1-D valve operationmodeling of the turbine can be simplified into a of valve as shown in Figure 1.due First, numerical system, is carried outhydraulic by using MOC. Gradual closing of to thevalve valveoperation during the maximum levelout and analysis of the transients, occurring due of the system, water is carried in by theusing reservoir and gradual opening of the valve during the minimum water level in the reservoir MOC. Gradual closing of the valve during the maximum water level in the reservoir and aregradual considered as the twovalve worstduring cases, the which give the maximum water level in as thethe opening of the minimum water level inand the minimum reservoir are considered surge tank, respectively. Secondly, major hydraulic transient behaviors in the system are studied and two worst cases, which give the maximum and minimum water level in the surge tank, respectively. analyzed. Secondly, major hydraulic transient behaviors in the system are studied and analyzed. continuous Water 2019, 11, 715optimization Figure 1. General layout of the hydropower system. Figure 1. General layout of the hydropower system. Finally, the necessary objective functions and constraints are developed regarding the transient Finally, the necessary objective functions and constraints are developed regarding the transient behavior of the system, and optimization is performed using the Particle Swarm Optimization (PSO) behavior of the system, and optimization is performed using the Particle Swarm Optimization (PSO) method to find the optimum diameters of the surge tank (DS ) and orifice (DO ). The results obtained method to find the optimum diameters of the surge tank (DS) and orifice (DO). The results obtained are compared with the contemporary methods of finding DO and DS , and discussions are presented. are compared with the contemporary methods of finding DO and DS, and discussions are presented. 2. Material and Methods 2. Material and Methods 2.1. Governing Equations 2.1. Governing Equations The 1-D governing equations for the physics of hydraulic transients are the equations of motion The 1-D governing equations for the physics of hydraulic transients are the equations of motion and continuity, also called the Saint-Venant equations [12]. These are hyperbolic quasilinear partial and continuity, also called thehaving Saint-Venant equations [12]. they These hyperbolic quasilinear differential equations, and, no analytical solution, areare most accurately solved bypartial MOC. differential equations, and, having no analytical solution, they are most accurately solved by MOC. fV V ∂H ππ ∂V ππΗπΗ ππ» (1) (1) πg ∂x ++ ∂t++ 2D= 0,= 0, ππ₯ ππ‘ 2π· 2 ∂V ππ»∂H π cππ + + = 0, = 0, π gππ₯∂x ππ‘ ∂t (2) (2) where H is the piezometric head; V is the mean flow velocity; f is the Darcy-Weisbach friction factor; D is the diameter of the pipe; g is the acceleration due to gravity; and c is the wave speed. Water 2019, 11, 715 4 of 19 where H is the piezometric head; V is the mean flow velocity; f is the Darcy-Weisbach friction factor; D is the diameter of the pipe; g is the acceleration due to gravity; and c is the wave speed. WaterMethod 2019, 11,of715 2.2. Characteristics 4 of 19 For 1-Dofnumerical modeling of the hydropower system, the most widely used method called 2.2. Method Characteristics Method of Characteristics (MOC) is applied, as proposed by Wylie et al. [12], which converts the For 1-D numerical modeling of the hydropower system, the most widely used method called equations of motion and continuity into two ordinary differential equations that are then solved as a Method of Characteristics (MOC) is applied, as proposed by Wylie et al. [12], which converts the simple form, Equations (3) and (4), by the finite-difference method. equations of motion and continuity into two ordinary differential equations that are then solved as a simple form, Equations (3) and (4), by the : HP = CP − BQmethod. (3) C+finite-difference P i i πΆ : π» = πΆ − π΅π (3) C− : HPi = CM + BQPi (4) πΆ : π» = πΆ + π΅π (4) CP = Hi−1 + BQi−1 − RQi−1 |Qi−1 | (5) (5) πΆ = π» + π΅π − π π |π | CM = Hi+1 − BQi+1 + RQi+1 Qi+1 (6) (6) πΆ = π» − π΅π + π π |π | c B= π (7) gA π΅= (7) ππ΄ f βx R = πβπ₯2 (8) π = 2gDA (8) 2ππ·π΄ where Hi and Qi are the piezometric head and discharge, respectively, at any time and location; A is where Hi and Qi are the piezometric head and discharge, respectively, at any time and location; A is the area of the pipe; and f is the coefficient of friction. the area of the pipe; and f is the coefficient of friction. As shown in Figure 2, the characteristic lines C+ and C− intersect at point P. Thus, with the known As shown in Figure 2, the characteristic lines C+ and C− intersect at point P. Thus, with the known initial values during a steady state condition, Equations (3)–(8) give the values of head and discharge initial values during a steady state condition, Equations (3)–(8) give the values of head and discharge at a particular time and position in the x-t grid. at a particular time and position in the x-t grid. Figure Figure 2. 2. Characteristic Characteristic lines lines in in x-t x-t grid. grid. 2.3. Boundary Conditions 2.3. Boundary Conditions In this study, we considered a simple model of a high-head hydropower system for hydraulic In this study, we considered a simple model of a high-head hydropower system for hydraulic transient analysis and optimization of its hydraulic characteristics. 1-D numerical analysis was carried transient analysis and optimization of its hydraulic characteristics. 1-D numerical analysis was out using MATLAB. Based on the layout of the hydropower system shown in Figure 1, the following carried out using MATLAB. Based on the layout of the hydropower system shown in Figure 1, the boundary conditions were established, as described by Wylie et al. [12]. following boundary conditions were established, as described by Wylie et al. [12]. 2.3.1. Reservoir 2.3.1. Reservoir The system consists of an upstream reservoir and it is assumed that water level of the reservoir Theconstant system consists of an upstream reservoir and it is assumed that water level of the reservoir remains during transient evaluation. remains constant during transient evaluation. Hπ»P1 = =Z πres (9) (9) thepiezometric piezometrichead headatatthe thereservoir reservoirand andZZ thewater waterlevel levelofofthe thereservoir. reservoir. where Hπ»P1 isisthe where resresisisthe 2.3.2. In-Line Surge Tank In the pressurized water conveyance system, the surge tank is a very important structure to effectively control the hazardous effects of hydraulic transients. In this study, the restricted orifice surge tank, as shown in Figure 3,is located at the end of the headrace tunnel. The equations defining Water 2019, 11, 715 5 of 19 2.3.2. In-Line Surge Tank In the pressurized water conveyance system, the surge tank is a very important structure to effectively control the hazardous effects of hydraulic transients. In this study, the restricted orifice surge tank, as shown in Figure 3, is located at the end of the headrace tunnel. The equations defining the boundary conditions of the restricted orifice surge tank are presented below [12]: HP = ZS + RS |QS |QS (10) ZS = W (QS + QS0 ) + ZS0 (11) Q1 = Q2 + QS (12) RS = 1 2gCd 2 AO 2 βt 2AS s Lt AS T = 2π gAt W= (13) (14) (15) where Q1 and Q2 are the discharges in pipe 1 and pipe 2, respectively, at the point of surge tank; QS is the discharge flowing into the surge tank; HP is the piezometric head in the pipe at the bottom of the surge tank; ZS is the water level in the surge tank, also referred to as surge head in further sections; QS0 and ZS0 are the previous discharge and water level in the surge tank, respectively; AO and AS are the area of orifice and area of surge tank, respectively; Cd is the coefficient of discharge; βt is the iteration time period considered for simulation; T is the time period for mass oscillation in the surge tank; and Lt and At are the length and area of the headrace tunnel, respectively. Equations (3), (4), Water(10)–(12) 2019, 11, 715 and are solved together to get the values of the variables at the surge tank boundary. 