Probabilistic Relational Models for Assessment of

Probabilistic Relational Models for Assessment of Reliability of Active Distribution Management Systems Johan König, Lars Nordström and Mathias Ekstedt Industrial Information and Control Systems Royal Institute of Technology Stockholm, Sweden {johank,larsn,mek101}@ics.kth.se Abstract— This paper presents the use of Probabilistic Relational Models (PRM) for reliability analysis of control systems for active distribution grids. The approach is based on two key concepts; first, it addresses both the reliability of primary system components and the supporting secondary, ICT-based systems. Secondly, the use of PRMs enables representation of architecture of the ICT systems, including for instance redundancy of hardware and allocation of software functions to several hardware devices. This later aspect is important, since allocation of software across different hardware platforms is a feature enabled by for instance the IEC 61850 standard. The increasing number of software dependent systems for controlling and supervising the power grid enhances the risk of software-caused failures. Thus, for reliable operation it is of high importance to not only concern primary component, but also the software and hardware of the secondary systems controlling it. A variety of methods exist for reliability analysis of secondary systems, however few address the issue of failing software together with failing primary components. The paper presents the underlying theory for Probabilistic Relational Models, and presents the steps necessary to use the technique. The paper is concluded with an example of application of the approach.

of the power grid’s behavior. As a result the operators must to a greater extent rely on the control and automation systems and their ability to provide correct functionality. Furthermore, a tighter coupling between the control and automation systems and the distribution grid increases the need for a total reliability concern, focusing on both parts. Not until recently has the focus on reliability mainly stressed the importance of power system equipment (e.g. breakers, transformers). However, closer interaction with control and supervisory system calls for a wider focus comprising both domains. Moreover, the increased importance of software systems performing critical and complex functions enhances the risk of software-caused failures.

Keywords-component; Reliability Analysis, Probabilistic Relational Models, Control and Automation Systems, Active Distribution Grids, Bayesian Networks.

B. Scope of the paper The scope of this paper is to present a novel method for reliability analysis of software intense control and automation systems that includes the reliability of primary system components. The framework is based on Probabilistic Relational Models - specifying a template for a probability distribution over an architecture model. As such, the methods may be adapted to study of other system architecture aspects, but the focus herein is on reliability.

I.

INTRODUCTION

A. Background To provide a safe and reliable infrastructure for electricity distribution, able to handle integration of renewable energy resources, present control and supervisory systems need to be revised. The former paradigm of unidirectional flow of power, radial distributed from centralized generation out to consumers is changing. Also, the penetration of decentralized, small scale, generation into the distribution grid enhances the risk of an uncontrollable reversed power-flow. To be able to provide safe and reliable operation, the present control and supervisory systems must be developed to reflect this advancement. The growth of complexity by an increased integration of control and automation systems, together with a closer interaction with grid operation, affects the prior understanding

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The introduction of standards such as CIM [1] and IEC 61850 [2] enables the possibility to adapt analysis frameworks according to their descriptive models, metamodels, which define objects and their relations. Further, the standardization has also allowed for functional allocation independent for the IT systems infrastructure, i.e. functions can be more freely allocated to any compliant device, leading to more complex system architectures.

C. Structure of the paper The remainder of this paper is structured as follows. Section II contrasts the proposed method with other reliability analysis methods. Section III describes Probabilistic Relational Models. Section IV is the locus of the main contribution, presenting an analysis framework where the methods of the previous section are put to use for reliability analysis of software intense control and automation systems and their interaction with primary components. Section V gives some practical examples of how the method can be employed.

PMAPS 2010

Section VI discusses the implications of the contribution and concludes the paper. II.

RELIABILITY ANALYSIS METHODS

A. Reliability analysis Reliability is defined as a measure of equipment or a system to perform its intended function under specified conditions for a specified period of time [3]. As stated here, reliability could either regard equipment or systems; active distribution management systems are often dependent on both hardware and software together with primary equipment. Thus, reliability assessment is becoming a cyber-physical concern. Reliability analysis in general is widely adopted within the power industry. Most frequently applied are methods focusing on primary system component e.g. transformers, breakers, wires etc, and the architecture which they constitute e.g. substations. In [4] issues regarding reliability of power systems are addressed and a variety of methods are applied to different systems and scenarios. A further illustration of the range of probabilistic methods for reliability analysis applied to power system is found in [5]. Applying these techniques to systems based on both hardware and software, e.g. control and supervisory system, must however be done with care. A major difference is the failure process, where failure models for hardware components traditionally consider the physical deterioration and wear out whereas software components are not affected by physical stress. Instead, the occurrence of software failure is a matter of flaws in the code; in other words being a design issue [6][7][8]. Thus, compared to hardware it is very difficult to judge the impacting factors that affects the reliability of software in an objectively manner. Unlike hardware software can be free of faults, and also once a fault is removed from the software, it will never cause the same fault again [6]. However, software components are typically developed and updated over time, resulting in the probability of introducing new faults. The subject of measuring and predicting reliability of software components is for example addressed in [7], [8], [9] and [10]. There are two main ways to look at software reliability; program-as-it-is, focusing on numbers of bugs it contains, or program-as-it-performs, focusing on the actual event that a fault occur [9]. Here we focus our analysis on the latter since reliability concerns the actual event of failure. B. Reliability analysis approaches A number of probabilistic methods for reliability analysis of control and automation systems within the power systems domain have previously been proposed and described in the literature, e.g. Fault Tree Analysis (FTA) [12][13], Reliability Block Diagrams (RBD) [12][14], Markov chain models [12][15] and Bayesian networks [12][16]. One of the most frequent adopted methods for reliability analysis is Fault Tree Analysis which translates the failure behavior of a physical system into a visual diagram and a logical model. The modeling structure of FTA does however only allow the modeler to visualize the system architecture from a relational perspective based on a primary component’s

