# Algorithm for Appropriate Design of Hydroelectric Turbines as Replacements for Pressure Reduction Valves in Water Distribution Systems

^{*}

## Abstract

**:**

## 1. Introduction

^{2}ratio, occupy large surfaces on rooftops to achieve a respectable amount of power output. Finally, the energy provided by hydro-turbine installations in WDSs is produced and mostly consumed locally as it is reasonable to assume that population density, water consumption and electricity demand are correlated [26,27].

## 2. Background

#### 2.1. Hydro-Turbine Performance-Characteristic H–Q Curves

^{3}/s), N (rpm) and H (m) determines the so-called “design point” or Best Efficiency Point (BEP) of the hydro-turbine, i.e., the point where the hydro-turbine is designed to operate at its highest efficiency. Euler’s fundamental equation for turbines relates the power absorbed from the water flow with the geometry and the velocities of the runner. Its general formulation is presented in Equation (2a). By rearranging Equation (2a) by means of introducing the geometrical properties of the runner, its mathematical formula shapes into Equation (2b). Equation (2b) applies to all reaction type turbines and becomes more accurate when blade thickness can be assumed negligible when compared with the annulus passage area over the entire blade zone [30]. The schematic representation of the velocity triangles at turbine inlet and outlet, along with the notation of the various velocity components and angles that are employed in Equation (2) are depicted in Figure 2.

^{2});

_{u}

_{1}, c

_{u}

_{2}is the tangential component of the absolute blade velocity at turbine inlet and outlet velocity triangles (m/s);

_{des}is the design flow rate (m

^{3}/s);

_{1}, u

_{2}are the peripheral blade velocity at turbine inlet and outlet velocity triangles (m/s);

_{2}is the outlet runner diameter (m);

_{1}, b

_{2}are the inlet and outlet blade width (m);

_{1}is the guide vane’s pre-swirl angle (in degrees);

_{2}is the blade outlet angle (in degrees).

_{o}) below which the hydro-turbine does not work as intended, but enters a mode of operation where it behaves neither as turbine nor as pump [33]. By changing the absolute flow angle at inlet to a bounded range and forming the equivalent η–Q and H–Q curves, the behavior of the same machine for different operating points (flow rates, heads and rotational speeds) can be assessed. Equation (3) suggests that the final characteristic H–Q curve, also known as the actual performance curve, results from the ideal Euler line if all forms of losses (i.e., hydraulic, volumetric, kinetic) are included. In practice, hydraulic losses (i.e., friction and shock losses) are the most consequential, and hence it is essential to introduce their mathematical expressions by Equations (4) and (5). Then, the hydraulic efficiency which accounts for these losses is calculated in Equation (6).

_{i}is the flow value away from the design point (m

^{3}/s);

_{h}is the hydraulic efficiency;

_{s}and ζ

_{f}are the shock and friction loss coefficients; respectively.

_{s}ranges between 0.5 and 0.7 [34] and ζ

_{f}includes the combined effect of both major (friction) and minor losses; their values are calculated in such a way that the actual curve of the estimated efficiency will cross the design point according to turbine and blade shape and structure. In practice, these various losses tend to generate characteristics whose maximum efficiency is reached at water flow values lower than the original design flow.

_{1}and since coefficients ζ

_{f}, and ζ

_{s}are considered constant at the fully turbulent region, hydraulic and shock losses are dependent solely on the flow [33].

#### 2.2. Similarity Laws

_{1}and N

_{2}represent different runner rotational speeds (rpm);

_{1}and Q

_{2}are the flow rates at speed N

_{1}and N

_{2}, respectively (m

^{3}/s);

_{1}and H

_{2}are turbine heads at speed N

_{1}and N

_{2}, respectively (m);

_{1}and P

_{2}are the mechanical power outputs at speed N

_{1}and N

_{2}, respectively (kW);

_{1}and D

_{2}represent the tip diameters of two different turbines (m).

