Distribution systems serve as connections between distribution substations and retail customers with low normal voltage, high feeder impedance and load density [1D. John, "McDonald, Bartosz Wojszczyk, Byron Flynn, and Ilia Voloh, Distribution Systems, Substations, and Integration of Distributed Generation", In: Encyclopedia Sustainability Sci. Technol., Springer, 2013., 2H. Lee Willis, "Power Distribution Planning Reference Book", crc press 2004
[http://dx.doi.org/10.1201/9781420030310] ]. Hence, there will be a rise in power and voltage losses that are particularly considered in planning, designing and operating problems of distribution systems.

Capacitors are able to reduce power losses and delay upgrading of feeders and source substations because of reduction in transmission power on devices. Many new technologies in capacitor designs have been introduced with less leakage powers and the price recently. Additionally, the investment in construction and operation is low-cost and installation places are also unlimited [3M. Jayalakshmi, and K. Balasubramanian, "Simple Capacitors to Supercapacitors-An Overview", Inter. J. Electr. Sci., vol. 3, pp. 1196-1217, 2008., 4J. Li, X. Cheng, A. Shashurin, and M. Keidar, "Review of Electrochemical Capacitors Based on Carbon Nanotubes and Graphene", Graphene, vol. 1, no. 01, p. 1, 2012.]. Hence, capacitors can be easily integrated in distribution systems and improve the economic and technical criteria. Many researches have been conducted to choose locations and capacity of capacitors to reduce power and voltage losses as well as operation cost in the distribution systems [5J.C. Carlisle, and A.A. El-Keib, "A review of capacitor placement techniques on distribution feeders", In: , the 29th Southeastern Symposium on System Theory: Pro., 1997, pp. 359-365.
[http://dx.doi.org/10.1109/SSST.1997.581664] , 6M.M. Aman, G.B. Jasmon, A.H. Bakar, and H. Mokhlis, "M. Karimi, “Optimum shunt capacitor placement in distribution system: A review and comparative study", Renew. Sustain. Energy Rev., vol. 30, pp. 429-439, 2014.
[http://dx.doi.org/10.1016/j.rser.2013.10.002] ]. The objective function to minimize power losses and voltage limits on load buses are introduced in [7M.A. Masoum, and M. Ladjevardi, "Optimal Placement, Replacement and Sizing of Capacitor Banks in Distorted Distribution Networks by GA", IEEE Trans. Power Deliv, vol. 19, no. 4, pp. 1794-1801, 2004., 8M. Dixit, P. Kundu, and H.R. Jariwala, "Optimal Allocation and Sizing of Shunt Capacitor in Distribution systemsfor Power Loss Minimization IEEE Students Con. on Electrical", In: , IEEE Students' Confrence on Electrical, Electronics and Computer Scienc, 2016, pp. 1-6.]. Furthermore, costs of devices are also considered in the objective function including total cost of energy losses and capacitor investment in order to enhance accuracy of calculation results [9A.A. Eajal, and M.E. El-Hawary, "Optimal Capacitor Placement and Sizing in Unbalanced Distribution Systems With Harmonics Consideration Using Particle Swarm Optimization", IEEE Trans. Power Deliv, vol. 25, no. 3, pp. 1734-1741, 2010.-11K. Tilakul, and P. Buasri, "Capacitor Location and Size Determination to Reduce Power Losses of a Distribution Feeder in Lao PDR", Int. J. Comput Electron. Eng., vol. 4, no. 1, p. 32, 2012.]. Constraints on capacity of equipment (such as feeders and transformers) and bus voltages are utilized to ensure the requirements of operations.

In above researches, capacitors are often considered in problems of seeking optimal setting and sizing in distribution systems. It is assumed that the structures and parameters of the systems remain unchanged. However, the distribution systems are always reformed and upgraded by increasing the section of feeders and the capacity of transformer substations to meet developments of loads. In other words, the peak load is utilized to calculate the operation of distribution systems that demand strongly changes by time in day and season per year. Therefore, the error of calculation can be significantly big, which causes the over compensation in time of low demand and the increase in power loss at this time.

