# Dynamic economic dispatch with demand-side management incorporating renewable energy sources and pumped hydroelectric energy storage

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## Abstract

This paper recommends chaotic fast convergence evolutionary programming (CFCEP) for solving real-world dynamic economic dispatch (DED) with demand-side management (DSM) incorporating renewable energy sources and pumped-storage hydroelectric unit. Here, solar–wind–thermal energy system has been considered taking into account pumped-storage hydroelectric unit and uncertainty of solar and wind energy sources. DSM programs reduce cost and boost up power system security. To investigate the upshot of DSM, the DED problem is solved with and without DSM. In the recommended technique, chaotic sequences have been applied for acquiring the dynamic scaling factor setting in FCEP. The efficiency of the recommended technique is revealed on two test systems. Simulation outcomes of the suggested technique have been matched against those acquired by fast convergence evolutionary programming (FCEP), colonial competitive differential evolution and heterogeneous strategy particle swarm optimization. It has been observed from the comparison that the recommended CFCEP technique has the ability to give better-quality solution.

## Keywords

Pumped-storage hydroelectric unit Ramp rate limits Demand-side management Wind power uncertainty Solar power uncertainty## List of symbols

- \( F_{C} \)
Cost function

- \( a_{si} ,b_{si} ,c_{si} ,d_{si} ,e_{si} \)
Cost coefficients of \( i \)th thermal generator

- \( { P}_{sit} \)
Output power of \( i \)th thermal unit at time \( t \)

- \( {P}_{si}^{\min } ,{P}_{si}^{\max } \)
Lower and upper generation limits for \( i \)th thermal generator

- \( UR_{i} ,DR_{i} \)
Ramp-up and ramp-down rate limits of the \( i \)th thermal generator

- \( { P}_{wkt} \)
Available wind power of \( k \)th wind-power-generating unit at time \( t \)

- \( { P}_{wk}^{\min } ,{ P}_{wk}^{\max } \)
Lower and upper generation limits for \( k \)th wind-power-generating unit

- \( { P}_{wrk} \)
Rated wind power of \( k \)th wind-power-generating unit

- \( {\rm K}_{wk} \)
Direct cost coefficient for the \( k \)th wind power generator

- \( O_{wkt} \left( {{ P}_{wkt} } \right) \)
Reserve cost function due to overestimation of \( k \)th wind power generator at time \( t \)

- \( U_{wkt} \left( {{ P}_{wkt} } \right) \)
Penalty cost function due to underestimation of \( k \)th wind power generator at time \( t \)

- \( u_{wk} ,o_{wk} \)
Penalty cost and reserve cost for the \( k \)th wind power generator

- \( f_{w} \left( y \right) \)
Weibull probability distribution function of wind power \( y \)

- \( { P}_{wkt}^{\min } \)
Minimum power output of \( k \)th wind power generator at time \( t \)

- \( v_{in} \)
Cut in wind speed

- \( v_{out} \)
Cut out wind speed

- \( v_{r} \)
Rated wind speed

- \( v_{wt} \)
Forecasted wind speed at time \( t \)

- \( { P}_{PVmt} \)
Power output from \( m \)th solar PV plant at time \( t \)

- \( { P}_{sr} \)
Equivalent rated power output of the PV generator

- \( G \)
Solar irradiation forecast

- \( G_{std} \)
Solar irradiation in the standard environment

- \( R_{c} \)
A certain irradiation point

- \( {\rm K}_{sm} \)
Direct cost coefficient for the \( m \)th solar PV plant

- \( O_{PVmt} \left( {{ P}_{PVmt} } \right) \)
Reserve cost function due to overestimation of the \( m \)th solar PV plant at time \( t \)

- \( U_{PVmt} \left( {{ P}_{PVmt} } \right) \)
Penalty cost function due to underestimation of the \( m \)th solar PV plant at time \( t \)

- \( u_{PVm} ,o_{PVm} \)
Penalty cost and reserve cost for the \( m \)th solar PV plant

- \( f_{PV} \left( x \right) \)
Log-normal probability distribution function of solar power \( x \)

- \( { P}_{PVmt}^{\min } \)
Minimum power output of \( m \)th solar PV plant at time \( t \)

- \( { P}_{PVmt}^{\max } \)
Maximum power output of \( m \)th solar PV plant at time \( t \)

- \( { P}_{ghjt} \)
Power generation of \( j \)th pumped-storage plant at time \( t \)

- \( { P}_{phjt} \)
Pumping power of \( j \)th pumped-storage plant at time \( t \)

- \( { P}_{ghj}^{\min } ,{ P}_{ghj}^{\max } \)
Minimum and maximum power generation limits of \( j \)th pumped-storage plant

- \( { P}_{phj}^{\min } ,{ P}_{phj}^{\max } \)
Minimum and maximum pumping power limits of \( j \)th pumped-storage plant

- \( Q_{ghjt} \left( {{ P}_{ghjt} } \right) \)
Discharge rate of \( j \)th pumped-storage plant at time \( t \)

- \( Q_{phjt} \left( {{ P}_{phjt} } \right) \)
Pumping rate of \( j \)th pumped-storage plant at time \( t \)

- \( Q_{spent,TOT,j} \)
Total water amount spent for generation of \( j \)th pumped-storage plant

- \( Q_{pump,TOT,j} \)
Total pumped water amount of \( j \)th pumped-storage plant

- \( Q_{net,spent,j} \)
Net spent water amount by \( j \)th pumped-storage hydroelectric unit during operation cycle

- \( V_{res,jt} \)
Water volume in upper reservoir of \( j \)th pumped-storage plant at time \( t \)

- \( V_{res,j}^{\min } ,V_{res,j}^{\max } \)
Minimum and maximum upper reservoir storage limits of \( j \)th pumped-storage plant

- \( V_{res,j}^{start} ,V_{res,j}^{end} \)
Specified starting and final stored water volumes in upper reservoir of \( j \)th pumped-storage plant

- \( Inc^{\max } \)
Maximum increased load at any hour (MW)

- \( L_{Base,t} \)
Forecasted base load at time \( t \)

- \( DR_{t} \)
Percentage of forecasted base load participated in DRP at time \( t \)

- \( Inc_{t} \)
Amount of increased load at time \( t \)

- \( Ls_{t} \)
Shiftable load at time \( t \)

- \( { P}_{Lt} \)
Total transmission line losses at time \( t \)

- \( t,T \)
Time index and scheduling period

- \( {\rm T}_{gen} \)
Set that contains all time intervals where pumped-storage plant operated in generation mode

- \( {\rm T}_{pump} \)
Set that contains all time intervals where pumped-storage plant operated in pumping mode

- \( { N}_{t} \)
Number of thermal-power-generating units

- \( { N}_{w} \)
Number of wind-power-generating units

- \( { N}_{PV} \)
Number of solar PV plants

- \( { N}_{Pump} \)
Number of pumped-storage plants

## Notes

## References

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