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

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## 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.

## References

- 1.McDonald JD, Wojszczyk B, Flynn B, Voloh I (2013) Distribution systems, substations, and integration of distributed generation. The encyclopedia of sustainability science and technology. Springer, BerlinGoogle Scholar
- 2.Willis HL (2004) Power distribution planning reference book. Marcel Dekker, New York CityCrossRefGoogle Scholar
- 3.Georgilakis PS, Hatziargyriou ND (2015) A review of power distribution planning in the modern power systems era: models, methods and future research. Electr Power Syst Res 121(2015):89–100CrossRefGoogle Scholar
- 4.Gonen T, Ramirez-Rosado IJ (1986) Review of distribution system planning models: a model for optimal multistage planning. In: IEEE proceedings C—generation, transmission and distribution, vol 133Google Scholar
- 5.Albadi MH, El-Saadany EF (2008) The role of distributed generation in restructured power systems. In: Proceedings \(40^{{\rm th}}\) North American power symposium, (2008) NAPS ’08. Calgary, AB, CanadaGoogle Scholar
- 6.Muneer W, Bhattacharya K, Cañizares C (2011) Large-scale solar PV investment models, tools and analysis: the Ontario case. IEEE Trans Power Syst 26:2547–2555CrossRefGoogle Scholar
- 7.Medina A, Hernandez JC, Jurado F (2006) Optimal placement and sizing procedure for PV systems on radial distribution systems, distribution systems. In: International conference on power system technology, Chongqing, ChinaGoogle Scholar
- 8.Ackermann T, Andersson G, Söder L (2001) Distributed generation: a definition. Electr Power Syst Res 57:195CrossRefGoogle Scholar
- 9.Zhou Z, Zhang J, Liu P, Li Z, Georgiadis MC, Pistikopoulos EN (2013) A two-stage stochastic programming model for the optimal design of distributed energy systems. Appl Energy 103:135–144CrossRefGoogle Scholar
- 10.Li Z, Zang C, Zeng P, Yu H (2016) Combined two-stage stochastic programming and receding horizon control strategy for microgrid energy management considering uncertainty. Energies 9:499Google Scholar
- 11.Wong S, Bhattacharya K, Fuller JD (2007) Comprehensive framework for long-term distribution system planning. In: Proceedings IEEE PES annual general meeting, Tampa, USA, p 2007Google Scholar
- 12.Algarni AAS, Bhattacharya K (2009) A novel approach to disco planning in electricity markets: mathematical model. In: Power systems conference and exposition, 2009. PSCE ’09. IEEE/PESGoogle Scholar
- 13.El-Khattam W, Hegazy Y, Salama M (2005) An integrated distributed generation optimization model for distribution system planning. IEEE Trans Power Syst 20(2):1158–1165CrossRefGoogle Scholar
- 14.Porkar S, Abbaspour-Tehrani-fard A, Poure P et al (2011) Distribution system planning considering integration of distributed generation and load curtailment options in a competitive electricity market. Electr Eng 93:23CrossRefGoogle Scholar
- 15.Wong S, Bhattacharya K, Fuller JD (2009) Electric power distribution system design and planning in a deregulated environment. IET Gener Transm Distrib 3(12):1061–1078CrossRefGoogle Scholar
- 16.Thang VV, Thong DQ, Khanh BQ (2011) A new model applied to the planning of distribution systems for competitive electricity markets. In: The fourth international conference on electric utility deregulation and restructuring and power technologies, Shandong, China, 2011Google Scholar
- 17.Su H, Zhang J, Liang Z, Niu S (2010) Power distribution network planning optimization based on life cycle cost. In: 2010 China international conference on electricity distribution, 13–16 Sept 2010Google Scholar
- 18.Atwa YM, El-Saadany EF, Salama MMA, Seethapathy R (2010) Optimal renewable resources mix for distribution system energy loss minimization. IEEE Trans Power Syst 25(1):360CrossRefGoogle Scholar
- 19.Soroudi A, Aien M, Ehsan M (2012) A probabilistic modeling of photo voltaic modules and wind power generation impact on distribution networks. IEEE Syst J 6(2):254CrossRefGoogle Scholar
- 20.International Energy Agency (2016) Trends 2016 in photovoltaic applications—report IEA PVPS T1-30, 2016Google Scholar
- 21.Rosenthal RE (2010) GAMS—a user’s guide. GAMS Development Corporation, WashingtonMATHGoogle Scholar