Moreover, the load of the distribution systems depends on demands of customers and weather that have arbitrary characteristics. Hence, the normal probability density function (PDF) is chosen for modeling the uncertainties of the load at each bus [13T. Zhang, X. Song, X. Meng, J. Yu, and X. Chen, "A probabilistic load model based on chi-square method for distribution network", In: , 2^nd IET Renewable Power Generation Conference (RPG 2013), 2013.
[http://dx.doi.org/10.1049/cp.2013.1738] , 14Z. Liu, F. Wen, G. Ledwich, and S. Member, "Optimal Siting and Sizing of Distributed Generators in Distribution Systems Considering Uncertainties", IEEE Trans. Power Deliv., vol. 26, no. 4, pp. 2541-2551, 2011.
[http://dx.doi.org/10.1109/TPWRD.2011.2165972] ]. In researches [15Y.M. Atwa, E.F. El-Saadany, M.M. Salama, and R. Seethapathy, "Optimal Renewable Resources Mix for Distribution systemsEnergy Loss Minimization", IEEE Transactions on Power Sytems, vol. 25, no. 1, pp. 360-370, 2010.-18Y.M. Atwa, E.F. El-Saadany, "Probabilistic approach for optimal allocation of wind based distributed generation in distribution systems", ET Renew. Power Gener., vol. 5, no. 1, pp. 79-88, 2011.
[http://dx.doi.org/10.1049/iet-rpg.2009.0011] ], the system’s loads are assumed to follow the hourly load shape divided into different levels using a clustering technique with various probabilities. Therefore, the probability forecasting on load distribution becomes a significant issue that notably affect the operation and reinforcement of the distribution systems. However, the optimal compensation calculation of capacitor at the same time with the upgrading of feeders and transformer substations has not been mentioned in above researches. Uncertainties of loads are not validated because the demand in each hour is always set to be unchanged.

In recent years, the electricity price usually determined by Time-of-Use (TOU) is the simplest dynamic price. The objective of TOU price is to encourage the reduction of energy consumption during peak hours and the electricity price varies each day. TOU price is currently based on three periods of energy use as shown in [19S. Wong, K. Bhattacharya, and J.D. Fuller, "Electric power distribution systemsdesign and planning in a deregulated environment", IET Gener. Transm. Distrib., vol. 3, no. 12, pp. 1061-1078, 2009.
[http://dx.doi.org/10.1049/iet-gtd.2008.0553] -22H.A. Thanhtung, Y. Zhang, V.V. Thang, and J. Huang, "Energy hub modeling to minimize residential energy costs considering solar energy and BESS", J. Mod. Power Syst. Clean Energy, vol. 5, no. 3, pp. 389-399, 2017.] as follows: i) Peak - the highest demand; ii) Intermediate - the moderate demand; iii) Base - the lowest demand.

The mathematical optimization is widely applied in planning and reinforcement of the distribution systems with a variety of new models and methods [23M.M. Aman, G.B. Jasmon, and A.H.A. Bakar, "Optimum Capacitor Placement and Sizing for Distribution System Based on an Improved Voltage Stability Index, N.3", Int. Rev. Elect. Eng. (IREE), vol. 7, pp. 4622-4630, 2012.]. Georgilall etal. and Tung etal. [24P.S. Georgilakis, and N.D. Hatziargyriou, "A review of power distribution planning in the modern power systems era: Models, methods and future research", Electr. Power Syst. Res., vol. 121, pp. 89-100, 2015., 25N.S. Tung, and S. Chakravorty, "Optimized Power Distribution Planning A Review", Int. J. Electron. Elect. Eng., vol. 2, 2014.
[http://dx.doi.org/10.12720/ijeee.2.4.332-335] ] present models for the planning of the distribution systems with concurrent participation of distributed generators or capacitors. An optimal model in [26A.A. Algarni, and K. Bhattacharya, "A Novel Approach to Disco Planning in Electricity Markets: Mathematical Model", In: Power Systems Conference and Exposition, pp. 1-5, 2009.] is implemented with the objective function minimizing cost for investment feeder, transformer substation, distributed generator, capacitor and energy consumption from the electrical market in planning period. Similarly [27S. Wong, K. Bhattacharya, and J.D. Fuller, "Electric power distribution systemsdesign and planning in a deregulated environment", IET Gen. Trans. Distribut., vol. 3, no. 12, pp. 1061-1078, 2009.], introduces two stage planning models aim at decreasing the quantity and time of calculation. The objective function that seeks minimum value of investment and operation cost is used in associated with constraints in two stage in order to ensure the operation of system.

