Probabilistic day-ahead simultaneous active/reactive power management in active distribution systems
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Abstract
Distributed generations (DGs) are main components for active distribution networks (ADNs). Owing to the large number of DGs integrated into distribution levels, it will be essential to schedule active and reactive power resources in ADNs. Generally, energy and reactive power scheduling problems are separately managed in ADNs. However, the separate scheduling cannot attain a global optimum scheme in the operation of ADNs. In this paper, a probabilistic simultaneous active/reactive scheduling framework is presented for ADNs. In order to handle the uncertainties of power generations of renewable-based DGs and upstream grid prices in an efficient framework, a stochastic programming technique is proposed. The stochastic programming can help distribution system operators (DSOs) to make operation decisions in front of existing uncertainties. The proposed coordinated model considers the minimization of the energy and reactive power costs of all distributed resources along with the upstream grid. Meanwhile, a new payment index as loss profit value for DG units is introduced and embedded in the model. Numerical results based on the 22-bus and IEEE 33-bus ADNs validate the effectiveness of the proposed method. The obtained results verify that through the proposed stochastic-based power management system, the DSO can effectively schedule all DGs along with its economic targets while considering severe uncertainties.
Keywords
Simultaneous active/reactive power scheduling Stochastic programming Uncertainty Loss profit value Distributed generations1 Introduction
The high growth of distribute energy resources (DERs) penetration into medium voltage (MV) distribution networks has caused the distribution system operators (DSOs) to face some management, economic and technical issues. In the near future, in order to technically and economically manage the electrical networks, DSOs will be forced to set up marketplaces in which DERs will able to sell their active and reactive powers [1, 2]. It means that an effective market operation is required for achieving an economic equilibrium to promote DERs integration [2]. In [3], a market-oriented method has been proposed to integrate an economical portfolio of DERs, storage systems and demand response as the network resources. An energy management problem in a smart grid containing DERs and responsive demands has been addressed in [4], in which an incremental welfare consensus algorithm has been introduced to solve the problem in a distributed and cooperative manner. In distribution networks, it has been paid less attention to the reactive power market and hence, there are few researches focusing on this issue [5, 6, 7]. In [5], a control framework has been presented that allows DER owner to attain benefits through offering the ancillary service of voltage regulation to the DSO and maximizing the production of its active power. A mixed integer convex programming model is adopted to solve the reactive power dispatch problem in a distribution network populated with DERs [6]. It has used a conic relaxation based on a branch flow formulation.
Regarding the research works on wholesale markets, the active and reactive power markets are cleared based on the both simultaneous and separate models [8]. In the separate model, first, the active power dispatches of producers are attained in the energy market and the obtained results are passed as inputs for clearing the reactive power market. In the simultaneous model, the active and reactive powers of producers are simultaneously dispatched. Because the active and reactive power of generators are correlated through their capability curves, the solutions of the simultaneous model are theoretically closer to the optimum in comparison with those achieved from the decoupled one [9, 10]. Some research papers have focused on the relationship between active and reactive powers during clearing of the reactive power market [11, 12, 13].
An important issue for reactive power pricing is to consider the lost revenue of generators due to reduced active power production. This lost revenue is called lost opportunity cost (LOC) [8, 14, 15, 16, 17]. A coupled market in the presence of plug-in electric vehicles (PEVs) has been proposed in [15]. The considered objective function that is minimized comprises offer costs related to energy market, total reactive power payments function and LOCs. A new paying structure of the LOC to the producers of the reactive power has been presented in [16] aiming at improving the coupled energy and reactive markets. The capability curve of synchronous generators has been used to carefully model the cost of production or absorption reactive power which includes availability cost, losses cost and LOC [17]. To the best of our knowledge, in distribution markets, no compensation term has yet considered for the lost cost of the sale energy opportunity of DGs due to the production or absorption of reactive power.
