Probabilistic Hosting Capacity for Active Distribution ...

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Probabilistic Hosting Capacity for Active Distribution Networks Hassan Al-Saadi, Student Member, IEEE, Rastko Zivanovic and Said F. Al-Sarawi, Member, IEEE

Abstract—The increased connection of distributed generation (DG), such as PhotoVoltaic (PV) and Wind Turbine (WT), has shifted the current distribution networks from being passive (consuming energy) into active (consuming/producing energy). However, there is still no consensus about how to determine the maximum amount of DGs that are allowed to be connected, i.e. how to quantify a so-called “Hosting Capacity” (HC). Therefore, this paper proposes a novel risk assessment tool for estimating network HC by considering uncertainties associated with PV, WT and loads. This evaluation is performed using the likelihood approximation approach. The paper, also, proposes a utilization of clearness index for localized solar irradiance prediction of PV. In addition, we propose the use of Sparse Grid Technique (SGT) as an effective means for uncertainty computation while the use of Monte Carlo Technique (MCT) is taken for a comparison purpose. Two actual distribution networks (11-buses and South Australian large feeder) are considered as case studies to demonstrate the usefulness of the proposed tool. Index Terms— Active distribution networks, risk assessment, Probabilistic Hosting Capacity (PHC), likelihood approximation approach, sparse grid, uncertainty computation.

NOMENCLATURE A. Indices and Sets b 𝑏-th branch index. i 𝑖-th receiving bus index. 𝑗-th sending bus index. 𝑗 DG-connection step. 𝑠 Time index (one hour). 𝑡 r Dimension index. Accuracy level’s index of univariate 𝑙𝑟 quadrature rules. Accuracy level’s index of multivariate ℓ quadrature rules. Index set of multi index vectors. 𝛩𝑧𝑑 Set of buses. 𝐵𝑈𝑆 Set of branches. 𝐵𝑅𝐴𝑁𝐶𝐻 This manuscript was received in February 10, 2017; and then accepted in April 12, 2017. Paper no. TII-17-0246. (Corresponding author: Hassan Al-Saadi) The authors are with the School of Electrical and Electronic Engineering, University of Adelaide, SA 5005, Australia (authors’ emails: {hassan.al-saadi, rastko.zivanovic, said.al-sarawi }@adelaide.edu.au). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

B. Indicators L DG injected Gen PV avg WT SG

Load. Distributed generation. Power injected. Generation. Photovoltaic unit. Average power. Small Scale Wind Turbine. Sparse grid.

C. Parameters Array surface area of one PV unit at s-step 𝐴𝑠 connection (m2 ). Complex current of a b-feeder during 𝑠-step at 𝐼𝑏,𝑠,𝑡 𝑡-time. 𝑏-feeder’s ampacity. 𝐼𝑏 𝑚𝑎𝑥 𝑉𝑖,𝑠,𝑡 𝑉𝑖 𝑚𝑎𝑥 ∆𝑠, 𝑆 𝜂 PV 𝛽 𝜃 DG 𝑁 𝑀 𝑘 𝓇 ℛ 𝜌 𝐷, 𝐷′ 𝜔, 𝜔𝑠𝑙𝑟 𝓃 𝐼𝑔𝑙𝑜𝑏𝑎𝑙 𝐼𝑜 𝑘𝑡 𝑚𝑖𝑛 𝑘𝑡 𝑚𝑎𝑥 𝑘̅𝑡 𝛾 𝛿L

Complex voltage of a 𝑖-th bus during 𝑠-step at 𝑡-time. Maximum operational voltage limit at 𝑖-th bus. Size of one stepwise DG connection and total s-steps, respectively. Photovoltaic efficiency. Inclination angle of a surface off a horizontal plane (°). Angle between the real and apparent power of DG output (°). Total number of DGs connected at 𝑖-th bus. Total number of abscissas. The diffuse fraction of the hourly radiation on a horizontal surface. Ratio of hourly to daily diffuse radiation (pu). Ratio of the beam irradiation on a tilted surface of 𝛽 angle to that on a horizontal surface (pu). Average ground reflectance (pu). Combinations of different parameters (𝛽, ℛ, 𝓇, 𝐿𝑙𝑜𝑐 , 𝐿𝑎𝑡 , 𝐿𝑙𝑎𝑡 , 𝐿𝑠𝑡 , 𝜌, 𝛾, 𝜔, 𝜔𝑠𝑙𝑟 , 𝓃). Angular displacement of the sun’s position and solar angular time (°). A particular day in a year. Global terrestrial solar radiation (kWh/m2 ). Extraterrestrial solar radiation (kWh/m2 ). Minimum boundary of 𝑘𝑡 . Maximum boundary of 𝑘𝑡 . Monthly average hourly clearness index (pu). Solar inclination (°). Standard deviation specified for the loads.

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2 𝐿𝑙𝑜𝑐 , 𝐿𝑙𝑎𝑡 , 𝐿𝑎𝑡 , 𝐿𝑠𝑡 𝑣 In , 𝑣 Out , 𝑣 R L,avg 𝑃𝑖,𝑡 WT,R 𝑃𝑖,𝑠

D. Variables 𝑂𝐿𝑉 𝑂𝑉𝑉 𝑘𝑡 injected 𝑃𝑖 𝑃𝑖DG 𝑃𝑖L injected 𝑄𝑖 𝑣

Longitude and latitude (°). Altitude (m) and standard meridian (°). Cut-in, cut-out and rated speed of WT (m/s). Average active power of load connected at 𝑖-th bus for a given t-time. Rated active power of WT connected to 𝑖-th bus during 𝑠-step. Number of overloading violation. Number of overvoltage violation. Hourly Clearness index. Active power injected at 𝑖-th bus from a combination of 𝑁-DGs. Generated active power at 𝑖-th bus from a single domestic DG. Load active power connected at 𝑖-th bus. Reactive power injected at 𝑖-th bus from a combination of 𝑁-DGs. Wind speed.

E. Functions Weight function. 𝑓 Universal function. 𝑔 WT WT active power as a function for 𝑃𝑖,𝑠,𝑡 (𝑣) 𝑣 at 𝑖-th bus during 𝑠-step and 𝑡-time. PV PV active power as a function for 𝑃𝑖,𝑠,𝑡 (𝑘𝑡 ) 𝑘𝑡 at 𝑖-th bus during 𝑠-step and 𝑡-time. 𝛽 Total irradiance, as a function for 𝑘𝑡 , on an 𝐼𝑡 (𝑘𝑡 ) inclined surface with 𝛽 to a horizontal plane. I. INTRODUCTION growth of renewable-related DG integration has Thehadrapid considerable impacts on distribution networks. The growth comes as global governmental efforts that are underway towards satisfying emission reduction, improved infrastructure reliability and energy independence. However, when DG penetration reaches high levels (about 20%–30% of a total generation), the intermittent nature of DG generation starts to have noticeable and negative effects on the entire network [1]. Research is currently underway to investigate effects such as voltage stability [2], network voltage rise and overloading [3], increase of system losses [4], rapid voltage change caused by sudden shift of DG generation [5] and power harmonic distortion [6]. In addition, other research also conduct system risk assessment study with high DG integration such as riskbased investigation [7], voltage sensitivity analysis [8], and system reliability evaluation [9]. As a result, a new era of distribution networks is emerging for more effective and reliable DG integration. Active distribution network (ADN) comes as a cheap alternative to cope with this increase, rather than reinforcing network assets which is the conventional solution. Several studies looked at the potential advantages of transferring the passive distribution network into ADN where in some cases it can be viewed as a virtual power plant supplying nearby loads [9, 10, 11, 12]. The goals are cost minimization, CO2 emissions reduction, energy saving, reliability improvement. The important feature of this kind of networks is the bi-

