# An optimization model for distribution system reinforcement integrated uncertainties of photovoltaic systems

## Abstract

In this paper, a two-stage model for the optimal planning of distribution systems considering uncertainties of photovoltaic systems is presented. The uncertainties in intermittent power of photovoltaic are taken as input variables by a probability distribution function. The proposed model determines the optimal sizing and timeframe of the equipments (feeders and transformer substations) in distribution systems. Therefore, the optimal displacement, sizing and installation period of PV are also specified. The objective function is minimizing the life-cycle costs of the planning project. The technical constraints are used to guarantee the operability of the distribution system including alternating current power flow, feeders upgrading section, substations upgrading capacity, limited of nodal voltage and PV capacity. The binary variables are also employed in the model in order to represent the cost function of the equipment as well as the investment and upgrade decisions. The algorithm is programmed in GAMS environment. The feasibility and effectiveness of the proposed model are verified in a 15-bus test system.

## Keywords

Distribution systems planning Photovoltaic (PV) Life-cycle cost (LCC) Optimization## List of symbols

*N*Set of buses in distribution system

*i*,*j*Bus (

*i*,*j*\(\in \)*N*)- \({N}_{\mathrm{L}}\)
Set of load buses in distribution system

- \({N}_{\mathrm{S}}\)
Set of substation buses in distribution system

- \({N}_{\mathrm{PV}}\)
Set of PV buses in distribution system

*t*,*T*Planning year and overall planning period (\(t \in T\))

*h*,*H*Hour and hours per day (\(h \in H\))

*s*, SSSeason and total seasons in a year (\(s \in \) SS)

*g*,*G*State and total sates each hour

*h**R*Discount rate (%)

- \({C}^{\mathrm{FF}}\)
Fixed capital cost of feeder ($/km)

- \({C}^{\mathrm{FC}}\)
Variable capital cost of feeder \((\$/\hbox {km\,mm}^{2})\)

- \({L}_{{i,j}}\)
Length of feeder (km)

- \({Y}_{{ i,j,t}}\), \(\theta _{{i,j,t}}\)
Magnitude and angles of admittance matrix element (pu)

- \({C}^{\mathrm{SF}}\)
Fixed capital cost of substation ($/substation)

- \({C}^{\mathrm{SC}}\)
Variable capital cost of substation ($/MVA)

- \({C}_{{i}}^{\mathrm{PV}}\)
New investment cost for PV

*i*($/M)- \(\rho _h^{\mathrm{PS}} \)
Active power purchased cost from market ($/kWh)

- \(\rho _h^{\mathrm{QS}} \)
Reactive power purchased cost from market ($/kVAh)

- \(\rho _{P\cdot h}^{\mathrm{DG}} \)
O&M cost of PV ($/kWh)

- \(\hbox {PD}_{{i,s,t,h}}\)
Active power demand buses (kW)

- \(\hbox {QD}_{{i,s,t,h}}\)
Reactive power demand buses (kVAr)

- \(P_{\mathrm{max}\cdot i}^{\mathrm{PV}} \)
Maximum power limit of PV

*i*(MW)- \(P_{\mathrm{max}\cdot i}^{*\mathrm{PV}} \)
New maximum power limit of PV in second stage (MW)

- \(F_{{ij},t}^*\)
Standard section of feeder in a planning year

*t*(mm\(^{2})\)- \(S_{{ij},t}^{*{F}} \)
Maximum capacity need upgrading of feeder (MVA)

- \(\Delta S_{\min }^{{ F}} \)
Capacity ramp-up limit of feeder (MVA)

- \(S_{\mathrm{max}\cdot ij,t}^{{ F}} \)
Maximum capacity limit of standard feeder (MVA)

- \(S_{{ i},t}^{*{S}} \)
Maximum capacity need upgrading of substation (MVA)

- \(\Delta S_{\min }^{{S}} \)
Capacity ramp-up limit for substation (MVA)

- \(S_{\mathrm{max\cdot }i,t}^{{S}} \)
Maximum capacity limit of standard substation in planning year

*t*(MVA)*J*Current density at thermal limit (A/mm\(^{2})\)

*M*Big number used maximum limit of variables in MIP (Mixed Integer Programming) and MINLP (Mixed Integer Nonlinear Programming) models

- \({U}_{\mathrm{max}}\)
Maximum voltage limit of bus (pu)

- \({U}_{\mathrm{min}}\)
Minimum voltage limit of bus (pu)

- \(\Delta {P}^{\mathrm{PV}}\)
Active power ramp-up limit of PV (MW)

- \(k_{s,h}^{\mathrm{PV}} \)
Output power factor of PV in season

*s*, hour*h*- \({k}_{\mathrm{P}}\)
Variation factor of the price of electricity

- \({D}_{{S}}\)
Total day per season

- \(\xi \)
Emission coefficient of traditional energies

- \(\beta \)
Emission tax

- \({C}_{{g}}\)
Factor of PV power in state

*g*- \(\lambda _{{g}}\)
Probability of PV power in state

*g*- \(t_{\text {d}}^{{ F}} ,t_{\text {d}}^{{S}} ,t_{\text {d}}^{\mathrm{PV}} \)
Time for depreciation equipment

- \(F_{{ij},t}\)
Upgrading section of feeders \((\hbox {mm}^{2})\)

- \(\Delta S_{{i},t}^{{S}} \)
Addition capacity for substations (MVA)

- \(P_{{ i},t}^{\mathrm{PV}}\)
New investment capacity of PV (MW)

- \(P_{{ i},s,t,h}^{{S}} \)
Active power purchased from electricity market (kW)

- \(Q_{{i},s,t,h}^{{S}} \)
Reactive power purchased from electricity market (kVAr)

- \(\Delta S_{{ ij},t}^{{F}}\)
Addition capacity of feeders (MVA)

- \(P_{{ i},s,t,h}^{\mathrm{PV}} \)
Active output power of PV (kW)

- \({U}_{{ i,s,t,h}}\)
Voltage of buses (pu)

- \(\delta _{{i,s,t,h}}\)
Voltage angle of buses (pu)

- \(\alpha _{{ij},t} \)
Binary variable of feeder upgrade decision (1/0)

- \(\gamma _{i,t} \)
Binary variable of substation upgrade decision (1/0)

## Notes

### Acknowledgements

The study was supported by Thai Nguyen University of Technology (TNUT), Thai Nguyen, Viet Nam.

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