Existence of significant levels of uncertain parameters is one of the major problems in decision-making process in power systems. Nowadays, the portion of energy generation from renewable resources such as wind and solar are growing and the intermittent nature of such resources is one of the main reasons for uncertainty. Meanwhile, load requirements and electricity prices can be considered as uncertain parameters. Generally, the uncertain parameters have limitless realizations and it is impossible to involve all of them in the decision-making process. In scenario-based decision-making methods, the realization space is approximated by limited scenarios with particular probabilities. Indeed, a set of scenarios is generated using probability density function (PDF) of the uncertain parameters. Although, the unlimited space of the uncertain parameters is transformed into an approximated one, these methods are efficient and simple to be performed [18]. A stochastic framework has been proposed to manage microgrids (MGs) in [19]. Firstly, a list of scenarios has been generated using the PDF of each uncertain parameter as well as roulette wheel mechanism (RWM), and secondly, in the scenario reduction process, for less computation, the most probable scenarios have been selected to remain. In each scenario, an adaptive modified firefly algorithm (AMFA) has been employed to solve the problem. In [20], a more comprehensive research work in comparison with the former, an improved multi-objective teaching-learning algorithm has been implemented to manage MGs. In [21], a stochastic bidding strategy of MGs in a coupled day-ahead energy and spinning reserve market has been proposed taking into account the uncertainties of renewable powers and loads. To do this, MG active power generation scenarios have been generated based on Latin hypercube sampling (LHS) technique and then reduced by backward method of the scenario reduction [21]. With this understanding, we need an efficient stochastic framework to cope with such uncertainties.
A distribution system is usually operated in a radial configuration, which comprises lines with relatively high resistance. Hence, active power dispatch of DGs in an ADN has considerably impact on buses voltage. Moreover, reactive power output of a generator correlates highly with active power through the capability curve. With this background, the separate active and reactive power dispatch scheduling that is prevalent in power systems can lead to non-optimal solution.
In this paper, a new stochastic market-based model for simultaneous day ahead active/reactive power dispatch scheduling are presented aiming at achieving coordinated volt/var control for ADNs. In the proposed model, distribution company (DisCo) is an intermediate entity between wholesale market and distribution system. DisCo purchases energy and reactive power from upstream market and sells them to the DSO via proposed distribution market. In the proposed model, unlike separate reactive power market, the LOC payment is not taken into consideration. Instead, a new payment index as loss profit value (LPV) is introduced and embedded in the model for a DG unit. It compensates for possible financial detriment arising from the reduction in energy sales profit in simultaneous market compared to the separate energy market. The uncertain parameters containing the output power of renewable energy resources including wind turbines (WTs) and photovoltaic (PV) units, wholesale active and reactive prices are considered. The scenario-based stochastic method is employed to approximate continuous environment of uncertain parameters. In this regard, in the phase of scenarios list generation, lattice Monte Carlo simulation (LMCS) and RWM are adopted to generate scenarios based on the PDF of uncertain parameters. Then, in the scenario reduction phase, the most probable and dissimilar scenarios are selected. The proposed probabilistic simultaneous active/reactive scheduling framework is modeled through a mixed-integer nonlinear optimization program implemented in generalized algebraic modeling systems (GAMS) software and solved with the DICOPT optimization solver.
- 1)
A probabilistic simultaneous active/reactive scheduling model that corresponds to ADNs is introduced.
- 2)
The uncertainties of renewable energy resources, wholesale active and reactive power prices are considered.
- 3)
A new index as loss profit value (LPV) term is formulated and added to the cost objective function of the model to minimize the profit difference of the DGs that they could gain in both simultaneous and separate markets.
The remainder of this paper is organized as follows. Scenario-based stochastic modeling is described in Section 2. In Section 3, the proposed stochastic simultaneous active/reactive power management is formulated. Numerical studies of the proposed modeling are implemented and analyzed in detail in Section 4 and finally, Section 5 is devoted to the conclusion.
2 Scenario-based stochastic modeling
2.1 Scenario generation and reduction
2.2 Scenario aggregation
3 Proposed stochastic simultaneous active/reactive power management
In order to handle the uncertainties of output power generation of WT and PV units as well as the energy and reactive power prices of upstream grid in the simultaneous active/reactive power scheduling problem, a two-stage stochastic programming framework is adopted. In the first stage of the proposed stochastic scheduling, different scenarios corresponding to the uncertain parameters during 24 h scheduling period are generated by a RWM and LMCS method. Moreover, in this stage, to mitigate the complexity and computational burden and to improve the performance of the presented model, a proper scenario reduction procedure is utilized. In the second stage, the simultaneous active/reactive power scheduling model is executed based on the selected scenarios (after applying scenario reduction method) as a mixed-integer nonlinear optimization problem (MINLP) optimization problem according to the occurrence probability of scenarios.