directionality of the power flow allowing the use of surplus energy; for example, exchanging the excess energy among other utility networks through bilateral contracts or pool markets. A central controller is usually responsible for determining the exported/imported energy to/from upstream networks such as the non-islanded mode in [13]. The controller behaves according to automation approaches based on system time-discretization where control actions might stand for coordinated voltage control of On-Line-Tap-Changer (OLTC), optimal network configuration, compensator voltage coordination, power factor (PF) control, energy curtailment, demand-side management, storage usage etc. [14]. The trend is heading towards intelligent and automated central control such as in [15]. Despite the considerable efforts to address these matters in the literature, distribution network operators (DNOs) still, according to CIGRE Working Group C6.19 [16], prefer not to adopt the new planning and operation techniques proposed during the last decade, rather they continue to use the traditional methods. The main reason could be the inability to answer, confidently, what is the future scenario of such a network going to look like? Especially with the presence of uncertainties in local power productions and demands [17]. For this reason, it is of paramount importance to quantify the network’s capacity for hosting an amount of DGs, which is so called “Hosting Capacity” (HC), before attempting to apply or investigate further approaches. The benefits brought from the determination of HC are: 1) identify the maximum allowable DG integration, 2) maintain network security and reliability, 3) help DNOs to have a transparent and fair discussion with DG owners (DGOs) regarding the rights and commitments of both parties, 4) avoid major network upgrade, 5) exploit efficiently the clean energy, 6) explore the existing methods for increasing DG penetration, 7) help in drawing HC standards and metrics, 8) better utilize the current infrastructure, 9) investigate potential adverse impacts of increasing DGs connection beyond HC, 10) adapt the current protection devices and regulations to cope with the concept of ADN, 11) incentivize trading locally in electricity markets, and many more. In terms of HC determination, stochastic processes were utilized in different ways in attempt to assess the intermittency of multi DGs integration. The probabilistic approaches are the most used schemes for both small scale WT and PV units. In [18], Electrical Power Research Institute (EPRI) used scenariobased analysis for evaluating the potential impact of high PV penetration, taking into account the frequency distributions of residential and commercial PVs of common-used sizes in the markets, and involving very large number of load flow calculations. The study emphasizes on the overvoltage violation as a limit for determining HC. The stochastic random selection from the frequency distributions in a variety of PV deployments resulted in conducting a worst-case scenario (high production, low demand) which was then further investigated by streamlining the possible rooms of extra PVs away from turbulent locations [19]. Randomizing locations of multi constant PVs was addressed in [20], giving the HC as a probability density function (PDF) of voltage violations. In [21], one day analysis was investigated for many distribution feeders in England with the overvoltage violation set to be the HC limits. Their HC is based on probabilistically classifying feeder parameters (such as No. of house connections, total path

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3 impedance, sum of wired line length etc.) with HC values, using constant PV sizes (3 kW). Similar work can be found in [22], as well, employing five HC categories (very weak to very strong) of network parameters with the use of Weibull-based model. In other work, data-based scenario was performed such as the idea of coincident hours presented in [23] that is dependent on the joint probability of hourly occurrences between the percentages of power generations and demands. Then, with the use of multi-period AC optimal power flow, HC maximization methods were developed taking the advantage of the available headrooms. The coincident hours method was, also, adapted in [24] for energy management system through real-time scheduling optimization and then in [25] for optimal energy losses minimization by considering a set of four hundred different unique scenarios over one year. However, for a shorttime assessment, using discrete data samples does not always give a precise answer, as the DG power outputs and demands are strictly continuous uncertain variables. In the meantime, the authors of [26] gave a clear description of how to identify the HC of a network, as they stated “The impact of distributed generation can be quantified only by using a set of performance indicators”. Their approach had been implemented on a MV network using only two giant wind turbines [27], then, the network performance was assessed using local virtual load flow software with two years of recorded data. In addition, timeseries analysis was performed with two performance indicators regarding power quality and overloading. The most recently proposed approach can be summarized as follows [28]: 1) Establish one or more performance indices; 2) Specify one or more standardized limits for step 1) such as in EN50160 [29] or any other standards defined by the local utility; 3) Find the functional relation between the indices and the increase of DG connections; 4) Identify network HC when step 2) is exceeded; The simplicity of this approach does not undermine the complexity of its application in the real world. In this paper, and for the sake of current DNOs’ regulations, we propose a probabilistic HC (PHC) approach that expands on the concept of HC discussed in [26] by the embedment of the system uncertainty models and with the utilization of stochastic sampling tools, making it a “ready-to-use” tool for decision/investment makers. The approach is time-dependent with network bidirectionality feature, and compatible with the automation approaches for ADNs. A Sparse Grid Technique (SGT) is adopted for system risk assessment and compared with Monte Carlo Technique (MCT). The probability (frequency) of occurrence is set to be the indictors for HC limits for time-run simulations with the exploitation of the likelihood approximation approach and quantile function. Specifically, this paper has the following contributions: 1. Proposing a PHC for ADN through a system risk assessment under the associated uncertainties. 2. Developing generation uncertainty model for PV based on clearness index that suits with the Australian meteorological conditions. 3. Easing the computational burden through the utilization of SGT.

The effectiveness of the proposed approach is demonstrated with a multi DG connections through the use of a real smallsize residential feeder and another large distribution network existing in South Australia. The paper is divided into five sections. Following this introductory section, the current problem of the increased DG connections is formulated with modelling system uncertainties. Then, in section III, the overall system risk assessment is described by establishment of Deterioration Indices (D.Inds), evaluation through MCT and SGT, and demonstration of the proposed PHC. In section IV, two case studies are carried out and the simulation results and discussions are presented. The final section summarizes the outcomes of this work. II. PROBLEM FORMULATION In this part, the problem in ADN when DG connection increases is described and modelled, providing assumptions first. In short, the increase in DG connections is discretized in a stepwise fashion that employs uncertainty models of PV, WT and load at each step. A. Assumptions  Any controllability and/or dispatchability in DGs and/or demands are not considered in which DNOs have no control over the power produced or consumed. However, possible control over both DG productions and demand consumptions across different times can be included in a different study aiming for one or more of the many goals mentioned in the introduction.  For the purpose of proposing a generic tool with a fair judgment, the approach does not consider the stochastic random distribution of DGs along the target feeder, instead, a set of predefined positions of possible domestic DGs is specified based on the distribution of residential houses. In the assessment of large networks, the distribution of residential suburban transformers is taken, instead of houses. The reason is that the future policy of grids is also uncertain, i.e. knowing that a house will have a DG or not is still uncertain, the same found in ([22], P.692). Another reason is that an amount of DGs is to be equally likely dedicated for each house or residential feeder in the purpose of a fair decision making.  Each uncertain variable is treated as a continuous random variable that is independently and identically distributed for each 𝑡-time. The assessment is run on an hourly basis in which wind speed and solar irradiance as well as demand (as will be discussed rigorously hereinafter) are barely dependent or correlated for the assessment of single hour. Therefore, dependencies among random variables are neglected for each 𝑡-time. However, and for long-term planning, dependencies are taken into consideration through the use of real average data of different time domains for days of different seasons or months. In the meanwhile, from the nodes’ prospective, the output profiles of domestic DGs connected along the feeder are proximately symmetrical due to the fact that a residential feeder is usually not too wide. Similar assumption applies for residential load profiles. The same has been assumed for WT in [30] and for PV in [31]. The last point is further elaborated on in section III.A.

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4 B. Power Flow Constraints The power flow constraints are postulated to cope with a conceptual principle that grid-connected DGs are treated as an active-reactive power injector with a constant PF. Therefore, the power balance at 𝑖-th bus can be expressed as follows: injected 𝑃𝑖 = −𝑃𝑖L + 𝑃𝑖DG = |𝑉𝑖 | ∑∀𝑗|𝑉𝑗 |(𝔅𝑖𝑗 cos 𝜃𝑖𝑗 + 𝒢𝑖𝑗 sin 𝜃𝑖𝑗 ) , (1) injected

𝑄𝑖

= −𝑄𝑖L + 𝑄𝑖DG = |𝑉𝑖 | ∑∀𝑗|𝑉𝑗 |(𝔅𝑖𝑗 sin 𝜃𝑖𝑗 − 𝒢𝑖𝑗 cos 𝜃𝑖𝑗 ),

injected

(2)

injected

where 𝑃𝑖 and 𝑄𝑖 represent the active and reactive power injected at 𝑖-th bus to either feed-in the load (𝑃𝑖L + 𝑄𝑖L ) or a DG (𝑃𝑖DG + 𝑄𝑖DG ) feeds energy back to the utility grid; 𝑉𝑖 and 𝑉𝑗 are voltages at 𝑖-th and 𝑗-th bus respectively; 𝔅𝑖𝑗 and 𝒢𝑖𝑗 are conductance and susceptance of the 𝑖𝑗-th line’s admittance; 𝜃𝑖𝑗 is the angle difference between the voltages at 𝑖-th and 𝑗-th bus. C. DG Connection Increase The term “DG connection” refers to the sum of the active power, according to the nameplate capacity of DGs, connected into the point of common coupling (PCC). In meanwhile, a DG penetration refers to the actual active power, which is uncertain in case of weather-based DG, injected by DGs into a PCC. Hence, to achieve a realistic assessment, the increase in the DG connection is treated in a stepwise style, where each step is denoted by s-step. Then, the actual DG power installations at each PCC can be formulated as follows: DG𝑛 DG (3) = ∑𝑁 𝑃𝑖,𝑠 𝑛=1 𝑃𝑖,𝑠 , DG is the total where 𝑁 represents the number of DGs and 𝑃𝑖,𝑠 active power injected at 𝑖-th bus, which is nominated to be PCC, within 𝑠-step from a combination of 𝑁-DGs. Therefore, a stepwise increase in 𝑠-steps leads 𝑃𝑖,1 < ⋯ < 𝑃𝑖,𝑠 < ⋯ < 𝑃𝑖,𝑆 in which 𝑃𝑖,𝑠+1 = 𝑃𝑖,𝑠 + ∆𝑠. Herein, the size of one step increase, ∆𝑠, total s-steps, 𝑆, and initial DG power, 𝑃𝑖,1 , are to be chosen by any expert on the subject. Within this increase, the reactive power of DGs, resulted from the interfacing passive components of the integrated converter, is considered and can be expressed using the following formula: 2