5 of 19 Figure 3. Restricted orifice surge tank. Figure 3. Restricted orifice surge tank. 2.3.3. Free Discharge Valve at the End (10) π» and = π closing + π |πof|πthe valve causes transients. For gradual In this model, linear gradual opening closure, the time period of closure of the valve is more than the pipe characteristic time, i.e., Tc > 2L . (11)c π = π(π + π ) + π The time of closure/opening of the valve for general surge analysis and the optimization procedure (12) π = π used + π closing time of guide vanes in hydropower is considered to be 10 s according to the practically stations. In surge analysis, the closing/opening of valve 1 is started at 5 s and the valve is completely = in Section 3.5, an elaborated analysis is conducted (13) closed/opened at 15 s in linear progression. π Later, 2ππΆ π΄ to demonstrate the transient changes for various closing/opening times of the valve, including both Δπ‘ π= (14) 2π΄ π = 2π πΏπ΄ ππ΄ (15) where Q1 and Q2 are the discharges in pipe 1 and pipe 2, respectively, at the point of surge tank; QS is the discharge flowing into the surge tank; HP is the piezometric head in the pipe at the bottom of the surge tank; ZS is the water level in the surge tank, also referred to as surge head in further sections; Water 2019, 11, 715 6 of 19 fast and gradual closure/opening. The equations determining the valve boundary conditions are as follows [12]. q QPN+1 = −BCV + CV = (BCV )2 + 2CV CP ( Q0 τ ) 2 2H0 (16) (17) where QPN+1 is the discharge at the valve; τ is the valve opening; Q0 is the steady state flow; and H0 is the operating head across the valve, and for a free discharge valve, H0 is the piezometric head at the inlet of the valve. 2.4. Stability Criteria The hydraulic transient causes pressure and discharge fluctuations in the surge tank. During such conditions, the surge tank is vulnerable to various instabilities. For a surge tank to be stable and effective, the following criteria must be fulfilled: 2.4.1. Minimum Area of Surge Tank During hydraulic transients, fluctuation of the water level occurs in the surge tank, which must be dampened in a reasonably short time to ensure the stability of the system. In addition to this, the maximum and minimum water level in the surge tank should be properly analyzed so that the system stays safe in the worst condition. For this, Thoma derived an equation called the Thoma equation, which gives the minimum area of a stable surge tank [40]. However, Jaeger later proved that this area is not sufficient for the stability of a surge tank and introduced a safety coefficient in the Thoma equation [41]. Lt At Ath = e (18) 2αgH1 e = 1 + 0.482 α= Zmax Ho H w o At 2 Q2 H1 = Ho − Hwo − 3Hwm (19) (20) (21) where Ath is the Thoma area; e is the safety coefficient; Lt and At are the length and area of the headrace tunnel, respectively; Ho is the gross head in the turbine; Hwo is the head loss in the headrace tunnel; Hwm is the head loss in the penstock; Zmax is the maximum water level in the surge tank; and Q is the discharge in the tunnel. 2.4.2. Vortex Control In the case of start-up of the pressurized water conveyance system, the surge tank acts as a temporary intake, supplying discharge to the system. This causes hydraulic transients in the system and the water level in the surge tank falls down to its minimum level. During such conditions, if the water level falls below a certain elevation, there is a very high risk of vortex formation or air entrainment, which can have devastating effects on the system. The following steps were followed to check the vortex formation in the system after the optimized values of DO and DS were obtained: 1. 2. The minimum possible down surge that can be created in the system is calculated, which is defined as objective function 2 in Section 2.5.3; From this minimum water surface elevation, the bottom surface elevation of the surge tank is subtracted to obtain the height of the minimum water column in the tank; Water 2019, 11, 715 3. 7 of 19 Then, the critical submergence for the possible suction vortex is calculated using the Gordon equation [42]: √ (22) Scr = cV d where Scr is the critical submergence depth; c is an empirical coefficient whose value ranges from 0.55 to 0.73 (c = 0.55 for symmetrical intake); V is the velocity at intake; and d is the diameter of the pipe at intake, and for this case, the velocity and the diameter of penstock pipe are considered to check vortex formation in the surge tank. The minimum water column depth in the surge tank must be greater than the critical submergence depth (Scr ) to prevent the formation of a harmful vortex in the system. 2.5. Optimization First, 1-D numerical modeling and analysis of the hydropower system were conducted for linear gradual closing and gradual opening of the valve, respectively, using MATLAB and the values of the piezometric head and discharge at the surge tank were obtained using MOC. From these obtained values, the following objective functions and constraints were developed for optimizing the diameter of the surge tank (DS ) and the diameter of the orifice (DO ). 2.5.1. Objective Function 1 The maximum water level in the surge tank (Max1) is reached during full load rejection, as shown in Figure 4a, when the system is working with full capacity and the reservoir is at the highest water level. This value is obtained in this study by 1-D numerical modeling of the system using MOC. This value gives the maximum possible water level in the surge tank and determines the height and performance of the surge tank during the worst transient condition and hence, plays a vital role in surge tank design. Minimization of this maximum value is taken as the first objective function in this study. Many researchers have derived empirical or stochastic equations for simplicity in the calculation of the maximum surge head in the surge tank [43–46], but MOC gives the most accurate results for surge analysis from which the maximum value can be easily obtained. Minimize: F1 (x1 , x2 ) where F1 (x1 , x2 ) is the function for the maximum water level for different diameters of the orifice (x1 ) and diameters of the surge tank (x2 ). 2.5.2. Objective Function 2 The minimum water level in the surge tank (Min1) is reached during full load acceptance, as shown in Figure 4b, when the system is put into operation with minimum capacity and the reservoir is at the lowest water level. This gives the value of the lowest possible water level in the surge tank, or the lowest submergence depth, which essentially determines the safety of the system from harmful vortices or air entrainments. This minimum value, obtained after surge analysis of the worst scenario of transients for the lowest possible water level in the surge tank using MOC, is assigned as objective function 2. The value of the lowest submergence depth should be increased to ensure effective working of the system, even in severe hydraulic transients. Minimize: −F2 (x1 , x2 ) where F2 (x1 , x2 ) is the function for minimum surge pressure for different diameters of the orifice (x1 ) and diameters of the surge tank (x2 ) (negative sign is for minimization). 