_{1}= D

_{2}= constant at different operational conditions, then Equations (7)–(9) are grouped together into Εquation (10):

_{h}, ζ

_{s}as constant (see Section 2.1), losses in head δh

_{hydraulic}and δh

_{shock}can be computed for different operating conditions (Q

_{2}, N

_{2,}H

_{2}, η

_{2}), with respect to the initial design (Q

_{1}, N

_{1,}H

_{1}, η

_{1}), as follows:

_{2}is the total efficiency for rotational speed N

_{2,}while η

_{1}represents the total efficiency of the corresponding working point for rotational speed N

_{1}.

- As a power generating machine by focusing on the working points which, according to the current flow rate, maintain the maximum efficiency regardless of the resulting head-pressure drop at the outlet;
- As a water pressure regulator that keeps track of the working points that achieve the requested head-pressure drop at its outlet with the available flow rate, regardless of the resulting efficiency or power output.

#### 2.3. Simulation Approach

#### 2.4. Hydro-Turbine Realistic Design

- Traditionally, impulse turbines like Pelton or Turgo are inadequate for PRV replacement because their runner is not submerged in water, and they provide atmospheric pressure outflow (not controllable). Crossflow turbines can reduce water velocity, but they have extremely limited capability to reduce head [48];
- The expected water intake Q is rather limited in the case of WDSs. Radial turbines tend to work more efficiently than Kaplan turbines in such flows;
- In practice, PRVs installed in WDSs reduce the pressure by 1–10 bars (≈10–100 m of Head), which ranges within the Francis region as it is depicted in Figure 1;
- Francis hydro-turbines, among all other types, have the ability to cover the widest range of heads and volumetric flows;
- Francis turbines are single-regulated, and thus, compared to double-regulated axial type turbines like Kaplan, present a robust, economical and less complicated alternative in terms of runner and generator construction.

_{q}values and thus lead towards the radial flow hydro-turbine region, which typically ranges between 20 and 80 (see Figure 1). The selection of Francis turbines among turbine types based on the specific speeds between 20 and 80 is a straightforward process, and is justified by classic hydropower analysis. All experimental results and Computational Fluid Dynamic (CFD) simulations indicate that turbine type selection and design based on n

_{q}leads to the best performance-highest efficiency.

## 3. Methodology, Turbine Installation and Network Simulation

#### 3.1. General Methodology

- Active, by keeping the downstream pressure to its setting value when the upstream pressure is higher than this setting value;
- Inactive, if the downstream pressure is lower than its setting value.

- PRV active periods;
- Head and pressure values of the upstream and downstream nodes of each PRV to determine the working head of GPV (head loss curve) for each time period;
- Data concerning the exact water flow value passing through the corresponding pipe at each time period, in order to select the design flow rate accordingly;
- The range of the minimum and maximum water velocity allowed in the WDS pipelines, according to the regulation of each country or region. The minimum velocity should be high enough to prevent sedimentation and the maximum velocity should not be too high, in order to avoid the erosion of the pipeline and high head losses.

_{total}) is the sum of elevation (H

_{e}), pressure (H

_{p}) and kinetic (H

_{k}) head, as Equation (13) suggests:

^{3}),

^{2});

_{1}), which in turn affects the velocity triangle at turbine’s inlet, as it was shown in Figure 2.

_{1}, as Equation (2b) suggests. When the rotational speeds are too far from the design point, the corrective nature of Equation (12) should be employed. In this paper, without affecting the validity of the method, the rotational speed N

_{2}that appears in Equations (7)–(12) is allowed to vary from 0.5N

_{des}to 2.5N

_{des}. The proper calculation of losses is described in Section 2.1. The previous methodology, by means of a concise algorithmic process that couples both turbine design (MATLAB) and turbine model simulation (EPANET), is summarized in the next section.