DG DG 𝑄𝑖,𝑠 = 𝑃𝑖,𝑠 DG 𝑄𝑖,𝑠

√1−(cos 𝜃𝑠DG )

𝜃𝑠DG

cos 𝜃𝑠DG

,

(4)

where the and are values specified for reactive power and constant PF, respectively, at 𝑖-th bus within 𝑠-step. D. PV Uncertainty Modelling 1) PV Output Power The output power of a PV unit is modelled as a linear function of the solar irradiance, providing empirical data, or as a non-linear function of the corresponding solar irradiance variations. The linear function assumes that the grid-connected PVs are equipped with a Maximum Power Point Tracker (MPPT) which can be described as [32]: PV (𝑘𝑡 ) = 𝐴𝑠 𝜂 PV 𝐼𝑡𝛽 (𝑘𝑡 ), 𝑃𝑖,𝑠,𝑡 (5) PV (𝑘 ) where 𝑃𝑖,𝑠,𝑡 𝑡 represents active power produced at 𝑖-th bus 𝛽

during 𝑠 -step at 𝑡 -time; 𝐼𝑡 (𝑘𝑡 ) represents the total solar radiance received on a PV array surface area, 𝐴𝑠 , with an

Fig. 1. PV generations using different PDFs during one day of August in Adelaide ((35oS), 𝐴𝑠 =1 𝑚2 , 𝜂PV = 0.7, 𝛽 = 15°.

inclination angle 𝛽 to a horizontal plane during 𝑡-time. The 𝜂 PV signifies the PV array’s efficiency. 𝛽 In (5), the 𝐼𝑡 (𝑘𝑡 ) is a non-linear function of clearness index, 𝑘𝑡 , (𝑘𝑡 will be rigorously explained in the following subsection). The total irradiance incident on an inclined surface with an angle 𝛽 can be estimated given the knowledge of 𝑘𝑡 and providing time-and-place characteristics (detailed derivations are given in the Appendix A). It can be expressed as: 𝑘𝑡 𝐷 ∙ 𝑘𝑡 + 𝐷 ′ 𝐷 > 0 & 𝐷′ ≥ 0 −𝐵(𝐵′ −𝑘 ) 𝛽

𝑡

1+𝑒

𝐼𝑡 (𝑘𝑡 ) = {𝐷 ∙ 𝑘 − 𝐷′ 𝑡

𝑘𝑡 ′ 1+𝑒 −𝐵(𝐵 −𝑘𝑡 )

𝐷 > 0 & 𝐷′ < 0 ,

(6)

0 𝐷 ≤0 where 𝐷 and 𝐷′ are combinations of different parameters (𝛽, ℛ, 𝓇, 𝐿𝑙𝑜𝑐 , 𝐿𝑎𝑡 , 𝐿𝑙𝑎𝑡 , 𝐿𝑠𝑡 , 𝜌, 𝛾, 𝜔, 𝜔𝑠𝑙𝑟 , 𝓃) based on the models involved in the Appendix A; 𝐵 and 𝐵′ are logistic function’s parameters. In an attempt to address the uncertainty knowledge of the solar irradiance, different probability distributions were employed such as beta [33], Weibull [34], normal [35]. However, to the best of our knowledge, a universally-approved PDF for irradiance description without specifying the local time and site characteristics has not been developed yet. 2) Solar Irradiance Variation The randomness of solar irradiance can be decomposed into deterministic variations such as sun’s position in the sky and stochastic variations such as physical cloud movements, vapor density, air mass, etc. In fact, the combination of deterministic and stochastic variations characterizes the so called clearness index, 𝑘𝑡 , which is defined as a measure of the atmosphere transparency acting in accordance with the daytime and seasonal variations and it is given by: 𝑘𝑡 = 𝐼𝑔𝑙𝑜𝑏𝑎𝑙 ⁄𝐼𝑜 , (7) where 𝐼global is the global solar radiation ( kWh/m2 ) which is uncertain and 𝐼𝑜 is the extraterrestrial solar radiation (kWh/m2 ) that reaches the earth before penetrating the atmosphere. The last one can be theoretically determined for a site with specifying the day of the year and the time of the day [36]. The domain of the 𝑘𝑡 is {𝑘𝑡 ∈ ℝ; 𝑘𝑡 𝑚𝑖𝑛 ≤ 𝑘𝑡 ≤ 𝑘𝑡 𝑚𝑎𝑥 } where TABLE I 𝐷 AND 𝐷′ PARAMETERS OF (6) FOR A DAY IN AUGUST Time 8:00AM 9:00AM 10:00AM 11:00AM 12:00PM 0.577 0.845 1.052 1.181 1.225 𝐷 -0.139 -0.177 -0.207 -0.225 -0.231 𝐷′

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5 𝑘𝑡 𝑚𝑖𝑛 and 𝑘𝑡 𝑚𝑎𝑥 are the minimum and maximum boundaries of 𝑘𝑡 . Basically, clearness index exhibits different probability distributions for different periods of time on a certain day of the year. These periods have been shortened over the time through extensive studies from hourly distribution [37, 38, 39, 40, 41] to 20 minutes [42, 43], to 5 minutes [44] or 1 minute [45] to very short named “instantaneous” [46]. It should be noted that the high resolution distributions came with limitations as they had not been tested on a variety of locations around the globe. In the literature, it has been found that the probability density of 𝑘𝑡 is distributed in a unimodal shape such as Boltzmann [38], double beta [47] and Single Gamma [39] or bimodal like biexponential [48], double normal [44], triple normal [46] and Weibull-logistic distribution [49]. The bimodal distributions were confirmed under some extra circumstances such as considering the reflection of cloud edges. We choose to use the unimodal-parameterized shape as follows [39]: 𝑓 solar (𝑘𝑡 |𝑘𝑡 𝑚𝑎𝑥 , 𝑘̅𝑡 ) = 𝐶𝑡 (1 −

𝑘𝑡 𝑘𝑡 𝑚𝑎𝑥

) 𝑒 𝜆𝑡𝑘𝑡 ,

(8)

𝐶𝑡 = 𝜆𝑡 = ξ𝑡 =

𝜆𝑡 .𝑘

̅ 𝑡) (𝑘𝑡 𝑚𝑎𝑥 −𝑘

𝑘𝑡 𝑚𝑎𝑥

,

.

E. WT Uncertainty Modelling 1) WT Output Power Fundamentally, the fluctuations in a WT-generated power are dependent on wind speed variations. The WT manufacturers and designers impose cut-in speed, 𝑣 In , and cut-out speed, 𝑣 Out as well as the rated speed, 𝑣 R , for protection and safety issues. Therefore, it is possible to use a piecewise function to describe the output power of WT with multiple wind speed domains as follows [51]: 0 𝑣 ≤ 𝑣 In or 𝑣 ≥ 𝑣 Out In WT,R 𝑣−𝑣 𝑣 R −𝑣 Out WT,R 𝑃𝑖,𝑠

𝑣 In < 𝑣 < 𝑣 R R

Out

,

(9)

𝑣 𝑉𝑖 𝑚𝑎𝑥 ∀𝑖 ∈ 𝐵𝑈𝑆, (16) where 𝑉𝑖,𝑠,𝑡 is a complex voltage of a 𝑖-th bus in the distribution system during 𝑠-step at 𝑡-time and 𝑉𝑖 𝑚𝑎𝑥 is a maximum operational voltage limit at 𝑖-th bus. As explained with the previous criterion, 𝐷. Ind2(𝑘𝑡 , 𝑣, 𝑃𝑖𝐿 |𝑠, 𝑡) can be defined, in the same way, to be an indicator of 𝑂𝑉𝑉𝑠,𝑡 conditional to t-time and s-step. Its expectation for multi random variables, clearness index, 𝑘𝑡 , wind speed, 𝑣, and load 𝑃𝑖L , can be computed as follows: 𝐷. Ind2(𝑘𝑡 , 𝑣, 𝑃𝑖L |𝑠, 𝑡) = ∭ 𝑂𝑉𝑉𝑠,𝑡 (𝑘𝑡 , 𝑣, 𝑃𝑖L |𝑠, 𝑡) ∙ 𝑓 solar (𝑘𝑡 ) Ω

∙ 𝑓 wind (𝑣) ∙ 𝑓 load (𝑃𝑖L ) ∙ 𝑑𝑘𝑡 𝑑𝑣 𝑑𝑃𝑖L , (17) where Ω is the domain of the involved random variables. B. Uncertainty Evaluation Technique The existing techniques for evaluating a randomized performance are classified into analytical and numerical. In theory, the analytical method is more preferable as it can deliver an exact solution, but at expenses of system realism, such as involving less data dimensions or making some variables in the system to be constant. In this paper, we exploit Monte Carlo Technique (MCT) and Sparse Grid Technique (SGT) for finding the likelihood approximation of 𝐷.Inds. 1) Monte Carlo Technique The Monte Carlo Technique is used to find the statistical expectation of (14) and (17) as well as the density distribution of (12) and (15). The use of MCT is basically based on the convolution concept of random variables. The random variables convoluted are jointly independent and serially correlated via using real average data of different time domains for days of different seasons or months. MCT is useful to find the joint density distribution of 𝑂𝐿𝑉 or 𝑂𝑉𝑉 using their quantiles (outliers plus quartiles) with a high number of simulations. Despite the high accuracy in estimation, MCT is