2.5.3. Objective Function 3 Whenever hydraulic transients occur in the system, oscillation of pressure and discharge is observed, which is gradually dampened by friction and various surge controlling devices, like the air chamber, surge tank, pressure release valves, etc. Damping of surge waves is a very crucial characteristic of the surge tank and effective surge tanks are required to dampen the surge waves as Water 2019, 11, 715 8 of 19 quickly as possible. In this study, we performed surge analysis for the worst condition of transients that gives the maximum water level in the surge tank, as explained in Section 2.5.1, extracted two successive values of maximum water level as shown in Figure 4a, and calculated the percentage of damping of surge waves using Equation (23). D= Max1 − Max2 × 100% Max1 (23) where D is the percentage of surge pressure damping; and Max1 and Max2 are the first and second maximum water level during full load rejection, respectively, as shown in Figure 4a. Minimize: −F3 (x1 , x2 ) 3 (x1 , x2 ) is the function for the percentage of damping obtained for different values Water where 2019, 11,F715 8 of of 19 diameter of the orifice (x1 ) and diameter of the surge tank (x2 ) (negative sign is for minimization). These three functions are combined into a single usingfunction normalization. threeobjective objective functions are combined into objective a singlefunction objective using normalization. Figure 4. Water level vs time graph for a restricted orifice surge tank during linear gradual closure of the valve (a) and linear gradual opening opening of of the the valve valve (b). (b). 2.6. Normalization 2.6. Normalization The objective functions chosen in this study is highly conflicting and the range The nature natureofofthe thethree three objective functions chosen in this study is highly conflicting and the of their values is also very diverse, as shown in Section 3.2. Hence, normalization was done so that range of their values is also very diverse, as shown in Section 3.2. Hence, normalization was done all so objective functions fell into a similar range, and could be combined under a single objective function. that all objective functions fell into a similar range, and could be combined under a single objective The unity-based normalization was used,was which brings all brings the values in the range 1]. of [0, 1]. function. The unity-based normalization used, which all the values in of the[0, range = FπΉiN ((π₯ x1 ,,xπ₯2 )) = (x1 ,, π₯x2 )) − πΉ − FπΉi,min Fi (π₯ , πΉ − πΉ Fi,max − F , , i,min (24) (24) (25) πΉ(π₯ , π₯ ) =X ∑πΉ (π₯ , π₯ ) F(x1 , x2 ) = FiN (x1 , x2 ) (25) where F(x1,x2) is the normalized objective function; FiN(x1,x2) represents the individual objective where F(x1 ,x2 ) is the normalized objective function; FiN (x1 ,x2 ) represents the individual objective functions; and Fi,max and Fi,min are the maximum and minimum values of each objective function within functions; and Fi,max and Fi,min are the maximum and minimum values of each objective function the given range, respectively. within the given range, respectively. 2.7. Variables and Constraints 2.7. Variables and Constraints This study intends to find a well-optimized surge tank which effectively overcomes all the This study intends to find a well-optimized surge tank which effectively overcomes all the hydraulic transient complications. The diameter of the orifice (DO) and the diameter of the surge tank hydraulic transient complications. The diameter of the orifice (DO ) and the diameter of the surge (DS) are considered as variables during the optimization. The lower and upper limits of these tank (DS ) are considered as variables during the optimization. The lower and upper limits of these diameters are determined as follows: diameters are determined as follows: According to the “Chinese design code for surge chamber of hydropower stations”, the According to the “Chinese design code for surge chamber of hydropower stations”, the minimum minimum and maximum area of the orifice must be 25% and 45% of the area of the headrace tunnel, and maximum area of the orifice must be 25% and 45% of the area of the headrace tunnel, respectively, respectively, for optimum performance of the surge tank [19,47]. Hence, we obtained, for optimum performance of the surge tank [19,47]. Hence, we obtained, • 3.1 π ≤ π· ≤ 4.1 π For the diameter of surge tank (Ds), the minimum value is obtained by the Thoma area, as discussed in Section 2.4.1, and the maximum value is taken to be 12 m based on the preliminary observations, as presented in Section 2.8, and the results obtained during optimization. Thus, we have, Water 2019, 11, 715 9 of 19 3.1 m ≤ DO ≤ 4.1 m For the diameter of surge tank (Ds), the minimum value is obtained by the Thoma area, as discussed in Section 2.4.1, and the maximum value is taken to be 12 m based on the preliminary observations, as presented in Section 2.8, and the results obtained during optimization. Thus, we have, 6.3 m ≤ DS ≤ 12 m Besides these, the difference between the maximum piezometric head at the bottom tunnel of the surge tank and the maximum water level of the surge tank plays a crucial role in designing an efficient surge tank. In this study, the DO and DS are accepted only when the difference between the maximum head at the bottom tunnel and the maximum water level in the surge tank is less than 1 m. When the difference is higher, the surge tank is not effective and the headrace tunnel becomes more vulnerable to Water 2019, effects 11, 715 [47]. 9 of 19 transient max(HP ) − max(ZS ) ≤ 1 m where π» is piezometric head at the bottom tunnel of of the surge where HPthe is the piezometric head at the bottom tunnel the surgetank tankand andπZS isis the the water water level level in the surge tank. in the surge tank. 2.8. Acceptance 2.8. Acceptance Region Region The constraints constraints described Section 2.7 2.7 define define the the total total population population for for optimization optimization and and also also The described in in Section distinguish valid and invalid regions based on the difference of the maximum piezometric head at distinguish valid and invalid regions based on the difference of the maximum piezometric head at the bottom bottom tunnel tunnel of of the the surge surge tank tank and and the the maximum maximum water water level level in in the the surge surge tank. tank. Figure 5 shows shows the Figure 5 various values of D O and D S within the range of optimization discussed earlier. The green area various values of DO and DS within the range of optimization discussed earlier. The green area in in the the graph shows shows the the valid validregion, region,where wherethe thevalue valueofofthe thedifference difference maximum heads is less than 1 graph ofof thethe maximum heads is less than 1 m, m, and the area red area the graph the invalid the difference of the maximum and the red in theingraph showsshows the invalid region,region, havinghaving the difference of the maximum heads of headsthan of more than eventually 1 m. This makes eventually makes the optimization problem highly and more 1 m. This the optimization problem highly non-linear andnon-linear discontinuous discontinuous and objective the individual objective functions are highly conflicting in nature. and the individual functions are highly conflicting in nature. Figure 5. 5. Acceptable Acceptable (green) (green) and and unacceptable unacceptable (red) (red) region region in in the Figure the total total population. population. 