#### 3.2. Algorithmic Process

#### 3.2.1. Turbine Design and Performance Prediction Algorithmic Process

^{upstream}− H

^{downstream}) is stored. At the same time, the total number of hours i and the exact set of hours within the day m(i) that the PRV is active are stored, before the algorithm proceeds to the next time period t. If the upstream pressure is less than or equal to the pressure setting, then the PRV, and successively the GPV/turbine, is considered as inactive. In this case, the algorithm skips the current time period t and advances to the next one (t + 1). When the final time period is reached (t = 24 h), the process continues with the calculation of the flow rate and head loss profile that is achieved during the PRV’s active periods and is later requested by the GPV/turbine operation. Next, according to the current pipe flow rate and PRV’s head loss range, the methodology explained in Section 2 is applied to determine the proper dimensioning and turbine working point. This eventually leads to the construction of the actual H–Q curve for the suggested design point. This H–Q curve is also constructed for different angle values α

_{1}, which are close to the design point value ${\alpha}_{1}^{des}$, by applying Equation (2b). By implementing the similarity laws around the design point, by means of Equations (7)–(11), the performance map of the hydro-turbine, i.e., H–Q and η–Q curves for different N and α

_{1}, are created. From these curves, only the non-negative H

_{o}–Q

_{o}pair values are accepted (see “turbine region” in Figure 3). At this final step the process terminates, and its output is stored as a performance map and is ready to be imported to the EPANET environment.

#### 3.2.2. EPANET-MATLAB Interaction Algorithmic Process

^{requested}− Q

^{achieved})/Q

^{achieved}|, δH = |(H

^{requested}− H

^{achieved})/H

^{achieved}|, δη = |(η

^{requested}− η

^{achieved})/η

^{achieved}|, respectively. For the given k, each H–Q curve is swept in order to trace the R number of candidate combinations {[H

_{c}(k), Q

_{c}(k)]

^{1},…, [H

_{c}(k), Q

_{c}(k)]

^{R−1}, [H

_{c}(k), Q

_{c}(k)]

^{R}} that approach the requested H

_{PRV}(k), Q

_{PRV}(k) values, under the given δQ and δH convergence value. In the meantime, the corresponding [N(k), α

_{1}(k), η

_{h}(k)]

^{1},…, [N(k), α

_{1}(k), η

_{h}(k)]

^{R−1}, [N(k), α

_{1}(k) and η

_{h}(k)]

^{R}values are stored. At the next step, depending on the current operating mode, the hydro-turbine can optimize its performance in two ways. It can be set to either maximize its efficiency, by operating as a power generating machine irrespective of the head loss that is achieved, or to minimize the head loss deviation by operating as a pressure regulating “device” that precisely mimics the PRV action. The δQ and δη are specified only during the “max efficiency mode” and δQ and δH during the “PRV mode”. Results where the minimum head loss deviation is achieved for H

_{c}> H

_{PRV}or Q

_{c}> Q

_{PRV}are accepted only when their percentage difference is lower than 5%. In any case, from the R number of combinations the proper H(k)–Q(k) curve is selected and then imported in EPANET’s GPV model and a single-period simulation is launched. This procedure repeats for all turbine working periods i. Before its termination, the algorithmic process provides access to all important results, e.g., head loss absolute difference between PRV and turbine action, energy produced along with the hydraulic efficiency, rotational speed, and angle α

_{1}variation (see Section 4).

## 4. Application Example

- Presents both branched and loop configurations;
- Consists of classic infrastructure and components, such as reservoirs, tanks, pipelines, pumps, etc., with typical sizing;
- Supplies consumers who display representative water demand requirements and demand patterns;
- Experiences high pressure in certain areas;
- Features conventional control pressure mechanisms such as PRVs.