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7 computationally an expensive tool to run, computations chosen here to be 10,000 [61]. So we intend to use it for time and accuracy comparisons. 2) Sparse Grid Technique As our random variables are mutually independent, the orthogonality principle can be adopted utilizing a univariate quadrature rule for a multi-dimensional integration. Different univariate quadrature rules have been found in the literature such as Gauss-Patterson and Gauss-Legendre [62]. Applying these rules for multivariate problems requires the use of tensorization. However, with high dimensions, the classical tensorization will be computationally too expensive, similar to MCT, resulting in so called “the curse of dimensionality.” One solution is to use the SGT because it is highly dependent on the continuity and smoothness of the functions involved, which is well suited for our problem. To clarify more, let us consider Φ𝑙𝑟 represent a numerical (one-dimensional) integration involving a sequence of univariate quadrature rules for a function, 𝑔, given its weight function, 𝑓, at r-th dimension, as follows: 𝑀𝑙𝑟 Φ𝑙𝑟 = ∑ℎ𝑟 =1 𝑔(𝑥𝑙𝑟 ,ℎ𝑟 )𝑓(𝑥𝑙𝑟 ,ℎ𝑟 )𝑤𝑙𝑟 ,ℎ𝑟 , (18) where the index 𝑙𝑟 signifies the level of accuracy for a dimension 𝑟, 𝑟 = 1,2, … , 𝑑, which relies on the number of nodes 𝑀𝑙𝑟 ; 𝑥𝑙𝑟 ,ℎ𝑟 ∈ [0,1] represents nodes (abscissas) and 𝑤𝑙𝑟 ,ℎ𝑟 represent precomputed quadrature weights . It should be noticed that the exactness of Φ𝑙𝑟 increases with 𝑙𝑟 , where 𝑙𝑟 ∈ ℕ. Now, for d-dimensions, employing a classical tensor product requires 𝑂(𝑀𝑑 ) nodes, where 𝑀 is the number of nodes for one coordinate direction, which means the number of nodes increases exponentially with the increase of dimensions, especially when a high level of accuracy is needed. To overcome this limitation the SGT, first introduced by the Russian mathematician Smolyak in 1963 [63], requires a modest number of nodes, 𝑂(𝑀 ∙ (Log 𝑀)𝑑−1 ), with an accuracy as good as the classical tensor product technique. It is important to mention that the technical details of SGT are out of this paper’s scope. Later developments on Smolyak’s technique have been found such as delayed algorithm [64], generalized algorithm [65], dimension adaptive algorithm [66] etc. For more information, the reader is referred to corresponding literature. Therefore, involving the SGT on tensor product yields a weighted sum of product rules indexed by the level of accuracy as follows [67]: Ψ𝑑 (ℓ) = ∑ℓ𝑧=ℓ−𝑑+1 𝑈𝑧ℓ ∑𝑙∈Θ𝑑 (Φ𝑙1 ⨂ ⋯ ⨂Φ𝑙𝑑 ), (19) 𝑧 where Θ𝑑𝑧 denotes the index set of multi index vectors, {ℓ𝑟 }𝑑𝑟=1 , such that Θ𝑑𝑧 = ({ℓ ∈ {ℕ}𝑑 : |ℓ| = ∑𝑑𝑟=1 ℓ𝑟 = 𝑧 + 𝑑} & {∅: 𝑧 < 0}), and 𝑈𝑧ℓ = (−1)ℓ−𝑧 (𝑑−1 ). The extended formula of the ℓ−𝑧 tensor product is as follows: Φ𝑙1 ⨂ ⋯ ⨂Φ𝑙𝑑 = 𝑀𝑑 𝑀2 1 1 𝑑 ∑𝑀 𝑟1 =1 ∑𝑟2 =1 ⋯ ∑𝑟𝑑 =1(𝑤𝑟1 ⨂ ⋯ ⨂𝑤𝑟𝑑 ) ⋅ 𝑔(𝑥𝑟11 , ⋯ , 𝑥𝑟𝑑𝑑 ) ∙ 𝑓(𝑥𝑟11 , ⋯ , 𝑥𝑟𝑑𝑑 ). (20) The Kronrod-Pattrson rule has been chosen for univariate quadrature because of its nesting feature that adds more nodes to the preceding level of accuracy, 𝑙𝑟 − 1, thereby maximizing the exactness of the likelihood approximation.

C. Probabilistic Hosting Capacity Within the objective of network risk assessment, we propose the PHC for ADNs that is mainly dependent on the evaluation of D.Inds over a period of time discretized into 𝑡-time slices and for a limited range of s-steps. During that period and range, D.Inds are computed at each slice and step indicating the probabilistic expectations of violation(s). Then, the HC limits (certain amount of DG connection) can be defined through a graphical display, as will be demonstrated in the following case studies. In the meantime, given expectations of D.Inds, their metrics are conceptualized to be having the same meaning when it comes to determine the initial system deteriorations. In fact, the metric can be a pair sequence defined in {(𝐷. Ind1, 𝐷. Ind2): 𝐷. Ind1 ∈ [0, |𝐵𝑈𝑆|] 𝑎𝑛𝑑 𝐷. Ind1 ∈ [0, |𝐵𝑅𝐴𝑁𝐶𝐻|]}𝑠,𝑡 ; where | | means the cardinality of a set. In these sequences, a certain power penetration from DGs is considered to be imposing a threat on the utility’s assets and users’ devices when 𝑂𝐿𝑉 and/or 𝑂𝑉𝑉 occur. This implies the physical meaning of D.Inds in which an integer one refers to one violation that is expected to occur. This leads us to define HC limits such that HC limit1 to be identified where 𝐷. Ind1 equal or less than 1 is. Similarly, HC limit2 is where 𝐷. Ind2 equal or less than 1 is. For a diurnal risk assessment, a critical region exists if the HC limit1 proceeds HC limit2 (DNOs can tolerate HC limit1 to be more than 1 based on their policy circumstances). Nevertheless, there is always a threat identified to be serious in the region beyond HC limit2 (DNOs’ should not tolerate HC limit2 for the safety and efficiency of the system). Any DG installation is considered to be tolerable in the region below HC limit1, conditioned to that HC limit1 does not exceed HC limit2. For seasonal risk assessment, as different t-time for different season can yield different expected 𝐷. Inds, an abroad critical amount of DGs’ installation is to be identified. In this critical area, the deterioration is more likely to exist and implying a system threat that can be a negotiable ground between DNOs and DGOs considering the variations of different seasons. IV. CASE STUDIES AND DISCUSSIONS To demonstrate the usefulness of the PHC as a risk assessment tool, two case studies are analyzed in this paper, namely: Case 1 and Case 2. Considering an assumption of maximum delivery power is linearly proportional to the array surface area of PV, one squared meter of a panel (1m2 ) is treated to be delivering 1kW along with the common standards of PV manufacturers. Also, considering that the installations of DG in the residential areas tend to be of more PVs than WTs, the amount of WTs is taken to be always a third of PV connections. Moreover, according to the localized weather conditions in South Australia, monthly average wind speed and solar irradiation with involved parameters for June and December that are characterized in their models as in (6) and (10) are shown the Table II and Table III, respectively. The technical characteristics for PV and WT are explained in the Table IV. The constant quantities are 𝛿 L to be 15% with 68% confidence interval for all loads. PF is to be 0.95 for DGs and loads. The domain of 𝑘𝑡 is 0 to 1. For simplicity, we kept the

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8

Point of customer connection

TABLE II WINTER AND SUMMER WT MONTHLY DATA Wind speed June December parameters 𝐶1 2.02 pu 2.01 pu 𝐶2 4.719 m/s 5.615 m/s TABLE III WINTER AND SUMMER PV DAILY DATA

Fig. 2 Distribution network supplying 36 residential loads with DG installations.

Time Shortest day Longest day

DG connection (kW) Fig. 3 Statistical distribution (boxes) of 𝑂𝐿𝑉 and D.Ind1 (green circles).