2.9. Particle Swarm Optimization 2.9. Particle Swarm Optimization The Particle Swarm Optimization (PSO) is a nature-inspired, metaheuristic stochastic optimization The Particle Swarm Optimization (PSO) is a nature-inspired, metaheuristic stochastic tool developed by Kennedy et al. [34]. PSO imitates the social behavior of swarm in which, initially, optimization tool developed by Kennedy et al. [34]. PSO imitates the social behavior of swarm in a random set of the population is selected, and then, in each iteration, each particle explores and moves which, initially, a random set of the population is selected, and then, in each iteration, each particle to a new location based on the best solution of that individual particle, and the best solution of all the explores and moves to a new location based on the best solution of that individual particle, and the particles. A parameter called velocity is added to each particle’s position in each iteration, which moves best solution of all the particles. A parameter called velocity is added to each particle’s position in it towards the global minimum or maximum. Due to its simplicity and robust nature, PSO can easily each iteration, which moves it towards the global minimum or maximum. Due to its simplicity and optimize complex problems and has been widely used in engineering optimization works. robust nature, PSO can easily optimize complex problems and has been widely used in engineering n o n o optimization works. (V ) = θ × (V ) + c r (P ) − (X ) + c r (G ) − (X ) (26) j i i j i−1 (π ) = π × (π ) 1 1 + π π (π π =π , ) − (π ) − 2 2 j i−1 best,j i π (π ) = (π ) + π π (πΊ −π π π + (π ) π = 1, 2, . . . . , π; πππ π = 1, 2, . . . . , π best i j i−1 ) − (π ) (26) (27) (28) Water 2019, 11, 715 10 of 19 θi = θMax − θMax − θMin i N (X j )i = (X j )i−1 + (V j )i (27) (28) i = 1, 2, . . . , N; and j = 1, 2, . . . , M where (Xj )i and (Vj )i are the position and velocity of the jth particle in the ith iteration; c1 and c2 are cognitive and social learning rates; r1 and r2 are random numbers in the range of 0 and 1; (Pbest,j )i is the best result obtained for the jth particle till the ith iteration; (Gbest )i is the best result among all particles till the ith iteration; θi is the inertia weight for the ith iteration, developed by Shi et al. [48], to dampen Water 2019, 11, 715over each iterations for efficient optimization; θMax and θMin are the maximum 10 of 19 the velocities and minimum values of inertia weight, respectively; and M and N are the total number of population and Figure 6 shows therespectively. flowchart for PSO used for the optimization of the hydraulic characteristics iterations used in PSO, of theFigure generalized system. Inused this study, the difference of the maximum head at 6 showshydropower the flowchart for PSO for the when optimization of the hydraulic characteristics of the bottom tunnel at the point of the surge tank and the maximum water level in the surge tank is the generalized hydropower system. In this study, when the difference of the maximum head at the less than or equal to 1 m (π» ≤ 1 m), the solution and corresponding D O and D S are considered bottom tunnel at the point of the surge tank and the maximum water level in the surge tank is less acceptable, more than and 1 m, they are considered If this head than or equalwhile, to 1 mwhen (Hd ≤π»1 m),isthe solution corresponding DO and Dunacceptable. S are considered acceptable, difference is smaller, it ensures efficient working of the surge tank [47]. During optimization, avoid while, when Hd is more than 1 m, they are considered unacceptable. If this head difference istosmaller, the solutions of theworking unacceptable a technique of reducing the position is applied, as shown in it ensures efficient of theregion, surge tank [47]. During optimization, to avoid the solutions of the Figure 6. Whenever the new position of any particle is calculated using Equation (28), if the new unacceptable region, a technique of reducing the position is applied, as shown in Figure 6. Whenever position intoof the unacceptable region, then theEquation average of theif previous position falls and into current the new falls position any particle is calculated using (28), the new position the position is taken as the new again, the region of the position is checked. unacceptable region, then theposition, average and of the previous position andnew current position is taken Every as the time, the velocity the new position reduced by halfisuntil it fallsEvery into the acceptance region. the new position, andof again, the region of is the new position checked. time, the velocity of theIfnew new position does not lie in the acceptance region for 50 iterations, a random new position is assumed position is reduced by half until it falls into the acceptance region. If the new position does not lie in the within the range the reduction new positions reduces the acceptance regionoffor 50variables. iterations,This a random new in position is assumed withinthe thespeed rangeofofacquiring the variables. global minimum or maximum, but more importantly, it enables the program to stay in the acceptable This reduction in new positions reduces the speed of acquiring the global minimum or maximum, region during all calculations. but more importantly, it enables the program to stay in the acceptable region during all calculations. Figure 6. Flowchart of particle swarm optimization. Figure 6. Flowchart of particle swarm optimization. 3. Results and Discussions 3.1. Surge Analysis Water 2019, 11, 715 11 of 19 3. Results and Discussions 3.1. Surge Analysis The surge behavior in the surge tank reveals important characteristics of transients occurring in a pressurized water conveyance system. The given generalized hydropower system is first analyzed numerically for gradual closure and opening of the valve respectively using the MOC. This method has been experimentally validated by several researchers in real world problems [6,11,14,49–51]. In recent years, many model tests for the hydropower systems have been conducted in the laboratory, mainly including the hydraulic characteristics of the surge tank [49,50]. The models provided complete head loss coefficients for water flowing into or out of the surge tank, which revealed the complete hydraulic transient processes during load rejection or load acceptance, and also display great agreement with the results obtained by MOC. Thus, MOC is widely accepted as a reliable tool for accurate numerical analysis involving computer modeling, especially for surge analysis. The system consists of a headrace pipe of length 570 m (head loss coefficient R = 6.81687 × 10−6 ) and a penstock pipe of length 1000 m (head loss coefficient R = 9.3451 × 10−6 ). It is divided into L 157 sections, each of a 10 m length, with 158 nodal points. Applying Courant’s condition, i.e., βt = cN , the time step of analysis is taken to be 0.008 s. An upstream reservoir is located at node 1 in the system. The maximum and minimum operational water level of reservoir are 520 m and 500 m, respectively, and the valve discharge is 132.4 m3 /s. The restricted orifice surge tank with DS = 9 m and DO = 4.3 m is at the end of the headrace tunnel at node 58 and a free discharge valve at the end of the penstock at node 158. The time of closure and opening of the valve was 10 s and the total time of analysis was 200 s. Initially, a steady state condition was modeled, and then the closing/opening of valve was started at 5 s and the valve was completely closed at 15 s in linear progression. Surge analysis was carried out to reveal important hydraulic characteristics occurring due to gradual closure and opening of the valve. Later, PSO was used to optimize the diameter of the orifice and the diameter of the surge tank to improve the