3.1 Objective function
Different portions of (5) of stochastic simultaneous active/reactive power scheduling are explained in the following.
3.1.1 Expected value of total energy costs
3.1.2 Expected total cost function of reactive power of DGs
In most of distribution systems, the DSO enters into contracts with DGs that are demanded to provide a minimum reactive power support. Here, we assumed that the DG units must operate between a mandatory leading and lagging power factor \(\cos (\varPhi_{mnd } )\) at every operation points [7]. By considering the explanations and using the capability curve of the DG, a three-component reactive power pricing structure is extracted as described in the following. At an assumed active power P_{DG,i}, we can distinguish three operation sections for reactive power of the DG as follows [27]:
Section 1 (\(- Q_{DG,i}^{mnd}\) to \(Q_{DG,i}^{mnd}\)): The reactive power of this section is according to the minimum reactive power support requirement. In this section, the DSO pays the DG only by an availability price (ρ_{0}) in $/h being a fixed component.
Section 2 (\(Q_{DG,i}^{\hbox{min} }\) to \(- Q_{DG,i}^{mnd}\)) and Section 3 (\(Q_{DG,i}^{mnd}\) to Q_{A,i}): In these sections, the DG provides an extra amount of reactive power beyond the Section 1, without requiring the adjustment its scheduled active power. Since the windings losses increase, the DG should be paid the cost of the losses for its reactive power service. Therefore, the reactive power cost function in Sections 2 and 3 should cover the losses cost besides the availability cost. We define a couple losses pricing components as price ρ_{1} in $/Mvarh for Section 2 and price ρ_{2} in $/Mvarh for Section 3 [28], respectively.
3.2 Expected total cost of reactive power purchased from the upstream grid
3.3 Expected total LPV of DGs
In the separate reactive power market, if a DG is needed to reduces its active power output that formerly is determined in the separate energy market to provide required reactive power of system, the LOC cost is paid. Nevertheless, if active power dispatch schedule of a DG in the proposed simultaneous energy and reactive market is less than the corresponding one in the separate energy market, the DG is not paid the LOC cost. Therefore, the reactive power cost function of DG units includes only availability and losses costs. A DG might receive lower profit of energy sale when it participates in the simultaneous market compared to the separate energy market. LPV index which is paid by DSO to the DG owner compensates for this probable lost revenue arising from reduced energy sale profit. Thus, the LPV payment for each DG unit is defined as the difference of profit value that a DG can receive from energy production in separate market and simultaneous market. Here, it is assumed that the DG unit submits its selling bid price based on the active power marginal cost. In deterministic approach, the LPV index for a DG is formulated based on the difference of active power production by the DG in the separate energy market and simultaneous market, and the MCP of these two markets as follows.
3.4 Constraints
- 1)Power flow equations$$P_{G,n,s}^{h} - P_{D,n}^{h} = \left| {V_{n,s}^{h} } \right|\mathop \sum \limits_{m = 1}^{{N_{Bus} }} \left| {V_{m,s}^{h} } \right|\left| {Y_{nm} } \right|{ \cos }\left( {\theta_{n,s}^{h} - \theta_{m,s}^{h} - \varphi_{nm} } \right)$$(22)where \(V_{n,s}^{h}\) is the voltage at node n in scenario s and hour h;\(P_{G,n,s}^{h}\) and \(Q_{G,n,s}^{h}\) show active and reactive power generation of node n in scenario s and hour h, respectively; \(P_{D,n}^{h}\) and \(Q_{D,n}^{h}\) are active and reactive power demand of node n and hour h, respectively.$$Q_{G,n,s}^{h} - Q_{D,n}^{h} = \left| {V_{n,s}^{h} } \right|\mathop \sum \limits_{m = 1}^{{N_{Bus} }} \left| {V_{m,s}^{h} } \right|\left| {Y_{nm} } \right|{ \sin }\left( {\theta_{n,s}^{h} - \theta_{m,s}^{h} - \varphi_{nm} } \right)$$(23)
- 2)The constraints related to the active and reactive power output of DGs and DisCo are expressed as follows:$$0 \le P_{DisCo,s}^{h} \le P_{DisCo}^{ \hbox{max} }$$(24)$$0 \le P_{DG,j,s}^{h} = \mathop \sum \limits_{b = 1}^{{N_{b} }} P_{DG,j,b,s}^{h} \le P_{DG,j}^{ \hbox{max} } \quad j = 1,2, \ldots , N_{SG}$$(25)$$0 \le P_{DG,w,s}^{h} \le P_{DG,w,s}^{F,h} \quad w = 1,2, \ldots , N_{Res}$$(26)where \(P_{DisCo}^{ \hbox{max} }\) and \(P_{DG,j}^{ \hbox{max} }\) signify the maximum active power provided by upstream grid and dispatchable DG j, respectively; \(Q_{DisCo}^{ \hbox{min} }\) and \(Q_{DisCo}^{ \hbox{max} }\) are provided minimum and maximum reactive power by upstream grid, respectively.$$Q_{DisCo}^{ \hbox{min} } \le Q_{DisCo,s}^{h} \le Q_{DisCo}^{ \hbox{max} }$$(27)
The operation constraints (11)–(17) associated to reactive power of DGs.