DG connection (kW) Fig. 4 Statistical distribution (doted circles and boxes) of 𝑂𝑉𝑉 and D.Ind2 (red circles). L,avg

same 𝑃𝑖 for any 𝑡-time involved. For SGT, the 𝑙𝑟 is 13. EN50160 is considered as voltage standards. The following studies were conducted using Matlab and OpenDSS, an advanced software for distribution network modelling introduced by EPRI, and run on Intel® i7-2600 8G RAM 4-core @ 3.4GHz processor. A. Case 1 and Discussion In this case, the method is implemented with s-step, 0.5 kW, increase across all connected DGs. An actual low-voltage distribution network consists of 11 buses with an adjustable distribution transformer, 11/0.4 kV, shown in Fig. 2 (technical details can be found in [68]). The minimum and maximum voltage adjustability of the transformer are to be within 0.93pu and 1.03pu, respectively, for certain times in the day such as 0.93pu around noon as in this case. This is in order to allow the upstream-reverse power to flow in specified times of a year when DG generation is expected to be relatively larger than demand. The probability density distributions of (12) and (15) are illustrated in Figs. 3 and 4, respectively, by using the quantile function including quartiles (25%, 50% and 75% quantile) and outliers with a 95% confidence interval at 12:00PM for the longest day. It is clearly shown that the probability of a single ampacity violation starts at 14.4kW in Fig. 3, while a single voltage violation is estimated at 18.2kW according to Fig. 4. However, according to the median (50% quantile) of these

PV 𝜂PV 0.7 pu

𝐷 𝐷′ 𝐷 𝐷′

10:00AM

11:00AM

12:00PM

0.8649 -0.2754 1.1735 -0.0246

0.987 -0.2928 1.288 -0.0409

1.0286 -0.2987 1.327 -0.0465

TABLE IV PV AND WT CHARACTERISTICS WT 𝛽 𝑣 In 𝑣 Out 15° 2 𝑚/𝑠 25 𝑚/𝑠

𝑣R 16.1 𝑚/𝑠

distributions, the deteriorations start to be noticeable at 30kW in Fig. 3 while 37.7kW in Fig. 4, indicating a larger HC for the network. In the meantime, D.Inds are included in these figures to show the role of expectations compared with the quantiles. Based on D.Inds, the system starts almost earlier in which D.Ind1 is expected to be 1 (single ampacity violation) at 13.3kW and D.Ind2 (single voltage violation) is expected at 16.5kW. Despite the ability of adopting both expectations and quantiles in all figures, it is obvious that expectations give a more robust evaluation. This is as the quantiles may ignore some extreme events while all events are to be included in the expectations, including the extreme ones. For this reason, D.Inds have just been considered in the following evaluations. The proposed risk assessment tool is performed as shown in Fig. 5, where two days (shortest and longest in a year) are considered. In Fig. 5, the ring symbol represents the use of MCT and plus symbol represents SGT. The D.Ind1 and D.Ind2 are distinguished by having different colors, green and red, respectively. The computational time for the MCT is 2.47 second to perform a single evaluation while for the SGT is 0.48 second. In Fig. 5a, different hour times are considered (at 10:00AM, 11:00AM and 12:00PM) for the two days. In the accuracy comparison, in Fig. 5a, the top left figure for the longest day computed using MCT shows HC limit1 (13.3kW) and HC limit2 (16.5kW), while HC limit1 (13.7kW) and HC limit2 (16.4kW) are estimated from the bottom left figure, that is computed suing SGT. This is as the green arrow symbolizes the HC limit1 when the value of D.Ind1 is 1, similarly, the red arrow symbolizes the HC limit2 when the value of D.Ind2 is 1. By considering only 12:00PM for the analysis purposes, the results of the shortest day show that the system deteriorates after 30.5kW of DG installations at each PCC (cf. (3) for the calculation of DG at each house). If this deterioration is tolerated by DNOs, up to 36.5kW DGs will be allowed to be connected, creating a critical area shaded by the half transparent yellow color. By comparing the longest day with shortest one in Fig. 5a for 12:00PM, a seasonal risk assessment is to be performed based on the outer HC limits as shown in Fig. 5b. The broader critical area is noticeable taking into account the

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9

Shortest day

Longest day

12:00PM

10:00AM

11:00AM

12:00PM

a)

tolerable

critical

serious

b) 𝐷. Ind1 using SGT 𝐷. Ind1 using MCT

𝐷. Ind2 using SGT 𝐷. Ind2 using MCT

HC limit1 HC limit2

Fig. 5 D.Inds (y-axis) vs. DG connections (kW) (x-axis) at different times showing three regions; tolerable (shaded blue), critical (shaded yellow) and serious (shaded red). a) diurnal risk assessment, b) seasonal risk assessment at 12:00PM.

seasonal differences of a year. Herein, the system critical deterioration starts after 13kW of installation up to a maximum amount of DGs which is 36.5kW. The resultant data shows three areas where the DNOs can establish a ground for negotiations according to the type of the threat (tolerable, critical and serious). In meanwhile, it is interesting to notice the stochastic nature of DGs with loads that is shown in a form of a small probabilities of violations (D.Inds less than 0.5), in the regions shaded with a transparent blue, indicating that the DGs have no noticeable impacts. Moreover, looking at the two top plots in Fig. 5a, these probabilities seem to be even less prior to the HC limits showing some improvements in the network security.

Fig. 6 11kV Distribution feeder supplying 59 residential and commercial (11/0.4 kV) distribution transformers.

B. Case 2 and Discussion A realistic 11kV distribution feeder in South Australia is used and the graphical diagram is shown in Fig. 6. The feeder is modelled in OpenDSS. COM interface (Component Object Model) is employed when applying our approach from Matlab. The technical details of this network can be found in Appendix B. In this case, the s-step increase is treated as a percentage of the overall load that is connected into a 11/0.4 kV distribution transformer. The risk assessment is considered on this network for two zones, considering the prospect connections of domestic DG only. The number of low voltage transformers in Zone A is 18 and for Zone B is 27. The 𝑡-time is an hour around 12:00PM. The assessments of one zone are performed separately at each time, i.e. no both zones being assessed simultaneously. It is up

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10 TABLE V TWO SUB-FEEDERS UNDER ASSESSMENT HC limits Zone A Zone B

Shortest day Longest day Shortest day Longest day

HC limit1

HC limit2

340% 165% 230% 98%

1610% 700% 365% 160%

to the utility to decide which zone is ready to be transformed into being ADN. Also, it is assumed that OLTC located at the substation transformer maintains the slack voltage to be 0.96pu for assessing Zone A and for assessing Zone B as well. The HC limit1 and limit2 for both zones are shown in Table V. It is obvious that Zone A is more capable to host DGs than Zone B as the Zone A is closer to the substation where the voltage profile is more stable than in Zone B. In general, it is apparent from both cases that the use of PHC is crucial for assisting DNOs to negotiate and regulate through a transparent discussion with DGOs. The region in-between these two limits is regarded to be critical where DNOs may allow their network to be at a vulnerable risk trading-off with profits of the exported energy. Although it is necessary to have in-deep discussions about why HC limit1 leading HC limit2, the most possible reason is that a reversed flowing power in ADNs may overload the downstream lines before a voltage violation occurs. This is because the downstream lines are usually modelled to be lighter than upstream ones. Another reason might be the use of EN50160 as different standards might deliver different outcomes. The region shadowed with red, which is beyond HC limit2, is regarded to be surely causing a serious threat for both network assets and customer devices. A decision for no allowance of more DG connections should not be compromised. In addition, the hourly assessments can be complemental to the automation approaches for the optimal operation of ADNs. This is as the system deteriorations are more likely to take place at the time around noon. Also, a possible maximization of network HC without jeopardizing system performance needs further investigations at these hours. The inclusion of ways to increase HC can be beneficial for adopting ADNs such as the use of storage, centralized or decentralized energy curtailments, load shifting techniques etc. V. CONCLUSION In this paper, PHC is presented and discussed as a risk assessment tool for estimating the amount of DGs to be connected into distribution networks. Two deterioration indices, based on ampacity and voltage violations, are formulated for a probabilistic evaluation of the system under the presence of uncertainties associated with PVs, WTs and loads. For PV uncertainty modelling, localized solar irradiances are considered with a major attention paid in the formulation of the clearness index that meets Australian weather conditions. For the computation of these indices, the likelihood approximation approach is adopted. The paper proposes the utilization of the SGT instead of MCT. Comparisons with MCT are given and the study shows that faster results with an acceptable accuracy are achievable when using SGT.

Two case studies are presented to demonstrate the effectiveness of the method. Based on different times in a day and in a year, different HCs are identified for two different distribution networks. Different regions are identified where the amount of DG connection may be classified into tolerable, critical and serious. The method does not rely on the worst-case scenario, instead, it delivers the likelihoods of the system to deteriorate, giving several choices for DNOs to discuss and regulate their policies. The PHC is meant to serve DNOs in monitoring and maintaining distribution system reliability and efficiency as well as complementing the concept of ADN. In future work, the duration, magnitude, real-time line rating and other factors could be included to better understand network HC. The PHC can be utilized in the automation approaches of ADN, exploiting the feature of hourly assessments. More, the effect of OLTC on network HC and HC maximization warrants a further investigation under the proposed tool. APPENDIX A To begin with, the amount of solar radiance in (kW/m2 ) 𝛽 incident on a inclined surface, 𝐼𝑡 , can be expressed as follows [36]: 1+cos 𝛽

𝛽

1−cos 𝛽

̅ 𝐻

𝐼𝑡 = [ℛ + ( − ℛ) 𝑘 + 𝜌 ]×𝓇 𝑜 , (A.1) 2 2 3600 where 𝜌 is the ground reflectance and 𝑘 is the diffuse fraction ̅𝑜 is the of the hourly radiation on a horizontal surface and 𝐻 daily-average extraterrestrial radiation that arrives at the atmosphere perpendicular to the surface of the earth (kW/m2 ), the 𝓇 is the ratio of hourly to daily diffuse radiation (pu). The ℛ is the ratio of the beam irradiation on a tilted surface of 𝛽 TABLE VI METEOROLOGICAL STATIONS WITH THE DEVELOPMENT OF THE DIFFUSE FUNCTION

Year [70] [71]

1977 1990

[72]

2004

[73]

Meteorological stations where data are collected o

1

fitting Function Piecewise linear. Piecewise linear.