transient controlling capacity, and design a more efficient and effective surge tank. Figure 7 depicts the most important characteristics of the hydraulic transient in the hydropower system due to valve maneuver, which plays a vital role in surge tank design. Figure 7a shows the water level in the surge tank, during the maximum water level in the reservoir and full load rejection condition. Hence, the maximum value of surge head shown by the graph indicates the maximum possible surge head reached for the given condition of valve closure. Initially, the water level is constant for 5 s, which represents the steady state condition. After the start of valve closure at 5 s, the water level gradually rises to its maximum value and starts oscillating to consecutive maximum and minimum values due to water hammer action. Damping of surge waves in this condition occurs gradually and its value, as given by Equation (23), is calculated to be 1.378%. Similarly, Figure 7b shows the water level in the surge tank during the minimum water level in the reservoir and full load acceptance condition and hence it reveals the minimum possible water level for the given time of valve opening. The initial horizontal line represents the steady state for 5 s, valve opening is started at 5 s, and the water level drops to its lowest value and then starts oscillating. Discharge is kept constant, regardless of the water level in the reservoir. Damping of surge waves during the valve opening condition occurs very fast, while during valve closing, surge waves dampen gradually and hence, damping of surge waves during the valve closing condition should be properly analyzed. The theoretical value of the time period of surge waves, given by Equation (15), is calculated to be 69.524 s, and from this analysis, this value is found to be 69.888 s. This shows that the modeling and analysis results are fairly accurate. 3.2. Effects of Changing Do and Ds The maximum and minimum water level in the surge tank as shown in Figure 8 and damping of surge waves is the most important characteristic, which determines the efficient design and stability of the surge tank. The sensitivity of these three parameters is analyzed with varying diameters of orifice (Do) and diameters of surge tank (Ds), as shown in Figure 8. As the DO is increased, the value of system due to valve maneuver, which plays a vital role in surge tank design. Figure 7a shows the water level in the surge tank, during the maximum water level in the reservoir and full load rejection condition. Hence, the maximum value of surge head shown by the graph indicates the maximum possible surge head reached for the given condition of valve closure. Initially, the water level is Water 2019, for 11, 715 19 constant 5 s, which represents the steady state condition. After the start of valve closure at 512s,ofthe water level gradually rises to its maximum value and starts oscillating to consecutive maximum and minimum values due to water hammer action. Damping of surge waves in this condition occurs maximum head gradually increases and minimum head gradually decreases. Thus, the lower value of gradually and its value, as given by Equation (23), is calculated to be 1.378%. Similarly, Figure 7b DO is more favorable, which was also concluded by Diao et al. [19]. Contrary to that, when the DS shows the water level in the surge tank during the minimum water level in the reservoir and full load is increased, the maximum water level gradually decreases and the minimum water level gradually acceptance condition and hence it reveals the minimum possible water level for the given time of increases. Thus, it can be concluded that a higher value of DS is favorable to minimize hydraulic valve opening. The initial horizontal line represents the steady state for 5 s, valve opening is started transient effects in the system. The percentage of damping of surge waves during gradual valve closure at 5 s, and the water level drops to its lowest value and then starts oscillating. Discharge is kept first increases, reaches a maximum, and then decreases with increasing DO , and damping gradually constant, regardless of the water level in the reservoir. Damping of surge waves during the valve decreases with an increase in DS . In addition to this, the difference of the maximum piezometric head opening condition occurs very fast, while during valve closing, surge waves dampen gradually and at the bottom tunnel of the surge tank and the maximum water level in the surge tank is required to hence, damping of surge waves during the valve closing condition should be properly analyzed. The be below 1 m for an effective surge tank. All these features make the system highly conflicting and theoretical value of the time period of surge waves, given by Equation (15), is calculated to be 69.524 discontinuous, thus, a robust optimizing tool is required to obtain global optimum values12ofofthe Water 11,this 715 and 19 s, and2019, from analysis, this value is found to be 69.888 s. This shows that the modeling and analysis variables within the defined constraints. results are fairly accurate. 3.2. Effects of Changing Do and Ds The maximum and minimum water level in the surge tank as shown in Figure 8 and damping of surge waves is the most important characteristic, which determines the efficient design and stability of the surge tank. The sensitivity of these three parameters is analyzed with varying diameters of orifice (Do) and diameters of surge tank (Ds), as shown in Figure 8. As the DO is increased, the value of maximum head gradually increases and minimum head gradually decreases. Thus, the lower value of DO is more favorable, which was also concluded by Diao et al. [19]. Contrary to that, when the DS is increased, the maximum water level gradually decreases and the minimum water level gradually increases. Thus, it can be concluded that a higher value of DS is favorable to minimize hydraulic transient effects in the system. The percentage of damping of surge waves during gradual valve closure first increases, reaches a maximum, and then decreases with increasing DO, and damping gradually decreases with an increase in DS. In addition to this, the difference of the maximum piezometric head at the bottom tunnel of the surge tank and the maximum water level in the surge tank is required to be below 1 m for an effective surge tank. All these features make the system highly conflicting and discontinuous, and thus, a robust optimizing tool is required to obtain Figure 7. Surge analysis in surge tank during transients caused by gradual closing (a) and opening (b) Figure 7. Surge analysis in surge tank within during the transients caused by gradual closing (a) and opening global optimum values of the variables defined constraints. of valve. (b) of valve. Figure 8. Maximum Maximum water water level level(a),(d); (a,d); minimum and percentage of of damping (c,f) for Figure 8. minimumwater waterlevel level(b,e); (b),(e); and percentage damping (c),(f) different diameters of orifice (Do) and diameters of surge tank (Ds). for different diameters of orifice (Do) and diameters of surge tank (Ds). Three inin thethe given system, as discussed in Section 2.5, Three major majorcharacteristics characteristicsofofhydraulic hydraulictransients transients given system, as discussed in Section are taken as the objective functions for optimizing D and D . For normalizing the three individual O 2.5, are taken as the objective functions for optimizing DSO and DS. For normalizing the three individual objective functions into a single objective function, the maximum and minimum values of each individual function are