Besides, the objective function based on life cycle cost is also widely utilized in the problems of planning and operation of the distribution systems. The model with the objective function minimizing the life cycle cost in entire distribution system planning period is introduced in [28H. Su, J. Zhang, Z. Liang, and S. Niu, "Power Distribution Network Planning Optimization Based on Life Cycle Cost", In: China International Conference on Electricity Distribution, pp. 1-8, 2010.-29L. Liu, H. Cheng, Z. Ma, Z. Zhu, J. Zhang, and L. Yao, "Life Cycle Cost Estimate of Power System Planning", In: International Conference on Power System Technology (POWERCON 2010), pp. 1-8, 2010.
[http://dx.doi.org/10.1109/POWERCON.2010.5666437] ]. These models aim at finding the best distribution system planning scheme to maximize overall benefits and costs in the life cycle of the system. Constraints on maximum allowable capacity of equipment (feeder and transformer) and balanced power flow of system are considered to guarantee the economic and technical indicators. Similarly, the life cycle cost is persistently used in objective functions in [30J.R Yi-Song, and J.Q Long-Li, "Comparative study Of 10kV distribution network structure based on Life Cycle Cost analysis", In: China International Conference on Electricity Distribution (CICED 2012), pp. 1-5, 2012.
[http://dx.doi.org/10.1109/CICED.2012.6508551] -31G-H. Qiao, R. Zheng, S. Lei, Z. Bo, J-G. Xing, and X-L. Zhan, "A New Life Cycle Cost Model of Power Transformer and Its Comprehensive Sensitivity Analysis", In: International Conference on Power System Technology (POWERCON 2014), pp. 1342-1348, 2014.
[http://dx.doi.org/10.1109/POWERCON.2014.6993659] ] to determine the optimal transformer capacity. The model integrates the dynamic change of money with time, environmental cost, preventive test cost, sensitivity coefficient and the parameters of transformers.

In this research, an optimal model combining the uncertainties of demand described by a Probability Distribution Function (PDF) will be proposed. The optimal displacement, sizing and installation period of capacitors are simultaneously determined with optimal sizing and timeframe of feeders and transformer substations. The objective function is minimum life cycle cost of the planning scheme as well as the technical constraints are utilized to guarantee operation of the distribution systems and capacitor. The electricity price in the system and the electrical market is TOU price.

Next sections of the paper are organized as follows: an uncertainty modeling of demand is introduced in Section 2. A mathematical model with objective function and constraints are presented in Section 3. Section 4 shows calculated results from the IEEE 9-bus test distribution system. Finally, the conclusion is demonstrated in Section 5.

The uncertainties of load at each bus are stochastic parameters often modeled by the normal PDF as shown in equation (1) with mean of the distribution μ, standard deviation σ and variance σ² [13T. Zhang, X. Song, X. Meng, J. Yu, and X. Chen, "A probabilistic load model based on chi-square method for distribution network", In: , 2^nd IET Renewable Power Generation Conference (RPG 2013), 2013.
[http://dx.doi.org/10.1049/cp.2013.1738] -16N. Neyestani, M.Y. Damavandi, M. Shafie-khah, G. Chicco, and J.P. Catalão, "Stochastic Modeling of Multienergy Carriers Dependencies in Smart Local Networks With Distributed Energy Resources", IEEE Trans. Smart Grid, vol. 6, no. 4, pp. 1748-1762, 2015.
[http://dx.doi.org/10.1109/TSG.2015.2423552] ].

