Detailed Audit of the Energy Efficiency in Water Systems: New Performance Indices

Abstract

In this study, a new methodology is proposed with the aim of quantifying the energy benefits resulting from different operation strategies. A new set of performance indices is obtained to discern among the multiple effects on the energy use of each single intervention on the network operation. The new methodology is applied to two different cases: a pressure reducing strategy in a water district and a variable pumping strategy in the water supply of a network. The two presented case studies demonstrated the applicability of the proposed methodology. The new performance indices were found to be a reliable measure for the choice of the best technical solution.

Appendix A

Based on Eqs. (18) and (19), Eq. (21) can be reformulated as:

$$\begin{aligned} \epsilon _2 \ q_0 = \alpha \ (h_0 + \xi \ h_0)^{\beta } - \alpha \ (h_0)^{\beta } \end{aligned}$$

(A1)

Hence:

$$\begin{aligned} \epsilon _2 \ q_0 = \alpha \ (1 + \xi )^{\beta } \ (h_0)^{\beta } - \alpha \ (h_0)^{\beta } \end{aligned}$$

(A2)

$$\begin{aligned} \epsilon _2 \ q_0 = \alpha \ (h_0)^{\beta } \ ((1 + \xi )^{\beta } - 1) \end{aligned}$$

(A3)

$$\begin{aligned} \epsilon _2 = \frac{\alpha \ (h_0)^{\beta } }{q_0}\ ((1 + \xi )^{\beta } - 1) \end{aligned}$$

(A4)

$$\begin{aligned} \epsilon _2 = \epsilon _1 \ ((1 + \xi )^{\beta } - 1) \end{aligned}$$

(A5)

Equation (12) can be reformulated according to Eq. (A6).

$$\begin{aligned} \begin{aligned} P_n&= \gamma \ \Bigl (q_0 \ z_n + \epsilon _1 \ q_0 \ z_n + \epsilon _2 \ q_0 \ z_n + q_0 \ h_0 + \epsilon _1 \ q_0 \ h_0 + \epsilon _2 \ q_0 \ h_0 + q_0 \ \xi \ h_0 + \\&\quad + \epsilon _1 \ q_0 \ \xi \ h_0 + \epsilon _2 \ q_0 \ \xi \ h_0 \Bigr ) \end{aligned} \end{aligned}$$

(A6)

Hence:

$$\begin{aligned} P_n = \gamma \Bigl ( q_0 \ (z_n + h_0) + q_0 \ (z_n + h_0) \ (\epsilon _1 + \epsilon _2) + q_0 \ h_0 \ (\xi + \epsilon _1 \ \xi + \epsilon _2 \ \xi ) \Bigr ) \end{aligned}$$

(A7)

$$\begin{aligned} P_n = \gamma \Bigl ( q_0 \ (z_n + h_0) + q_0 \ (z_n + h_0) \ \epsilon _1 \ (1 + xi)^{\beta } + q_0 \ h_0 \ (\xi + \epsilon _1 \ \xi \ (1+ \xi )^{\beta }) \ \Bigr ) \end{aligned}$$

(A8)

The leakage ratio $\lambda$ in Eq. (26) is given by the following quantity:

$$\begin{aligned} \lambda = \frac{(q_{leak}^{h_n})}{q_n }= \frac{q_{leak}^{h_0}+q_{leak}^{\Delta h_n}}{(q_{min} + q_{leak}^{h_{min}} + q_{leak}^{\Delta h_n} )} \end{aligned}$$

(A9)

Based on Eqs. (20) and (21), Eq. (A9) can be reformulated as:

$$\begin{aligned} \lambda = \frac{\epsilon _1 \ q_{min} + \epsilon _2 \ q_{min}}{q_{min} + \epsilon _1 \ q_{min} + \epsilon _2 \ q_{min} } \end{aligned}$$

(A10)