required, which are calculated using the PSO in each objective function within the given range of variables. Table 1 shows the obtained minimum and maximum values, which can also be verified by analyzing the nature of graphs presented in Figure 8. Water 2019, 11, 715 13 of 19 objective functions into a single objective function, the maximum and minimum values of each individual function are required, which are calculated using the PSO in each objective function within the given range of variables. Table 1 shows the obtained minimum and maximum values, which can also be verified by analyzing the nature of graphs presented in Figure 8. Table 1. Maximum and minimum values of each objective function. For Maximum Values For Minimum Values Objective Functions Values of Functions Do (m) Ds (m) Values of Functions Do (m) Ds (m) Function 1 Function 2 Function 3 541.550 m 488.740 m 2.038 % 4.1 3.1 3.552 6.3 12 6.3 524.594 m 473.056 m 0.562 % 3.1 4.1 3.1 12 6.3 12 Water 2019, 11, 715 13 of 19 3.3. Optimization with PSO 3.3. Optimization with PSO As described earlier, the maximum water level, the minimum water level, and damping are chosen As described earlier, the maximum water level, the minimum water level, and damping are as objective functions for optimization, with diameter of the orifice (DO ) and diameter of the surge chosen as objective functions for optimization, with diameter of the orifice (DO) and diameter of the tank (DS ) as variables. All three objective functions are reduced to minimization problem and a single surge tank (DS) as variables. All three objective functions are reduced to minimization problem and objective function is designed by using normalization. The cognitive and social learning rates are taken a single objective function is designed by using normalization. The cognitive and social learning rates to be 1 (c1 = c2 = 1). Initially, random values of the variables are provided, and then, 500 iterations are taken to be 1 (c1 = c2 = 1). Initially, random values of the variables are provided, and then, 500 are carried out with a population size of 10. Due to the small range of variation of variables, a small iterations are carried out with a population size of 10. Due to the small range of variation of variables, population is considered and the discontinuous nature of the problem, as discussed in Section 2.8, a small population is considered and the discontinuous nature of the problem, as discussed in Section required a large number of iterations before the optimized values were obtained. The following 2.8, required a large number of iterations before the optimized values were obtained. The following graph in Figure 9 shows the convergence of the values towards the optimum minimal solution after graph in Figure 9 shows the convergence of the values towards the optimum minimal solution after optimization with PSO. optimization with PSO. Figure Figure 9. 9. Graph Graph showing showing the the minimization minimization of of results results in in each each iteration iteration using using PSO. PSO. After After performing performingPSO PSOfor forthe thegiven givensystem, system,the theoptimized optimizedvalues valuesofofDDOO and and D DSS were were obtained obtained to to be 4.1 m and 11.362 m, respectively. These optimized dimensions of the surge tank produced significant be 4.1 m and 11.362 m, respectively. These optimized dimensions of the surge tank produced improvements in the maximum minimum level in the surge tank, damping of significant improvements in theand maximum andwater minimum water level in the while surge the tank, while the surge waves was not improved, as shown in Table 2. In order to improve the damping of surge damping of surge waves was not improved, as shown in Table 2. In order to improve the damping waves, a weighting factor of 0.25, 0.25, wasand multiplied each objective function, respectively, of surge waves, a weighting factor of and 0.25,0.5 0.25, 0.5 wasin multiplied in each objective function, and optimization performed again. The optimized values were obtained as Do = 3.684as m Do and= respectively, and was optimization was performed again. The optimized values were obtained Ds = 6.434 m and better damping of surge waves was obtained, while the values of the maximum 3.684 m and Ds = 6.434 m and better damping of surge waves was obtained, while the values ofand the minimum water level in the surge tank were worse than the original case. For these two optimization maximum and minimum water level in the surge tank were worse than the original case. For these cases and the original case, carried outanalysis and theisresults areout presented Figure are 10. two optimization cases andsurge the analysis original is case, surge carried and theinresults The values of each objective function for all three cases are tabulated in Table 2 and improvements presented in Figure 10. The values of each objective function for all three cases are tabulated in Table obtained are analyzed.obtained are analyzed. 2 and improvements Table 2. Comparison and improvements in each objective function after optimization. Objective Without Optimization Normalized Optimization Normalized Optimization with Weighting Water 2019, 11, 715 14 of 19 Table 2. Comparison and improvements in each objective function after optimization. Objective Functions Function 1 Function 2 Function 3 715 Water 2019, 11, Water 2019, 11, 715 Without Optimization (Original) Normalized Optimization Do = 4.1 m and Ds = 11.362 m Normalized Optimization with Weighting Do = 3.684 m and Ds = 6.434 m Do = 4.3 m and Ds = 9 m Obtained Values Improvement Obtained Values 533.643 m 480.609 m 1.378% 528.937 m 485.095 m 0.973% 4.706 m 4.486 m −0.405% 539.142 m 474.618 m 2.009% Improvement −4.706 m −5.991 m 0.631% 14 of 19 14 of 19 Figure 10. Comparison of of surge surge analysis for different Figure 10. 10. Comparison Comparison analysis for for different different cases cases of of valve valve closing closing (a) (a) and and opening opening (b). (b). Figure of surge analysis cases of valve closing (a) and opening (b). After examining the results shown in Figure 10 and Table 2, the first case of optimization is After examining examining the the results results shown shown in in Figure Figure 10 10 and and Table Table 2, 2, the the first first case of of optimization optimization is is After concluded to be better than the other one. An improvement of 4.706 m andcase 4.486 m in maximum concluded to to be better better than than the the other other one. one. An improvement improvement of of 4.706 4.706 m m and and 4.486 4.486 m in maximum maximum and and concluded and minimumbepossible water levels in theAn surge tank, respectively, were obtainedmininthis case for the minimum possible water levels in the surge tank, respectively, were obtained in this case for the given minimum possible water levels in the surge tank, respectively, were as obtained in m this case given given hydropower system. Hence, the optimized values are taken Do = 4.1 and Dsfor = the 11.362 m hydropower system. Hence, the optimized values are taken as Do = 4.1 m and Ds = 11.362 m and hydropower system. the showing optimizedpiezometric values are head taken at asthe Do bottom = 4.1 mtunnel and Dsof=the 11.362 m tank and and surge analysis is Hence, conducted surge surge analysis analysis is conducted showing showing piezometric piezometric head head at at the the bottom bottom tunnel of of the the surge surge tank tank and and surge and water levelisinconducted the surge tank referring to these values, as shown intunnel Figure 11. The difference of water level in the surge tank referring to these values, as shown in