(1)

In other words, there are a number of states of load in each time segment. The selection of the number of states plays an important role because a small number of states may affect badly accuracy of calculations while a large number increases the complexity of the problem and calculation time. Hence, in this research, the requirement of the distribution systems is hypothesized to follow the hourly load characteristics of the IEEE-RTS using a clustering technique [17J.M. Pinheiro, C.R. Dornellas, and A.C. Melo, "Probing the new IEEE reliability test system (RTS-96): HL-II assessment", IEEE Trans. Power Syst., vol. 13, pp. 171-176, 1998.
[http://dx.doi.org/10.1109/59.651632] -18Y.M. Atwa, E.F. El-Saadany, "Probabilistic approach for optimal allocation of wind based distributed generation in distribution systems", ET Renew. Power Gener., vol. 5, no. 1, pp. 79-88, 2011.
[http://dx.doi.org/10.1049/iet-rpg.2009.0011] ]. In each time segment, the load is divided into ten levels (states) with different probabilities to guarantee a reasonable balance between accuracy and speed as shown in Table 1.

Table 1
Load data in states.

The uncertainty of the electrical demand considered in this research causes an increase in the complication of the problem. The mathematical model must take an uncertain load including many states into account with the different values of load and probabilities in each state. The proposed model simultaneously examines cost for upgrading of feeders, transformer substations and investment capacitors. Furthermore, the active and reactive power flow constraints in each state are proposed to calculate power flow and losses of system in all operation states with high accuracy. The model applies binary variables to represent the cost characteristic as well as investment and upgrading decision of equipment. Details of the model are presented as follows:

3.1. Objective Function

In the study, an objective function minimizing the life cycle cost of the investment project during calculation period is proposed as written in equation (2).

(2)

where: the life cycle cost is calculated at the base year by discount rate r; the upgrading cost of feeders CF_t; the upgrading cost of transformer substations CT_t; the investment cost of capacitors CC_t; the loss cost in the capacitor itself CL_t and the energy purchased cost CE_t and residual value of equipment at the end of the reinforcement period.

The upgrading cost of feeders including fixed capital cost C_F0 and the variable capital cost C_F is presented in equation (3) with the binary variable α_ij.t to represent the cost characteristic and decision variable for feeder upgrading F_{ij, t}.

(3)

where N is total number of system buses; T is total number of planning year; i and j are indices for all buses in the system; t is index for examination year.

Similarly, the upgrading cost of transformer substation CT_t including fixed capital cost C_T0 and variable capital cost C_T is presented in equation (4) with a binary variable γ_i.t to perform the cost characteristic, transformer upgrading decision variable ΔS^T_{i, t}-N_T. N_T is total number of transformer substation buses.

(4)

The equation (5) represents a new investment cost of capacitor in year t including fixed capital cost C_C0 and variable capital cost C_C with the binary variable β_i,t to represent the cost characteristic and decision variable for capacitor investment Q_{i, t}^C [12M. Dixit, P. Kundu, and H.R. Jariwala, "Incorporation of distributed generation and shunt capacitor in radial distribution system for techno-economic benefits", Eng. Sci. Technol. Int. J., vol. 20, pp. 482-493, 2017.
[http://dx.doi.org/10.1016/j.jestch.2017.01.003] ]. Q_{i, t}^C is affected by voltage and determined as (10). N_C is total number of capacitor installation buses.

(5)

The loss cost in the capacitor itself integrated uncertainties in year t are shown in equation (6) including decision variable for investment capacitor Q_{i, t}^C.

(6)

where k_C is the active power loss factor of the capacitor itself; d^s is the total number of days in each season; N_H is total number of hours in each day; h is index of each hour and N_s is total seasons in a year.

The energy purchased cost CE_t considering uncertainties of loads is presented in equation (7) with active power P_{i, t, s, h, g} and reactive power Q_{i, t, s, h, g} purchased through transformer substation.

(7)

where P_h^P, P_h^Q are active power purchased price at hour h; K_p is increase coefficient of electrical price in year t; The load probability in state g; λ_g; N_G is total number of states; g is index of each state.

The life cycle of each equipment is different from each other and the equipment is not concurrently invested so at the end of the calculation period the residual value of them is presented in equation (8). Hence, the residual value is the current value calculated at the base year in objective function.

(8)

where t_inv^F, T_L^F are installation time and life cycle of feeder; t_inv^T, T_L^T are installation time and life cycle of transformer substation; t_inv^C, T_L^C are installation time and life cycle of capacitor.