2012

Toronto (43 N) . Oslo (60o N), Albany (43o N), Cape Canaveral (28o N), Copenhagen (56o N), Hamburg (54o N), and Valentia (52o N). In 78 Meteorological stations in China. Sao Paulo (23o S).

[74]

1994

Madras (13o N).

[75]

2001

[76]

2004

Perpignan (43o N), Lisbon (39o N), Coimbra (40o N), Evora (39o N), Faro (37o N), Carpentras (44o ), Pau (43o N), Athens (38o N), Madrid (40o N), Seville (37o N) and Porto (41o N). Sao Paolo (23o S)

Piecewise 4th-order polynomial. Piecewise 3rd-order polynomial.

Adelaide (35o S), Darwin (12o S), Geelong (38o S), Maputo (26o S), Bracknell (51o S), Lisbon (39o S), Macau (22o S), and Uccle (51o N) [69] 2010 Adelaide (35o S), Uccle (51o N), Darwin (12o S), Bracknell (51o N), Lisbon (39o N), Macau (22o N) and Maputo (26o S). 1 The latitude of geographical region in degrees. [50]

2008

Piecewise linear. Piecewise linear.

Piecewise 2ndorder polynomial. Logistic.

Logistic.

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11 angle to that on a horizontal surface computed for the northern hemisphere, when the place altitude, 𝐿𝑎𝑡 , (0° ≤ 𝐿𝑎𝑡 ≤ 90°), by [36]: cos(𝐿𝑎𝑡 −𝛽) cos 𝛾 cos 𝜔+sin(𝐿𝑎𝑡 −𝛽) sin 𝛾 ℛ= , (A.2) cos 𝐿𝑎𝑡 cos 𝛾 cos 𝜔+sin 𝐿𝑎𝑡 sin 𝛾

and for the southern hemisphere (−90° ≤ 𝐿𝑎𝑡 ≤ 0°) computed by: ℛ=

cos(𝐿𝑎𝑡 +𝛽) cos 𝛾 cos 𝜔+sin(Lat +β) sin γ cos 𝐿𝑎𝑡 cos 𝛾 cos 𝜔+sin 𝐿𝑎𝑡 𝑠𝑖𝑛 𝛾

,

(A.3)

where 𝛾 solar declination or the angular position of the solar noon (−23.45° ≤ 𝛾 ≤ 23.45°) for a particular, 𝓃-th, day in a year (0 ≤ 𝓃 ≤ 365), 𝜔 is the angular displacement (in degrees) of the sun’s position at 15o per hour. By gathering locational data of 𝑘𝑡 and k for different sites around the globe, it was found that the diffuse fraction 𝑘 is in correlation with clearness index, 𝑘𝑡 . Different correlations are

obtained forming different functions and aiming to achieve more accuracy and convergence rate. These came as different studies found different types of functions for fitting their data. It should be mentioned that the sampled data is collected for different sites, generating different fitting functions as they are shown in Table VI. From this table, it is obvious that there is no consistency of modelling diffuse fraction 𝑘 as different studies show different function types for fitting their observed data. However, the logistic function was confirmed in [50] to be applicable for different geographical regions in the northern and southern hemisphere. The same model was then further complicated in [69] by taking into consideration the effects of apparent solar time, persistence of global radiation level and daily clearness index. Another important usefulness of logistic based diffuse fraction 𝑘 is a single logistic equation for all

TABLE VII DISTRIBUTION NETWORK IN SOUTH AUSTRALIA WITH 89 LINES Node j Node i R (ohm) X(ohm) Rating (A) L* (kW) L (kAV-R) Node j Node i R (ohm) X(ohm) ss 1 0.5233 0.2522 500 45 46 0.0744 0.1561 1 2 0.1084 0.1435 500 46 47 0.2013 0.0970 2 3 0.1624 0.1415 90 47 48 0.4025 0.1940 3 4 0.1932 0.0931 20 46 49 0.0170 0.0289 4 5 0.1006 0.0485 11 142.5 46.84 49 50 0.0314 0.0151 3 6 0.0628 0.0547 70 49 51 0.1826 0.3104 6 7 0.0974 0.0849 14 190 62.45 51 52 0.1610 0.0776 6 9 0.0909 0.0792 60 51 53 0.0548 0.0931 8 9 0.0692 0.0334 60 47.5 15.61 53 54 0.0365 0.0621 9 10 0.1811 0.0873 60 54 55 0.0368 0.0321 10 11 0.2013 0.0970 60 55 56 0.1732 0.1509 11 12 0.1610 0.0776 13 56 57 0.0217 0.0189 12 13 0.0435 0.0210 13 28.5 9.37 57 58 0.8855 0.4268 13 14 0.2938 0.1416 8 95 31.22 56 59 0.4830 0.2328 11 15 0.4186 0.2018 50 59 60 0.1610 0.0776 15 16 0.2415 0.1164 50 142.5 46.84 60 61 0.1610 0.0776 16 17 0.1811 0.0873 40 60 62 0.4025 0.1940 17 18 0.1811 0.0873 30 28.5 9.37 53 63 0.0913 0.1552 18 19 0.0805 0.0388 30 63 64 0.0183 0.0310 19 20 0.0644 0.0794 30 64 65 0.4025 0.1940 20 21 0.0939 0.1158 8 28.5 9.37 64 66 0.8050 0.3880 21 22 0.1342 0.1654 8 28.5 9.37 53 67 0.1732 0.1509 22 23 0.0617 0.0761 5 19 6.24 67 68 0.8660 0.7544 23 24 0.0349 0.0430 5 15.2 5.00 68 69 0.0805 0.0388 20 25 0.1530 0.0737 20 68 70 0.1826 0.3104 25 26 0.0805 0.0388 20 9.5 3.12 70 71 0.1610 0.0776 26 27 0.1006 0.0485 11 9.5 3.12 70 72 0.0351 0.0596 27 28 0.3019 0.1455 10 19 6.24 72 73 0.0570 0.0968 28 29 0.2013 0.0970 8 73 74 0.0953 0.0830 26 30 0.2415 0.1164 11 19 6.24 73 75 0.0913 0.1552 30 31 0.0612 0.0754 8 19 6.24 75 76 0.0913 0.1552 17 32 0.0612 0.0295 16 28.5 9.37 76 77 0.0548 0.0931 32 33 0.1771 0.0854 13 28.5 9.37 77 78 0.0457 0.0776 33 34 0.2898 0.1397 11 142.5 46.84 78 79 0.2013 0.0970 2 35 0.1811 0.0873 11 142.5 46.84 76 80 0.1299 0.1132 35 36 0.0805 0.0702 11 142.5 46.84 80 81 0.0537 0.0468 35 37 0.0563 0.0490 20 142.5 46.84 81 82 0.3897 0.3395 37 38 0.1610 0.0776 13 82 83 0.1610 0.0776 38 39 0.0805 0.0388 13 142.5 46.84 82 84 0.1516 0.1320 35 40 0.0548 0.0931 335 84 85 0.0572 0.0275 40 41 0.0563 0.0490 8 95 31.22 85 86 0.1047 0.0504 40 42 0.0365 0.0621 315 190 62.45 84 87 0.1127 0.0543 42 43 0.0247 0.0419 315 87 88 0.1208 0.0582 43 44 0.0585 0.0509 31 475 156.12 87 89 0.1208 0.0582 43 45 0.0822 0.1397 260 190 62.45 * L means a load connected at node i.