#### 4.1. PRV Replacement with Turbines

^{3}/h, with a head loss ranging between 3 and 16 m. The flow through the pipe that feeds GPV-3 ranges between 9.72 and 54 m

^{3}/h, with a head loss ranging between 22 and 26 m. The head loss requirements for GPV-2 and GPV-3 are summarized in Figure 10 by subtracting the corresponding upstream and downstream head values of Figure 8. The significant fluctuations in both flow and head indicate that the hydro-turbines are not expected to always work under design point conditions. In Figure 11, a set of characteristic H–Q curves and η–Q curves are presented for the cases of GPV-2 and GPV-3. These curves are constructed for design rotational speed N

_{des}and values α

_{1}and η

_{h}around their corresponding design point values α

_{des}and ${\eta}_{h}^{des}$. Under this consideration, the H–Q curves’ starting point is calculated in Equation (14) by setting Q = 0 to Equations (2.b), (4) and (5), respectively, and deploying Equation (3):

_{s}is considered constant at the fully turbulent region (see Section 2.1), H

_{min}has a fixed, but different, value for each GPV case. In this paper, due to the constant variation in the turbines’ operating conditions, multiple equivalent curves were constructed for different runner rotational speeds.

#### 4.2. Turbine Placement at a Location Suffering from High Pressure without PRV Installed

_{des}and inlet angles α

_{1}and efficiency values n

_{h}around the design point, is displayed in Figure 14.

#### 4.3. Results and Discussion

_{h}) and the rotational speed (N) at which the requested head loss values are accomplished by the turbine. Each bar in these graphs corresponds to the time periods during which the PRVs were active (i.e., downstream water pressure is above the defined PRV setting) and therefore the time periods during which the hydro turbine is requested to operate.

_{h}that each hydro-turbine was originally designed for, the mean efficiency η

_{mean}that derives from all time periods does not deviate significantly. For both GPV-2 and GPV-3 cases, η

_{mean}is higher than the efficiency values (30–60%) that are typically encountered in micro sites, as is suggested by [18,55]. Especially for the GPV-3 case, the resultant efficiency and power production are considerably greater than PAT systems operating in similar flow rate conditions [10]. For GPV-2 the resultant mean efficiency value is larger than the design efficiency since the corresponding turbine, for the first half of its working period (t = 1–3), operates at flow rate, head and rotational speed values that are far from the selected design point. At these operating conditions the resultant hydraulic efficiency is significantly larger (70–80%) than its design point value (52%), which is approximately achieved during the second half of its working period (t = 4–6).

_{1}with respect to the original design point value α

_{des}is displayed for each time period and GPV. It is clear that for the GPV-3 case the selected design point is considered adequate, since the α

_{1}variation is most of the time close to the design value, and thus close to BEP. Conversely, for the GPV-2 case, the energy potential is so low that any related improvement in design and performance is considered unprofitable.

## 5. Conclusions

^{achieved}) by matching it with the total available head that is requested by the PRV (H

^{requested}) with low errors (2% maximum). At the expense of computational time, these errors could potentially be further improved if a smaller step during consecutive values of α

_{1}and N is used. This work proves that a Francis turbine in a WDS can handle extreme variations in flow rate and head by fully exploiting the available head, with high efficiency in a wide range of working points and not just close to the BEP. The average efficiency displayed during the PRV replacement scenarios (GPV-2 and GPV-3) is superior to efficiencies presented in relevant research for micro sites.

- From a WDS standpoint, the replacement of a PRV with a suitable hydro-turbine does not affect the former network operational conditions, i.e., flow and pressure, or disrupt its functionality; therefore, it does not add any operating challenges to the actual network. The limitation of the excessive pressure head in certain regions can only work beneficially for the network as it leads to the restriction of water hammer occurrences that are often considered as the main pipeline failure cause;
- From a consumer standpoint, the service level is preserved when replacing a PRV with a hydro-turbine, or even upgraded in cases where a PRV is absent and a hydro-turbine is purposefully installed in a certain consumer area, in order to secure that this area will witness normative water pressure values. Ιn this latter case, the resulting pressure drop can also lead to scaled-down pipeline head losses, i.e., less total energy losses and reduced pipeline stress.