Figure 11. The difference of water levelhead in the tankcases referring tothe these values, as condition shown inisFigure The difference of maximum for surge these two during valve closure found 11. to be 0.999 m and the maximum head head for these these two two cases cases during during the the valve valve closure closure condition condition is is found found to to be be 0.999 0.999 m m and and the the maximum graphs for thesefor two heads are almost overlapping, which indicates sound functioning of the surge graphs for these two heads are almost overlapping, which indicates sound functioning of the surge graphs for these heads are almost overlapping, which functioning thebottom surge tank. Thus, in thistwo optimization, both water level in the surgeindicates tank andsound piezometric head atofthe tank. Thus, in this optimization, both water level in the surge tank and piezometric head at the bottom tank. Thus, in this optimization, water level inperformance the surge tank andsurge piezometric tunnel are effectively controlled both for the optimum of the tank. head at the bottom tunnel are are effectively effectively controlled controlled for for the the optimum optimum performance performance of of the the surge surge tank. tank. tunnel Figure 11. 11. Piezometric Piezometric head head at at the the bottom bottom tunnel tunnel of of surge surge tank tank and and water water level level in in surge surge tank tank during during Figure gradual closing (a) and opening (b) of valve. gradual closing (a) and opening (b) of valve. 3.4. Vortex Vortex Formation Formation Check Check 3.4. The minimum minimum possible possible surge surge head head in in the the surge surge tank tank during during hydraulic hydraulic transients transients in in the the given given The system after optimization was found to be 485.095 m. Elevation of the bottom of the surge tank was system after optimization was found to be 485.095 m. Elevation of the bottom of the surge tank was 450.00 m and hence the minimum water column height in this system became 35.095 m. From Water 2019, 11, 715 15 of 19 3.4. Vortex Formation Check The minimum possible surge head in the surge tank during hydraulic transients in the given system after optimization was found to be 485.095 m. Elevation of the bottom of the surge tank was 450.00 m and hence the minimum water column height in this system became 35.095 m. From Equation (22), the critical submergence depth of flow was calculated to be 6.638 m, which is way below the minimum water column height in the surge tank. Therefore, the optimized surge tank effectively prevents the formation of715 a vortex in the system during the worst hydraulic transient condition. Water 2019, 11, 15 of 19 3.5. 3.5. Time Time of of Closure Closure and and Opening Opening of of Valve Valve The have significant significant effects The closure closure and and opening opening time time of of the the valve valve can can have effects on on the the transients transients occurring occurring in the system [23–25,28]. Generally, in hydropower projects, gradual closure and opening valve in the system [23–25,28]. Generally, in hydropower projects, gradual closure and opening ofof valve is is adapted for adjusting the condition of flow in the system and power demands. In this study, adapted for adjusting the condition of flow in the system and power demands. In this study, all the all the analysis is conducted withclosing linear closing and opening of the in 10 s.12 Figures andthe 13 analysis is conducted with linear and opening of the valve in valve 10 s. Figures and 1312 show show the sensitivity analysis for valve closing/opening time in linear progression for the optimized sensitivity analysis for valve closing/opening time in linear progression for the optimized values of values DS .closing/opening The valve closing/opening time is changed 10interval s to 60 sofat10an interval O and DO andof DSD . The valve time is changed from 10 s to 60from s at an s. For time of s. For time10ofs to closure from 10 s to 30water s, thelevel maximum water level and is almost the same and it of 10 closure from 30 s, the maximum is almost the same it gradually decreases gradually decreases a further increase in closing time, shown in 12. Similarly, for the for a further increasefor in closing time, as shown in Figure 12.as Similarly, forFigure the valve opening case, the valve opening case, the minimum water level gradually increases with an increase in valve opening minimum water level gradually increases with an increase in valve opening time, as shown in Figure time, as shown in Figure 13. Hence, as valvethe maneuver time increases, the maximum level 13. Hence, as valve maneuver time increases, maximum water level decreases and thewater minimum decreases and the minimum water level increases, and thus the effects of transients are slightly reduced. water level increases, and thus the effects of transients are slightly reduced. Generally, the effect of Generally, the effectclosing/opening of changing gradual of the effect valve in hassurge negligible effect in but surge changing gradual time closing/opening of the valve has time negligible analysis [15], in analysis [15], but in this case, the effect is significant. Hence, valve or guide vanes’ closing/opening this case, the effect is significant. Hence, valve or guide vanes’ closing/opening time is also an time is alsoparameter an important for the hydraulic optimization the for surge for and this study important forparameter the hydraulic optimization of the surge of tank thistank study shouldand be should be considered future optimization cases. An elaborate study the effects of changing closure considered in future in optimization cases. An elaborate study on the on effects of changing closure time time for maximum piezometric at bottom the bottom tunnel of the surge tank maximum water level for maximum piezometric headhead at the tunnel of the surge tank andand maximum water level in in the surge tank is performed, as shown in Figure 14. The pipe characteristic time for this case is 1.667 s the surge tank is performed, as shown in Figure 14. The pipe characteristic time for this case is 1.667 and closing s and closingofofthe thevalve valvebelow belowthis thistime timeindicates indicatesfast fastclosure closureofofthe thevalve. valve.The The maximum maximum pressure pressure head at the bottom tunnel of the surge tank is very high for fast closure and it quickly decreases as soon head at the bottom tunnel of the surge tank is very high for fast closure and it quickly decreases as as gradual closureclosure occursoccurs and gradually becomes almost almost constant after theafter valve reaches soon as gradual and gradually becomes constant theclosing valve time closing time 15 s. Contrary to that, the maximum water level inlevel the surge during fast closure is lower it reaches 15 s. Contrary to that, the maximum water in thetank surge tank during fast closure is and lower increases linearly till 2.5 s of valve closing time, and afterwards, the changes in the maximum water and it increases linearly till 2.5 s of valve closing time, and afterwards, the changes in the maximum level the surge are very small. difference of theseoftwo pressure heads for a closing of waterinlevel in thetank surge tank are very The small. The difference these two pressure heads for a time closing more than 10 s is less than 1 m, which indicates effective functioning of the surge tank. The effects time of more than 10 s is less than 1 m, which indicates effective functioning of the surge tank. The of fast closing of the valve more in tunnel pressure changes, while while for thefor surge tank, effects of fast closing of theare valve aresignificant more significant in tunnel pressure changes, the surge they comparatively less significant. tank,are they are comparatively less significant. Figure 12. 