Objective Calculators

Three separate modules of PEPSO are used to calculate independently three objective values: electricity cost, pollution emission and total penalty of each solution. Before calculating objective values, solutions should be simulated hydraulically and the energy consumption, peak power demand, water level in tanks, water pressure and velocity at junctions and pipes and pump schedule characteristics (e.g. number of pump switches) at all time steps are stored in a temporary file.

The electricity cost calculator module receives total energy consumption of all the pumps that are connected to a meter at each optimization time step. The calculator then multiplies the energy consumption value with the corresponding energy consumption

charge to calculate the energy consumption cost associated with that electricity meter. It similarly uses the peak power demand of all pumps that are connected to the meter to calculate associated power demand cost.

The pollution emission calculator module uses the energy consumption of each electricity meter at each optimization time step. It multiplies the energy consumption value (kWh) by the corresponding emission factor (lb/kWh) to calculate the emission pollution weight (lb). It should be noted that values of emission factors depend on the time of energy consumption and location of the electricity meter (pump station). LEEM reports the marginal emission factor which is equal to the amount of pollution emission due to one unit increase in energy consumption of the whole region. We cannot multiply the total energy usage in the region by marginal emission factor to calculate the pollution emission of the region. For this purpose, we need to use pollution emission data of the all the energy generators that provide energy of the region (not only the marginal generator). However, it is justifiable to assume that the total energy consumption of a WDS is relatively negligible in comparison with the total energy consumption of a region and is not able to change the marginal generator. In this case, the marginal emission factor can directly be multiplied by the total energy consumption of the WDS to calculate its pollution emission. Theoretically the resulted emission value is not equal to the total pollution emission of the real system. However, the calculated emission value by this method (by using marginal emission factors) can be used for comparing different operational scenarios. The difference between resulted emission values of scenarios shows change of the total emission of the real system due to change of the operational scenario.

The third module calculates associated penalties of each solution (pump schedule). Based on the user request, the total penalty may include pump operation

penalty, tank level penalty, junction pressure penalty and velocity penalty. So it is possible for users to turn off a constraint and the associated penalty will not be calculated and will not affect the optimization process. For each time step, the penalty is calculated as a violation value of the parameter raised to a predefined factor. By default, the factor is two. This helps to amplify the importance of the deviation from the acceptable range as the deviation increases. For instance, if the acceptable range of tank level is from 1 to 5.5 meter, and if the tank level goes up to 7 meters the violation can be calculated as |7 - 5.5| = 1.5 meter and the penalty is 1.52 _{= 2.25. If, at another time step, tank level is 0 meter}

the violation is |0 - 1| = 1 meter and penalty is 12 _{= 1. It can be seen that although the}

violation of the first case is 1,5 times more than the violation of the second case, the penalty of the first case is 2.25 times more that the second case. This way of using the power factor to increase penalty when the violation is large, help ensure that PEPSO will recognize unfeasible solutions. For instance, if there are 100 junctions in a system with 3 meters excess pressure for each of them, this solution is physically more acceptable in respect to the same system that has pressure violation on just one junction, but the amount of violation is 200 meter. Although 100 × 3 = 300 meters violation is more than 1 × 200 = 200 violation, but the 200 meters violation may cause pipe breakage, so the second scenario is not as feasible and acceptable. In this case raising the penalty to a power greater than one (e.g. two) let PEPSO see that 100 × 32 _{= 900 is way smaller than1}

× 2002 _{= 40000.}

Although penalty power factor of two is a default value of PEPSO, some simulation showed that sometimes the effect of the power factor of two is considerably severe, and 1.5 may be a more reasonable value that may better guide PEPSO to discover more feasible solutions.

Penalties that are associated with low limit violations are treated differently than penalties that are associated with high limit violations. These are treated differently and stored separately because they have different implications, and different policies should be implemented to reduce them. For instance, if PEPSO faces a high penalty value that is associated with the excess pressure, it might need to turn some pumps off to reduce the pressure of the system. On the other hand, if the same amount of penalty is related to insufficient pressure, PEPSO might need to turn some pumps on to increase pressure and tackle the issue.

PEPSO provides a flexible option for calculating penalties of strategic junctions and pipes that allows users to control the effect of violation of pressure or velocity of each component on the optimization process. For instance, In a WDS, controlling pressure of one junction might be more important than the others. In this case, users can increase pressure constraint importance multiplier of the desired junction to increase the effect of its penalty on the optimization process. As it was explained in section 2.2.1.5, these multipliers can be adjusted for each strategic junction and pipe and will be multiplied by calculated penalty of each junction or pipes before adding them up to calculate the total penalty. By default, these multipliers are 1.0, which means the violation of all strategic junction and pipes have the same effect on the optimization process.

If pump operation shows some violation regarding the defined pump constraints (e.g. number of pump switches in a day), the total penalty value will be increased one unit. Most of the time other penalties like tank level or junction pressure penalties are more important than pump operation penalties. So small pump operation penalties will not affect the optimization process unless other penalties are negligible.