- is fed the electrical grid with predictable and nearly constant amount of power;
- is consumed locally;
- does not create any visual or audible nuisance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BEP | Best efficiency point |

BPT | Brake-pressure tank |

CFD | Computational Fluid Dynamics |

DDA | Demand driven analysis |

GPV | General purpose valve |

GVO | Gate vane opening |

PAT | Pump as turbine |

PDA | Pressure driven analysis |

PRV | Pressure reduction valve |

RES | Renewable energy source |

WDS | Water distribution system |

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**Figure 2.**Basic form of inlet and outlet velocity triangles of a typical reaction turbine [31].

**Figure 3.**Typical H–Q curve for reaction hydro-turbines [20].

**Figure 11.**Turbine constant speed performance curves as a function of water flow rate and angle α

_{1}values for the cases of GPV-2 and GPV-3.

**Figure 13.**Upstream total head and water pressure before the GPV-4 installation and GPV-4 head loss requirement.

**Figure 14.**Turbine constant speed performance curves as a function of water flow rate and angle α

_{1}values for the case of GPV-4.

**Figure 16.**Turbine mechanical power output, efficiency, and rotational speed for each active time period.

Network Characteristics | Value |
---|---|

Number of total pipes | 1043 |

Number of branch pipes | 473 |

Number of junction nodes | 917 |

Number of reservoirs | 2 |

Number of tanks | 13 |

Number of pumps | 13 |

Number of serviceable population | 30,681 |

Ratio (Residential/total costumers) | 0.92 |

Ratio (Residential/total water usage) | 0.84 |

Estimated Annual Water Loss (%) | 6 |

Water cost per 3.78 m^{3} | 6–8 |

Pipe Diameter (mm) | Total Length (km) | # of Pipes | Pipe Material for Locations |
---|---|---|---|

25.4 | 0.413 | 9 | x |

50.8 | 17.79 | 105 | x |

76.2 | 26.74 | 42 | x |

101.6 | 104.34 | 188 | GPV-4 (vitrified clay) |

152.4 | 207.49 | 307 | PRV-2 + PRV-3 (PVC) |

203.2 | 49.54 | 293 | x |

304.8 | 6.34 | 53 | x |

Design Point Data. | GPV-2 | GPV-3 | GPV-4 |
---|---|---|---|

volumetric flow rate, Q (m^{3}/h)head, H (m) rotational speed, N (rpm) | 1.58 | 36 | 6.84 |

2.2 | 18.5 | 13.4 | |

1190 | 2410 | 1950 | |

specific speed, n_{q}hydraulic efficiency, η _{h} (%) | 14 | 27 | 12 |

52 | 87 | 55 | |

Number of blades guide vane opening angle, α _{1} (°)inlet diameter, D _{1} (mm)outlet diameter, D _{2} (mm) | 6 | 6 | 6 |

4.85 | 10.3 | 3.8 | |

102 | 152 | 152 | |

34.7 | 66.4 | 50 | |

D_{2}/D_{1} ratiomean efficiency, η _{mean} (%) | 0.34 | 0.43 | 0.33 |

62 | 72 | 50 |

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## Share and Cite

**MDPI and ACS Style**

Bideris-Davos, A.A.; Vovos, P.N.
Algorithm for Appropriate Design of Hydroelectric Turbines as Replacements for Pressure Reduction Valves in Water Distribution Systems. *Water* **2023**, *15*, 554.
https://doi.org/10.3390/w15030554

**AMA Style**

Bideris-Davos AA, Vovos PN.
Algorithm for Appropriate Design of Hydroelectric Turbines as Replacements for Pressure Reduction Valves in Water Distribution Systems. *Water*. 2023; 15(3):554.
https://doi.org/10.3390/w15030554

**Chicago/Turabian Style**

Bideris-Davos, Admitos A., and Panagis N. Vovos.
2023. "Algorithm for Appropriate Design of Hydroelectric Turbines as Replacements for Pressure Reduction Valves in Water Distribution Systems" *Water* 15, no. 3: 554.
https://doi.org/10.3390/w15030554