12. Surge Surgeanalysis analysisfor forgradual gradualclosure closure valve with varying of closure ofvalve the valve Figure of of thethe valve with varying timetime of closure of the (τc). (τc). Water 2019, 2019, 11, 11, 715 715 Water Water 2019, 11, 715 16 16of of 19 19 16 of 19 Figure 13. Surge analysis for gradual opening of the valve with varying time of opening of the valve Figure Surgeanalysis analysisfor forgradual gradual opening valve with varying of opening the Figure 13. 13. Surge opening of of thethe valve with varying timetime of opening of theofvalve (τo). valve (τo). (τo). Figure 14. Maximum piezometric head at the bottom tunnel tunnel of of the the surge surgetank tank(H (HP.max P.max)) and maximum Figurelevel 14. Maximum head at the bottom of the surge tank (HP.max) and maximum water in the surgepiezometric tank (ZS,max fordifferent different timestunnel ofclosure closure ofthe thevalve. valve. S,max) )for times of of water level in the surge tank (ZS,max) for different times of closure of the valve. 4. Conclusions 4. Conclusions 4. Conclusions Surge tanks are very important structures for a pressurized water conveyance system, Surge tanks are very important structures for a pressurized water conveyance system, which whichSurge prevents the effects of hydraulic in the system. Valve or guide which vane areharmful very important structures fortransients pressurized water prevents thetanks harmful effects of hydraulic transients ina the system. Valve orconveyance guide vanesystem, operation is a operation is aharmful very frequent in a hydropower or otherValve water conveyance withisaa prevents the effectsphenomenon ofahydraulic transients in the system. guide with vanesystem very frequent phenomenon in hydropower or other water conveyanceorsystem aoperation surge tank, surge tank, which causes harmful transients and the system should be robust enough to cope with very frequent in a hydropower or other waterbeconveyance systemtowith tank, which causes phenomenon harmful transients and the system should robust enough copea surge with such such hazardous conditions. The efficient design of a should surge tank is vital to ensuretomaximum benefit which causes harmful transients and the system be robust enough cope with hazardous conditions. The efficient design of a surge tank is vital to ensure maximum benefit such and and safety in the system. In this study, surge is carried out toensure examine vital parameters of hazardous conditions. ofanalysis a surge is vital maximum benefit and safety in the system. InThe thisefficient study, design surge analysis is tank carried out to to examine vital parameters of hydraulic transient caused by opening and closing of the valve. Further, optimization is done using safety in transient the system. In this study, surge analysis to examine vital parameters of hydraulic caused by opening and closing of is thecarried valve. out Further, optimization is done using PSO to investigate the best values of DO and Dclosing and enhance the performance of the surgeistank. Effects hydraulic transient caused by opening and of the valve. Further, optimization done using S PSO to investigate the best values of DO and DS and enhance the performance of the surge tank. Effects of various parameters hydraulic transients are studied and favorable conditions are discussed. PSO to investigate the during best values of DO and D S and enhance performance of the surge tank. Effects of various parameters during hydraulic transients are the studied and favorable conditions are The following are the conclusions made in this study: of various parameters during hydraulic transients are studied and favorable conditions are discussed. The following are the conclusions made in this study: discussed. The following are the conclusions made in this study: modelofofaageneralized generalizedhydropower hydropowersystem systemconsisting consistingof of an an upstream upstream reservoir, reservoir, an an orifice orifice 1.1. AAmodel 1. surge Asurge model of a generalized hydropower system consisting of an upstream reservoir, an tank, conduit pipes, and an outlet valve was designed and numerically analyzed using tank, conduit pipes, and an outlet valve was designed and numerically analyzed orifice using surge tank, conduit pipes, and an outlet valve was designed and numerically analyzed MOCininMATLAB. MATLAB.Surge Surgeanalysis analysisby byMOC MOCrevealed revealed the the variation variation of of water water level level in in the the surge surgeusing tank MOC tank MOC in MATLAB. Surge analysis by MOC revealed the variation of water level in the surge tank in different conditions and assisted in defining and modeling the required hydraulic parameters in different conditions and assisted in defining and modeling the required hydraulic parameters in different conditions and assisted in defining and modeling the required hydraulic parameters for optimization of the surge tank; for optimization of the surge tank; for optimization of the surge tank; Water 2019, 11, 715 2. 3. 4. 17 of 19 The maximum and minimum possible water level in the surge tank and damping of surge waves were considered as the important parameters for surge tank optimization. Analyses in this study show that these transient behaviors are highly conflicting in nature for different values of DO and DS . In addition, for certain values of DO and DS , the difference of maximum piezometric head at the bottom tunnel of the surge tank and maximum water level in the surge tank becomes unacceptable. Hence, a proper optimization method is necessary to investigate the best values of DO and DS to enhance the efficiency of the surge tank and minimize the effects of transients in the worst conditions; The Particle Swarm Optimization successfully optimized the values of DO and DS with a significant improvement in the maximum and minimum water level in the surge tank with reasonable damping of surge waves and the recommended surge tank was free from vortex formation. The overall performance of the surge tank is better than the contemporary methods; Based on the sensitivity analysis of the valve’s closing and opening time, significant changes were observed in surge analysis for different closing and opening times of the valve. This indicates that valve or guide vanes’ closing/opening time should also be considered during the hydraulic optimization of the surge tank in further cases. The difference of maximum piezometric head at the bottom tunnel of the surge tank and maximum water level in the surge tank was studied for various times of closure of the valve and it is concluded that the closing time of the valve results in more significant pressure changes in the penstock pipe and headrace tunnels, while in the surge tank, the effect is comparatively lower. This study establishes a foundation for the design of a hydraulically optimized restricted orifice surge tank using PSO. In future research, more parameters like location of the surge tank, valve closure time and mechanism, bifurcation of penstocks, etc., could be included in optimization. In addition to that, optimization of other types of orifices or other surge controlling devices could also be done. Author Contributions: K.P.B. and J.Z. conceptualized the research. K.P.B. defined the methodology and carried out the analysis and S.P. helped in numerical modeling. 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