3.2. Constraints

The technical constraints are utilized to guarantee the operability of the distribution systems and capacitor including alternating current (AC) power flow, feeder upgrading section, transformer substation upgrading capacity, limitation of bus voltage and capacitor size.

3.2.1. Constraints for Power Flow

Considering uncertainties in this research will lead to calculating the power flow for every state of load. Therefore, an AC nonlinear power flow model supplemented a state index g is written in equation (9). The effects of reactive power on the calculation of power and voltage losses are interested in the model because the accuracy of calculation results is improved.

(9)

where PD_i,t,s,h,g, QD_i,s,t,h,g are active and reactive power of load i; U_{i, s, t,h,g}, δ_i,s,t,h,g are module and angle of bus voltage; Y_{ij, t}, θ_{ij, t} are magnitude and angle of admittance matrix element formulated by the feeder impedances in year t; Q^C_i,s,t,h,g is reactive power of capacitor at each state as equation (10).

(10)

3.2.2. Limits on Dimension of Capacitors

The participation of capacitor changes the power flow and affects the technical and economic indicators of distribution systems. To ensure annual installation of capacity corresponding to the equipment parameters, the size of capacitor invested at each bus in each year needs to follow constraints in (11). Additionally, the planning period is short-term; therefore, at each load bus, only a capacitor system is chosen in overall the plan horizon to reduce the installation cost as shown in equation (12).

(11)

(12)

where Q_max, Q_min are maximum and minimum capacity limit of capacitor; β_i,t is binary variable.

3.2.3. Upgrading Section Constraints of Feeder

The power carrying capability of feeders is imposed by thermal limits taken into the consideration of new feeder investments. The constraints for feeder upgrading are represented in equation (13) with capacity supplemented at each year, ΔS_ij,t^F. ΔS_ij,t^F is set when the capacity value is greater than the capacity limit used at year t-1. Thus, the feeder capacity needs to be upgraded at year t to meet in order to supply power of the load as equation (14).

(13)

(14)

where ΔS_max^F, ΔS_min^F is maximum power running in feeder at year t; ΔS_max^F are maximum and minimum capacity limit added of feeders; S_{ij, t}^*F is standard size of feeder and α_ij,t is binary variable.

3.2.4. Addition Capacity Constraints for Substation

In order to use the maximum of existing substations capacity and satisfy upgrading power corresponding to the equipment parameters. The constraint (15) is utilized to set substation sizes added in year t, ΔS_i,t^S, with binary variable γ_i,t.

(15)

where S_max^S, ΔS_min^S are capacity limit added of transformer substation; S_i,t^*S is standard size and S_i,t^max is maximum capacity running in the transformer at year t.

3.2.5. Constraints on Bus Voltages

The distribution systems are connected to the transmission systems by transformer substations so the voltages at substation bus are usually stabilized and assumed to be constant. At load buses, the voltages are usually changed according to parameters of the equipment and the value of the loads. Thus, the limits on load bus voltage are shown in equation (16).

(16)

where N_L is total number of load buses; U_min, U_max are maximum and minimum limit of bus voltage.

The proposed model is a mixed integer nonlinear programming (MINLP) calculated in GAMS environment with BONMIN solver to find out an optimal solution [36R.E. Rosenthal, A user’s guide., Development Corporation: Washington, 2015.].

This paper presents a generalized formulation of distribution system reinforcement integrating the capacitor and uncertainties of loads. The problem is scheduling the upgrading equipment of distribution systems (feeders and transformer substation) together with installation capacitor to minimize life cycle cost of the investment project. The proposed model is based on mixed integer nonlinear programming (MINLP) with binary variables utilized to perform the nonlinear cost characteristics and installation size of the equipment. The proposed formulation is verified by the IEEE 9-bus radial structure with uncertainties of loads and TOU prices. The result of simulation shows that a significant saving of life cycle cost can be achieved in comparison with traditional schemes. Only upgrading of the feeders and transformer substation is demanded. The reduction of power loss and electrical energy loss as well as upgrading delay of equipment in during calculating period could be seen as an effect of capacitor and stochastic load to distribution system reinforcement. Moreover, the voltage at all buses is always ensured in technical requirements in all operation modes.