Rating (A) L (kW) L (kAV-R) 260 30 30 475 156.12 250 50 190 62.45 215 8 95 31.22 215 88 78 190 62.45 68 5 9.5 3.12 3 9.5 3.12 65 19 6.24 61 20 285 93.67 48 285 93.67 36 142.5 46.84 28 11 142.5 46.84 20 285 93.67 96 285 93.67 80 14 190 62.45 70 14 190 62.45 60 190 62.45 48 14 190 62.45 38 190 62.45 28 8 9.5 3.12 5 5 28.5 9.37 24 23.75 7.81 24 47.5 15.61 20 28.5 9.37 5 9.5 3.12 18 14 47.5 15.61 8 19 6.24 8 9.5 3.12 8 19 6.24 4 9.5 0.00

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12 provided domain of clearness index 𝑘𝑡 i.e. no constraints required, which can save time in computational involvement. In this paper, the logistic function is utilized to estimate the total solar irradiance on an inclined surface given locational characteristics as follows: 1 𝑘= , (A.4) 1+𝑒 −𝐵(𝐵′−𝑘𝑡 ) where 𝐵 and 𝐵′ are specified parameters for the logistic based diffuse fraction which can be computed in [50]. 𝛽 From (A.1) and (A.4), the 𝐼𝑡 can be expressed in terms of 𝑘𝑡 by specifying the 𝑡-th hour of the local time on the 𝓃-th day of a year: 𝑘𝑡 𝛽 𝐼𝑡 (𝑘𝑡 ) = 𝐷 ∙ 𝑘𝑡 + 𝐷′ , (A.4) −𝐵(𝐵′ −𝑘 ) 𝑡

1+𝑒

where: 1+cos 𝛽

𝐷 =[

2

𝐷 ′ = [ℛ + 𝜌.

− ℛ] × 𝓇 1−cos 𝛽 2

̅𝑜 𝐻 3600 ̅𝑜 𝐻

]×𝓇

3600

APPENDIX B Details of the South Australian feeder in Table VII. VI. REFERENCES

[13] J. M. Guerrero, M. Chandorkar, T. L. Lee and P. C. Loh, "Advanced control architectures for intelligent microgrids—Part I: Decentralized and hierarchical control," IEEE Trans. Ind. Electron., vol. 60, no. 4, pp. 1254-1262, Apr. 2013. [14] A. Ipakchi and A. Farrokh, "Grid of the future," IEEE Power Energy Mag., vol. 7, no. 2, pp. 52-62, Mar. 2009. [15] M. N. Kabir, Y. Mishra, G. Ledwich, Z. Y. Dong and K. P. Wong, "Coordinated control of grid-connected photovoltaic reactive power and battery energy storage systems to improve the voltage profile of a residential distribution feeder," IEEE Trans. Ind. Informat., vol. 10, no. 2, pp. 967-977, May 2014. [16] E.-B. K and e. al., "Survey on methods and tools for planning of active distribution networks," in Integr. Renew. Distrib. Grid, CIRED 2012 Workshop, IET, May 2012. [17] D. E. Olivares and e. al., "Trends in microgrid control," IEEE Trans. Smart Grid, vol. 5, no. 4, pp. 1905-1919, Jul. 2014. [18] M. Rylander and J. Smith, "Stochastic Analysis to Determine Feeder Hosting Capacity for Distributed Solar PV," EPRI Technical Report 1026640, Dec. 2012. [19] M. Rylander, J. Smith and W. Sunderman, "Streamlined method for determining distribution system hosting capacity," IEEE Trans. Ind. Appl., vol. 52, no. 1, pp. 105-111, Jan 2016. [20] M. Kolenc, I. Papič and B. Blažič, "Assessment of maximum distributed generation penetration levels in low voltage networks using a probabilistic approach." 64 (2015): .," Int. J. Electr. Power Energy Syst., vol. 64, pp. 505-515, Jan. 2015.

[1]

S. Papathanassiou and et al., "WG C6.24 capacity of distribution feeders for hosting DER," in Technical brochure 586, Paris, Jun 2014.

[21] V. Rigoni, L. F. Ochoa, G. Chicco, A. Navarro-Espinosa and T. Gozel, "Representative Residential LV Feeders: A Case Study for the North West of England," IEEE Trans. Power Syst., vol. 31, no. 1, pp. 348-360, Jan. 2016.

[2]

A. Ameli, S. Bahrami, F. Khazaeli and M. R. Haghifam, "A multiobjective particle swarm optimization for sizing and placement of DGs from DG owner's and distribution company's viewpoints," IEEE Trans. Power Del., vol. 29, no. 4, pp. 1831-1840, Aug. 2014.

[22] S. Breker, A. Claudi and B. Sick, "Capacity of low-voltage grids for distributed generation: Classification by means of stochastic simulations," IEEE Trans. Power Syst., vol. 30, no. 2, pp. 689-700, Mar. 2015.

[3]

P. Juanuwattanakul and M. A. S. Masoum, "Increasing distributed generation penetration in multiphase distribution networks considering grid losses, maximum loading factor and bus voltage limits," IET Gener. Transm. Distrib., vol. 6, no. 12, pp. 1262-1271, Dec. 2012.

[23] L. F. Ochoa, C. J. Dent and G. P. Harrison, "Distribution network capacity assessment: Variable DG and active networks," IEEE Trans. Power Syt., vol. 25, no. 1, pp. 87-95, Feb. 2010.

[4]

N. C. Hien, N. Mithulananthan and R. C. Bansal, "Location and sizing of distributed generation units for loadabilty enhancement in primary feeder," IEEE Syst. J., vol. 7, no. 4, pp. 797-806, Dec. 2013.

[5]

E. Zio, M. Delfanti, L. Giorgi, V. Olivieri and G. Sansavini, "Monte Carlo simulation-based probabilistic assessment of DG penetration in medium voltage distribution networks," Int. J. Electr. Power Energy Syst., vol. 64, pp. 852-860, Jan. 2015.

[6]

M. Patsalidesa, A. Stavrou, V. Efthymiou and G. E. Georghiou, "Towards the establishment of maximum PV generation limits due to power quality constraints," Int. J. Electr. Power Energy Syst., vol. 42, no. 1, pp. 285-298, Nov. 2012.

[7]

S. W. Alnaser and L. F. Ochoa, "Advanced network management systems: a risk-based AC OPF approach," IEEE Trans. Power Syst., vol. 30, no. 1, pp. 409-418, Jan. 2015.

[8]

H. M. Ayres and W. Freitas, "Method for determining the maximum allowable penetration level of distributed generation without steadystate voltage violations," IET Gen. Transm. Distrib, vol. 4, no. 4, pp. 495-508, Apr. 2010.

[9]

Z. Bie, G. L. Zhang, B. Hua, M. Meehan and X. Wang, "Reliability evaluation of active distribution systems including microgrids," IEEE Trans. Power Syst., vol. 27, no. 4, pp. 2342-2350, Nov. 2012.

[10] S. Abapour, K. Zare and B. Mohammadi-Ivatloo, "Dynamic planning of distributed generation units in active distribution network," IET Gen. Transm. Distrib., vol. 9, no. 12, pp. 1455-1463, Sep. 2015. [11] Q. Yang, J. A. Barria and T. C. Green, "Communication infrastructures for distributed control of power distribution networks," IEEE Trans. Ind. Informat., vol. 7, no. 2, pp. 316-327, May 2011. [12] P. Vrba and et al., "A review of agent and service-oriented concepts applied to intelligent energy systems," IEEE Trans. Ind. Informat., vol. 10, no. 3, pp. 1890-1903, Aug. 2014.

[24] P. Siano, C. Cecati, H. Yu and J. Kolbusz, "Real time operation of smart grids via FCN networks and optimal power flow," IEEE Trans. Ind. Informat., vol. 8, no. 4, pp. 944-952, Jun. 2012. [25] P. Siano, "Evaluating the impact of registered power zones incentive on wind systems integration in active sistribution networks," IEEE Trans. Ind. Infomat., vol. 11, no. 2, pp. 523-530, April 2015. [26] H. Bollen M and F. Hassan, Integration of Distributed Generation in the Power System, John wiley & sons, Aug. 2011. [27] N. Etherden and M. H. J. Bollen, "Dimensioning of energy storage for increased integration of wind power," IEEE Trans. Sustain. Energy, vol. 4, no. 3, pp. 546-553, Jan. 2013. [28] N. Etherden, M. Bollen, S. Ackeby and O. Lennerhag, "The transparent hosting-capacity approach–overview, applications and developments," in Int. Conf. Exhib. Electr. Distrib., Jun. 2015. [29] Voltage characteristics of electricity supplied by public distribution systems, CENELEC EN 50160, 2010. [30] M. Sedghi, A. Ahmadian and M. Aliakbar-Golkar, "Optimal storage planning in active distribution network considering uncertainty of wind power distributed generation," IEEE Trans. Power Syst., vol. 31, no. 1, pp. 304-316, Jan. 2016. [31] B. Bletterie, S. Kadam, R. Bolgaryn and A. Zegers, "Voltage control with PV inverters in low voltage networks—In depth analysis of different concepts and parameterization criteria," IEEE Trans. Power Syst., vol. 32, no. 1, pp. 177-185, Jan. 2017. [32] B. Kroposki, K. Emery, D. Myers and L. Mrig, "Comparison of photovoltaic module performance evaluation methodologies for energy ratings," in IEEE 1st World Conf. Photovoltaic Energy Convers., Waikoloa, Dec. 1994. [33] S. H. Karaki, R. B. Chedid and R. Ramadan, "Probabilistic performance assessment of autonomous solar-wind energy conversion systems," IEEE Trans. Energy Conv., vol. 14, no. 3, p. 766–772, Sep. 1999.