- 3)Bus voltage magnitude$$V_{ \hbox{min} } \le \left| {V_{n,s}^{h} } \right| \le V_{ \hbox{max} }$$(28)
- 4)Limit of transformers tapwhere \(U_{s}^{Tap,h}\), \(U_{ \hbox{min} }^{Tap}\) and \(U_{ \hbox{max} }^{Tap}\) are tap setting of under load tap changing (ULTC) transformer tap position in scenario s and hour h, minimum and maximum of ULTC, respectively.$$U_{ \hbox{min} }^{Tap} \le U_{s}^{Tap,h} \le U_{ \hbox{max} }^{Tap}$$(29)
- 5)Limit of steps of shunt capacitorswhere \({SC}_{{k},{s}}^{h}\), \({SC}_{k}^{{\text{min}}}\) and \({SC}_{k}^{\text{max}}\) denote step setting of switched capacitor k in scenario s and hour h, minimum and maximum step of switched capacitor k, respectively.$$SC_{k}^{ \hbox{min} } \le SC_{k,s}^{h} \le SC_{k}^{ \hbox{max} }$$(30)
4 Simulation results
4.1 Case study 1: 22-bus ADN
Reactive bids of DGs and upstream grid for case study 1
Generating unit | Reactive power bid | ||
---|---|---|---|
\(\rho_{0 }\)($/h) | \(\rho_{1}\)($/Mvarh) | \(\rho_{2}\)($/Mvarh) | |
PV | 0.068 | 13 | 13 |
WT | 0.082 | 15 | 15 |
Dispatchable DG | 0.095 | 17 | 17 |
Upstream grid | \(\rho_{Q,DisCo}^{{}}\)=16 $/Mvarh |
Optimization results of the stochastic simultaneous scheduling for case study 1
Portion of objective function | Value ($) |
---|---|
EC_{E} | 4695.66 |
EC_{Q,DG} | 110.52 |
EC_{Q,DisCo} | 187.11 |
ELPV_{DG} | 12.43 |
Total | 5005.72 |
Different components of P-Q stochastic simultaneous scheduling cost
Generating unit | Expected cost ($) | ||
---|---|---|---|
Energy cost | Reactive power cost | LPV | |
DisCo | 3501.29 | 187.11 | – |
Dispatchable DG | 759.72 | 41.50 | 12.43 |
WT | 328.91 | 30.16 | 0 |
PV | 108.74 | 38.87 | 0 |
Settlement results of the simultaneous and separate scheduling methods in stochastic framework for case study 1
Method | Energy cost ($) | Reactive power cost ($) | Total cost of energy and reactive power without LPV (for simultaneous scheduling method) and LOC (for separate scheduling method) ($) | Total costs ($) |
---|---|---|---|---|
Separate active and reactive power scheduling [22] | 4574.63 | 490.56 (with LOC) | 5015.77 | 5065.16 |
Simultaneous active and reactive power scheduling | 4695.66 | 297.63 | 4993.29 | 5005.72 |
4.2 Case study 2: IEEE 33-bus ADN
Reactive bids of the DGs and upstream grid for case study 2
Generating unit | Reactive power bid | ||
---|---|---|---|
\(\rho_{0 }\) ($/h) | ρ_{1} ($/Mvarh) | ρ_{2} ($/Mvarh) | |
DG1 | 0.082 | 15 | 15 |
DG2 | 0.068 | 13 | 13 |
DG3 | 0.078 | 17 | 17 |
WT | 0.095 | 19 | 19 |
PV1 | 0.096 | 12 | 12 |
PV2 | 0.092 | 14 | 14 |
Upstream grid | ρ_{Q,DisCo}=18 $/Mvarh |
Optimization results of the stochastic simultaneous scheduling for case study 2
Portion of objective function | Value ($) |
---|---|
EC_{E} | 4531.77 |
EC_{Q,DG} | 249.03 |
EC_{Q,DisCo} | 403.84 |
ELPV_{DG} | 17.82 |
Total | 5202.46 |
5 Conclusion
This paper presents an efficient stochastic programming for optimal active and reactive power scheduling to find the robust scheduling of DGs and the scheduled input power from the upstream grid. The proposed stochastic simultaneous active/reactive power scheduling consists of two stages. In the first stage, based on the RWM and LMCS approaches, some randomly scenarios corresponding to the intermittent renewable power generation and upstream grid prices are created and then properly reduced. In the second stage, the simultaneous active and reactive power scheduling methodology is utilized for the opted scenarios. In the proposed model, DisCos act as intermediate entities between wholesale market and distribution system. A DisCo purchases