Optimizing Photovoltaic Panel Quantity for Water Distribution Networks

TL;DR

The paper tackles optimal PV sizing for grid-connected pumped WDNs by coupling a probabilistic PV power model with a stochastic economic MPC for pump scheduling and a derivative-free Nelder-Mead search to minimize lifecycle costs. It introduces a multiplicative correction term in the PV power model to capture intraday fluctuations and uses year-long PV realizations to estimate total costs under uncertainty. Applied to Randers, Denmark, the method yields a around 14.5% reduction in network costs at an optimal PV capacity near 262 kW, with grid costs well captured by an exponential trend. The work demonstrates that adaptive pump scheduling paired with a probabilistic PV model can inform PV installation decisions and offers directions for faster surrogates and improved demand modeling.

Abstract

The paper introduces a procedure for determining an approximation of the optimal amount of photovoltaics (PVs) for powering water distribution networks (WDNs) through grid-connected PVs. The procedure aims to find the PV amount minimizing the total expected cost of the WDN over the lifespan of the PVs. The approach follows an iterative process, starting with an initial estimate of the PV quantity, and then calculating the total cost of WDN operation. To calculate the total cost of the WDN, we sample PV power profiles that represent the future production based on a probabilistic PV production model. Simulations are conducted assuming these sampled PV profiles power the WDN, and pump flow rates are determined using a control method designed for PV-powered WDNs. Following the simulations, the overall WDN cost is calculated. Since we lack access to derivative information, we employ the derivative-free Nelder-Mead method for iteratively adjusting the PV quantity to find an approximation of the optimal value. The procedure is applied for the WDN of Randers, a Danish town. By determining an approximation of the optimal quantity of PVs, we observe a 14.5\% decrease in WDN costs compared to the scenario without PV installations, assuming a 25 year lifespan for the PV panels.

Figures (5)

• Figure 1: Overall method for optimizing the PV Quantity. The Nelder Mead method is used to update the PV quantity iteratively. For each PV quantity, a synthetic historical PV power data set is generated using a physical PV model and historical weather measurements. The PV power model is then fitted to the synthetic data, and PV samples are generated from this model. An approximate WDN model (see \ref{['eq:reducedModel']}.) is simulated over the duration of PV samples treating the PV samples as the actual PV power generations powering the network. To simulate the network a stochastic controller-based pump scheduling method is used. Since future PV power values are unknown when pump schedules are calculated, the projections of future PV power production obtained from the PV model are used by the stochastic controller. Following the simulations, the total cost including the costs of power bought from the grid and PV installation, and maintenance costs throughout the lifespan of PVs is calculated. The PV quantity is updated according to the total cost using the Nelder Mead method.
• Figure 2: An illustration of the $g(\tau)$ fitted to the time series of the daily maximum power production $\max(X_{\tau})$ of the synthetic PV data generated with $P_{STC}=1$ kW.
• Figure 3: Water Distribution Network of Randers. The pumping stations to be controlled are shown in red and the remaining in blue. Tanks are shown with a 'T' shaped symbol in yellow.
• Figure 4: (a) Total cost of the WDN over the lifespan $\ell_{pv}=25$ of the PV panels including the installation and maintenance costs of the PV panels and the electricity costs of the WDN for a given amount of PVs, $J_{c}(x)+\ell_{pv}J_{o}(x)$. (b) The total costs of power bought from the grid and an exponential function fitted to the data. (c) Total cost of the WDN and the ratio of mean PV production of sampled yearly PV profile and the installed PV amount.
• Figure 5: Total costs for different lifespans of PVs $\ell_{pv}=25,\ell_{pv}=30,\ell_{pv}=35$. The red dashed lines represent the approximated optimal quantity of PV panels, obtained by fitting an exponential function to network costs and minimizing the expected total cost based on this exponential function.