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13 [34] F. A. Viawan, F. Vuinovich and A. Sannino, "Probabilistic approach to the design of photovoltaic distributed generation in low voltage feeder," in 9th Int. Conf. Probabilistic Methods Applied to Power Systems, Stockholm, Jun. 2006. [35] M. N. Kabir, Y. Mishra and R. C. Bansal, "Probabilistic load flow for distribution systems with uncertain PV generation," Appl. Energy, vol. 163, pp. 343-351, Feb. 2016.

[55] IEEE Standard for Interconnecting Distributed Resources with Electric Power Systems - Amendment 1, IEEE Std 1547a-2014 (Amendment to IEEE Std 1547-2003), pp.1-16, May 2014. [56] B. Noone, "PV Integration on Australian distribution networks: Literature review," Australian PV Association, UNSW, Australia, Sep. 2013.

[36] J. A. Duffie and W. A. Beckman, Solar Engineering of Thermal Processes, vol. 3, New York: Wiley, 2013.

[57] W. H. Kersting, "Series impedance of overhead and underground lines," in Distribution System Modeling and Analysis, CRC press, 2012 Jan 24, pp. 77-108.

[37] B. Y. Liu and R. C. Jordan, "The interrelationship and the characteristics distributionof direct, diffuse and total solar radiation," Sol. Energy, vol. 4, no. 3, pp. 1-19, Jul. 1960.

[58] G. Kerber, “Aufnahmefähigkeit von Niederspannungsverteilnetzen für die Einspeisung aus Photovoltaikkleinanlagen,” Doctoral dissertation, Technische Universität München, 2011.

[38] P. Bendt, M. Collares-Pereira and A. Rabl, "The frequency distribution of daily insolation values," Sol. Energy, vol. 27, no. 1, pp. 1-5, Jan. 1981.

[59] N. Etherden, V. Vyatkin and M. H. I. Bollen, "Virtual power plant for grid services using IEC 61850," IEEE Trans. Ind. Informat., vol. 12, no. 1, pp. 437-447, Feb. 2016.

[39] K. G. T. Hollands and R. G. Huget, "A probability density function for the clearness index, with applications," Sol. Energy, vol. 30, no. 3, p. 195–209, Dec. 1983.

[60] R. A. Shayani and M. A. G. De Oliveira, "Photovoltaic generation penetration limits in radial distribution systems," IEEE Trans. Power Syst., vol. 26, no. 3, pp. 1625-1631, Aug. 2011.

[40] G. Y. Saunier, T. A. Reddy and S. Kumar, "A monthly probability distribution function of daily global irradiation values appropriate for both tropical and temperate locations," Sol. Energy, vol. 38, no. 3, p. 169–177, Dec. 1987.

[61] S. Meliopoulos A, G. J. Cokkinides and X. Y. Chao, "A new probabilistic power flow analysis method," IEEE Trans. Power Syst., vol. 5, no. 1, pp. 182-190, 1990.

[41] J. A. Olseth and A. Skartveit, "A probability density model for hourly total and beam irradiance on arbitrarily orientated planes," Sol. Energy, vol. 39, no. 4, p. 343–351, Jan. 1987. [42] H. Suehrcke, "The effect of time errors on the accuracy of solar radiation measurements," Sol. Energy, vol. 53, no. 4, pp. 353-357, Oct. 1994. [43] H. Suehrcke and P. G. McCormick, "The frequency distribution of instantaneous insolation values," Sol. Energy, vol. 40, no. 5, pp. 413422, Dec. 1988. [44] M. Jurado, J. M. Caridad and V. Ruiz, "Statistical distribution of the clearness index with radiation data integrated over five minute intervals," Sol. Energy, vol. 55, no. 6, p. 469–473, Dec. 1995. [45] J. Tovar, F. J. Olmo and L. Alados-Arboledas, "One-minute global irradiance probability density distributions conditioned to the optical air mass," Sol. Energy, vol. 62, no. 6, pp. 387-393, Jun. 1998. [46] K. T. Hollands and H. Suehrcke, "A three-state model for the probability distribution of instantaneous solar radiation, with applications," Sol. Energy, vol. 96, pp. 103-112, Oct. 2013. [47] H. F. Assuncao, F. Escobedo J and P. Oliveira A, "Modelling frequency distributions of 5 minute-averaged solar radiation indexes using Beta probability functions," Theoret. Applied Climat., vol. 75, no. 3-4, pp. 213-224, Sep. 2003.

[62] T. Gerstner and M. Griebel, "Numerical Integration Using Sparse Grids," Numer. Algorithms, vol. 18, no. 3-4, pp. 209-232, Jan. 1998. [63] S. A. Smolyak, "Quadrature and interpolation formulas for tensor products of certain classes of functions," Soviet. Math. Dokl, vol. 4, pp. 240-243, Jan. 1963. [64] K. Petras, "Smolyak cubature of given polynomial degree with few nodes for increasing dimension," Numer. Math., vol. 93, no. 4, pp. 729753, 2003. [65] G. W. Wasilkowski and H. Woźniakowski, "Weighted tensor product algorithms for linear multivariate problems," J. of Complexity, vol. 15, no. 3, pp. 402-447, Sep. 1999. [66] T. Gerstner and M. Griebel, "Dimension–adaptive tensor–product quadrature," Computing , vol. 71, no. 1, p. 65–87, Aug. 2003. [67] F. Heiss and V. Winschel, "Likelihood approximation by numerical integration on sparse grids," J. Econometrics, vol. 144, no. 1, pp. 62-80, May 2008. [68] S. Conti and S. Raiti, "Probabilistic load flow using Monte Carlo techniques for distribution networks with photovoltaic generators," Sol. Energy, vol. 81, no. 12, pp. 1473-1481, Dec. 2007. [69] B. Ridley, J. Boland and P. Lauret, "Modelling of diffuse solar fraction with multiple predictors," Renew. Energy, vol. 35, no. 2, pp. 478-483, Feb. 2010.

[48] M. Ibanez, I. Rosell J and A. Beckman W, "A bi-variable probability density function for the daily clearness index," Sol. Energy, vol. 75, no. 1, pp. 73-80, Jul. 2003.

[70] J. F. Orgill and K. G. T. Hollands, "Correlation equation for hourly diffuse radiation on a horizontal surface," Sol. Energy, vol. 19, no. 4, pp. 357-359, Dec. 1977.

[49] H. F. Assuncao, F. Escobedo J and A. P. Oliveira, "A new algorithm to estimate sky condition based on 5 minutes-averaged values of clearness index and relative optical air mass," Theoret. Applied Climat., vol. 90, no. 3-4, pp. 235-248, Nov. 2007.

[71] D. T. Reindl, W. A. Beckman and J. A. Duffie, "Diffuse fraction correlations," Sol. Energy, vol. 45, no. 1, pp. 1-7, 1990.

[50] J. Boland, B. Ridley and B. Brown, "Models of diffuse solar radiation," Renew. Energy, vol. 33, no. 4, pp. 575-584, Apr. 2008. [51] J. H. Zhao, F. Wen, Z. Y. Dong, Y. Xue and K. P. Wong, "Optimal dispatch of electric vehicles and wind power using enhanced particle swarm optimization," IEEE Trans. Ind. Informat., vol. 8, no. 2, pp. 889899, Nov. 2012. [52] P. Lamaina, D. Sarno, P. Siano, A. Zakariazadeh and R. Romano, "A model for wind turbines placement within a distribution network acquisition market," IEEE Trans. Ind. Informat., vol. 11, no. 1, pp. 210219, Feb. 2015. [53] C. F. Walker and J. L. Pokoski, "Residential load shape modelling based on customer behavior," IEEE Trans. Power App. Syst, vol. 7, pp. 17031711, Jul. 1985. [54] IEEE Standard for Interconnecting Distributed Resources With Electric Power Systems, IEEE Std 1547-2003, pp.1-28, Jul. 2003.

[72] Z. Jin, W. Yezheng and Y. Gang, "Estimation of daily diffuse solar radiation in China," Renew. Energy, vol. 29, no. 9, pp. 1537-1548, Jul. 2004. [73] C. Furlan, A. P. de Oliveira, J. Soares, G. Codato and J. F. Escobedo, "The role of clouds in improving the regression model for hourly values of diffuse solar radiation," Appl. Energy, vol. 92, pp. 240-254, Apr. 2012. [74] J. Chandrasekaran and S. Kumar, "Hourly diffuse fraction correlation at a tropical location," Sol. Energy, vol. 53, no. 6, pp. 505-510, Dec. 1994. [75] A. De Miguel, J. Bilbao, R. Aguiar, H. Kambezidis and E. Negro, "Diffuse solar irradiation model evaluation in the north Mediterranean belt area," Sol. Energy, vol. 70, no. 2, pp. 143-153, Dec. 2001. [76] J. Soares and et al., "Modeling hourly diffuse solar-radiation in the city of São Paulo using a neural-network technique," Appl. Energy, vol. 79, no. 2, pp. 201-214, Oct. 2004.

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