energy and reactive power from upstream market and sells them to the DSO via proposed distribution market. Instead of the LOC payment for DGs, a new payment index as LPV was introduced to compensate for possible financial detriment arising from reduction in energy sales profit in simultaneous dispatch compared to the separate active power dispatch. The proposed approach is investigated on the modified 22-bus and IEEE 33-bus ADNs to demonstrate its applicability. The stochastic programming results denote that the P-Q stochastic simultaneous scheduling method attains better dispatch scheduling results from economical viewpoint compared to the separate scheduling approach. Considering the uncertainty prices and output power of renewable generations by DSO will lead to a higher operation scheduling cost of distribution system. The obtained results of proposed stochastic framework will be reliable for DSO. Furthermore, the exploitation of the stochastic approach can enhance the reliability of the optimal solution with capturing more uncertainty range.
References
- [1]Adefarati T, Bansal R (2016) Integration of renewable distributed generators into the distribution system: a review. IET Renew Power Gener 10(7):873–884CrossRefGoogle Scholar
- [2]Colmenar-Santos A, Reino-Rio C, Borge-Diez D et al (2016) Distributed generation: a review of factors that can contribute most to achieve a scenario of DG units embedded in the new distribution networks. Renew Sustain Energy Rev 59:1130–1148CrossRefGoogle Scholar
- [3]Poudineh R, Jamasb T (2014) Distributed generation, storage, demand response and energy efficiency as alternatives to grid capacity enhancement. Energy Policy 67:222–231CrossRefGoogle Scholar
- [4]Rahbari-Asr N, Ojha U, Zhang Z et al (2014) Incremental welfare consensus algorithm for cooperative distributed generation/demand response in smart grid. IEEE Trans Smart Grid 5(6):2836–2845CrossRefGoogle Scholar
- [5]Calderaro V, Galdi V, Lamberti F et al (2015) A smart strategy for voltage control ancillary service in distribution networks. IEEE Trans Power Syst 30(1):494–502CrossRefGoogle Scholar
- [6]Ding T, Liu SY, Yuan W et al (2016) A two-stage robust reactive power optimization considering uncertain wind power integration in active distribution networks. IEEE Trans Sustain Energy 7(1):301–311MathSciNetCrossRefGoogle Scholar
- [7]Samimi A, Kazemi A, Siano P (2015) Economic-environmental active and reactive power scheduling of modern distribution systems in presence of wind generations: a distribution market-based approach. Energy Convers Manag 106:495–509CrossRefGoogle Scholar
- [8]Zhong J, Bhattacharya K (2002) Reactive power management in deregulated electricity markets-a review. In: Proceedings of 2002 IEEE power engineering society winter meeting, New York, USA, 27–31 January 2002, pp 1287–1292Google Scholar
- [9]El-Samahy I, Bhattacharya K, Canizares C et al (2008) A procurement market model for reactive power services considering system security. IEEE Trans Power Syst 23(1):137–149CrossRefGoogle Scholar
- [10]El-Samahy I, Bhattacharya K, Cañizares CA (2006) A unified framework for reactive power management in deregulated electricity markets. In: Proceedings of 2006 IEEE PES power systems conference and exposition, Atlanta, USA, 29 October–1 November 2006, pp 901–907Google Scholar
- [11]Ongsakul W, Chayakulkheeree K (2006) Coordinated fuzzy constrained optimal power dispatch for bilateral contract, balancing electricity, and ancillary services markets. IEEE Trans Power Syst 21(2):593–604CrossRefGoogle Scholar
