Refining PSO Applied to Electric Energy Cost Reduction in Water Pumping - page 01

Materials and Methods

The pumping stations are responsible for much of the energy consumption and are the primary target in energy efficiency studies in the water supply sector. Studies suggest that on average, strategic operations with the objective of energy efficiency of hydraulic systems operations may lead to a 25% reduction in energy consumption. (Ramos, et al, 2012), (Moreira and Ramos, 2013).

Jowitt and Germanopoulos (1992) evaluated the electrical efficiency of motor-pump sets in a real network and considered the time variation of the demand. The authors used a simulation for an extensive period of time and used dynamic programming. The results were considered good, particularly the computational time spent in the study. Nevertheless, the treatments of the equations and restrictions of the hydraulic problem were arduous.

Cunha (2009) proposed real-time optimization of water delivery systems using binary optimization to represent the pump status problem. The GAlib generic algorithm was used as the optimization algorithm. According to Cunha (2009), although the search results for strategy routines (period of 24h) were good, the results were strongly influenced by penalties and operations of the optimization algorithm.

Ribeiro (2007) and Rodrigues (2007) modelled pumping systems using a frequency inverter and therefore searched for the best rotation for each hour of the day. Using a non-classical approach, the search for optimal rotations made it possible to work with continuous variables, which helped the search method. Both Ribeiro (2007) and Rodrigues (2007) used genetic algorithms as their optimization method. Ribeiro (2007) applied a fictitious network, whereas Rodrigues (2007) applied the model developed in a real network. Both authors obtained interesting values regarding the energy cost reduction. In the case of Ribeiro (2007), great improvement was also obtained by using variable-level reservoirs.

Finally, Al-Ani (2012) used MA-PSO (multi-agent PSO) with a bi-objective approach (pumping and maintenance costs minimization) to search for optimal routines for the motor-pump set. For the pump operation, this study modelled the stopping and start-up operations of pumps and thus only considered the working status of the machines. According to Al-Ani (2012), the algorithm proved to be efficient in reducing energy cost (approximately 9%) when applied to an actual network (Saskatoon West network).

By considering a previous study of the problem, the equations approached by this study are developed below, which presents the equations developed for continuous phase. Considering the change from nominal rotation (NR) to an inferior rotation (Ni) by frequency modifications, the equation can be written in dimensionless form, and the nominal frequency (f) of 60 Hz can be expressed as

For01     (1)

where fi is the frequency that defines rotation (Ni), and α is the associated dimensionless rotation.

The change from nominal rotation to any rotation modifies the pump operating point, which can be expressed by its physical similarity laws:

For02     (2)

where subscript p was used to specify the yield (η) associated with the pump, and subscript R is associated with the condition of nominal rotation. Q, H and P represent the flow rate, head and pump power, respectively.

The electric power P(α) required for pumping a flow rate Q for a specific head H for a given specific rotation α can be calculated as

For03     (3)

where γ is the specific weight of water. The yield ηt represents the total yield of the set with the motor, inverter, pump, etc. Moreno (2007) suggests the following composition for this yield:

For04     (4)

where ηp, ηm, ηsd, ηc, and ηl are the yields of the pump, motor, inverter, and cables, respectively, which are associated with the loss of head in the pump set-up line.

The minimum desired consumption of electrical energy can be expressed for a period (Pe) of equal time intervals (i) by

For05     (5)

Subject to

For06-10

where ci is the electric energy cost for time interval i. The power P(α)n,i is the electric power required for pumping to satisfy a demand Q by the system for a given period of time at a given head H of pump n to meet the operational requirements (Brentan, et al, 2013). Furthermore, pmin is the minimum dynamic pressure determined by the standard, pref is the pressure at a reference node at any time of the day, pmáx is the maximum static pressure determined by the standard, vmin is the minimum velocity in any tube of the facility determined by the standard, vref is the velocity in a reference tube at any time of the day, vmáx is the maximum velocity in any tube of the facility determined by the standard, Nk,mín is the minimum operational level of reservoir k of the facility, Nk is the level of reservoir k at any time of the day, and Nk,máx is the maximum operational level of reservoir k. Finally, nmanobra is the number of stops and startup pump operations performed in the day, and nmáx is the maximum number of manoeuvres allowed for the safety and maintenance of the equipment.