- [12]Gomes MHR, Saraiva JT (2008) Active/reactive bid based dispatch models to be used in electricity markets. Electr Power Syst Res 78(1):106–121CrossRefGoogle Scholar
- [13]El-Samahy I, Canizares CA, Bhattacharya K et al (2007) An optimal reactive power dispatch model for deregulated electricity markets. In: Proceedings of 2007 IEEE PES general meeting, Tampa, USA, 24–28 June 2007, pp 1–7Google Scholar
- [14]Bhattacharya K, Jin Z (2001) Reactive power as an ancillary service. IEEE Trans Power Syst 16(2):294–300CrossRefGoogle Scholar
- [15]Rabiee A, Farahani HF, Khalili M et al (2016) Integration of plug-in electric vehicles into microgrids as energy and reactive power providers in market environment. IEEE Trans Ind Inf 12(4):1312–1320CrossRefGoogle Scholar
- [16]Ahmadi H, Foroud AA (2016) Improvement of the simultaneous active and reactive power markets pricing and structure. IET Gener Transm Distrib 10(1):81–92CrossRefGoogle Scholar
- [17]Homaee O, Jadid S (2014) Investigation of synchronous generator in reactive power market–an accurate view. IET Gener Transm Distrib 8(11):1881–1890CrossRefGoogle Scholar
- [18]Aien M, Hajebrahimi A, Fotuhi-Firuzabad M (2016) A comprehensive review on uncertainty modeling techniques in power system studies. Renew Sustain Energy Rev 57:1077–1089CrossRefGoogle Scholar
- [19]Mohammadi S, Soleymani S, Mozafari B (2014) Scenario-based stochastic operation management of microgrid including wind, photovoltaic, micro-turbine, fuel cell and energy storage devices. Int J Electr Power Energy Syst 54:525–535CrossRefGoogle Scholar
- [20]Niknam T, Azizipanah-Abarghooee R, Narimani MR (2012) An efficient scenario-based stochastic programming framework for multi-objective optimal micro-grid operation. Appl Energy 99:455–470CrossRefGoogle Scholar
- [21]Shi L, Luo Y, Tu GY (2014) Bidding strategy of microgrid with consideration of uncertainty for participating in power market. Int J Electr Power Energy Syst 59:1–13CrossRefGoogle Scholar
- [22]Samimi A, Kazemi A (2016) Scenario-based stochastic programming for volt/var control in distribution systems with renewable energy sources. IETE Techn Rev 33(6):638–650CrossRefGoogle Scholar
- [23]Aghaei J, Karami M, Muttaqi KM et al (2015) MIP-based stochastic security-constrained daily hydrothermal generation scheduling. IEEE Syst J 9(2):615–628CrossRefGoogle Scholar
- [24]Amjady N, Rabiee A, Shayanfar HA (2010) A stochastic framework for clearing of reactive power market. Energy 35(1):239–245CrossRefGoogle Scholar
- [25]Gomes MH, Saraiva JT (2010) Allocation of reactive power support, active loss balancing and demand interruption ancillary services in MicroGrids. Electr Power Syst Res 80(10):1267–1276CrossRefGoogle Scholar
- [26]Zou K, Agalgaonkar AP, Muttaqi KM et al (2012) Distribution system planning with incorporating DG reactive capability and system uncertainties. IEEE Trans Sustain Energy 3(1):112–123CrossRefGoogle Scholar
- [27]Samimi A, Nikzad M (2017) Complete active-reactive power resource scheduling of smart distribution system with high penetration of distributed energy resources. J Mod Power Syst Clean Energy 5(6):863–875CrossRefGoogle Scholar
- [28]Zhong J, Bhattacharya K (2002) Toward a competitive market for reactive power. IEEE Trans Power Syst 17(4):1206–1215CrossRefGoogle Scholar
- [29]Homaee O, Zakariazadeh A, Jadid S (2014) Real-time voltage control algorithm with switched capacitors in smart distribution system in presence of renewable generations. Int J Electr Power Energy Syst 54:187–197CrossRefGoogle Scholar
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