dependency on relation subcomponents and their logical relation to the primary component. Applications of FTA analysis to control and protection systems are presented in [17], [18] and [19]; however the dependencies between hardware and software components are not taken into consideration or are abstracted. A Reliability Block Diagram is, similar to FTA, a method for architecture based reliability analysis. The concept behind RBD is to identify undirected relational paths between components within the architecture. First two nodes s and t are defined. Secondly, relational paths comprising system components between the nodes are identified. A system is said to be available if there exists at least one path comprising a chain of available components from s to t. Both FTA and RBD are, however, architecture based analyze methods limited in the sense that the analysis they allow only considers failure of stochastically independent components, thus not allowing capturing of inter-component dependencies as for example failure propagation. In [20], [21] and [22] is reliability analysis of control and protection systems using RBDs presented. Nor here are dependencies between hardware and software components taken into consideration or are abstracted. A third approach is state-based analysis. State-based analysis is not, as for architecture based analysis, limited to stochastically independent failure behaviors between components [8], thus allowing capturing of component failure propagation. Instead, state-based methods enumerate all possible failure states of the system. A drawback with statebased analysis using Markov chains is the model’s exponential growth when expanding the number of components. The applications of Markov chains for reliability analysis of control and protection systems are found in [23], [24] and [25]. For the modeling and analysis of uncertain phenomena, Bayesian networks are often proposed [26]. Bayesian networks are composed of nodes with associated values, and arcs between the nodes. The nodes’ probabilistic dependencies on each other are specified by the arcs and by so called conditional probability distributions. Hence, Bayesian networks provide support for modeling causality, causal uncertainty and empirical uncertainty. In [27] Bayesian networks are applied for reliability assessment of software based digital systems. Further, in [28] the same authors present a case study demonstrating the application of Bayesian networks on reliability estimation of software design on a protection relay and in [29] an a posteriori estimation of the failure rates for different software versions of the protection relay is presented. The adaptation of Bayesian networks has been done with success in various fields; however they often are inadequate for describing large and complex domains [30]. III.

PROBABILISTIC RELATIONAL MODELS

Probabilistic Relational models (PRM) extend Bayesian networks with the concept of objects, their properties, and relations between them [30]. A PRM specifies a template for a probability distribution over an architecture model. The template describes the metamodel for the architecture model, and the probabilistic dependencies between attributes of the architecture’s objects. A PRM, together with an instantiated architecture model of specific objects and relations, defines a

probability distribution over the attributes of the objects. The probability distribution can be used to infer the values of unknown attributes, given evidence of the values of a set of known attributes. A. Relational language An architectural metamodel defines a set of classes, = X1,…,Xn. Each class is associated with a set of descriptive attributes and a set of reference slots. The set of descriptive attributes of a class X is denoted (X). Attributes A of class X is denoted X.A and its domain of values is denoted V(X.A). The set of reference slots of a class X is denoted (X). We use X.ρ to denote the reference slot ρ of X. Each reference slot ρ is typed with the domain type Dom[ρ] = X and the range type Range[ρ] = Y, where Y . A slot ρ denotes a relation from X to Y similar to Entity-Relationship diagrams. For example, we might have a class Function with the reference slot AssignedTo whose range is the class Device. For each reference slot ρ we have an inverse reference slot ρ-1 denoting the inverse relation. Using the prior example, the class Device has an inverse reference slot AssignedTo-1 to the class Function. An architecture instantiation (or an architecture model) specifies the set of objects in each class X, the values for the attributes, and the reference slots of the objects. It specifies a particular set of components, functions, etc., along with values for each of their attributes and references. For future use, we also define a relational skeleton σr as a partial instantiation which specifies the set of objects in all classes as well as all the reference slot values, but not the attribute values. A probabilistic relational model Π specifies a probability distribution over all instantiations of the metamodel . This probability distribution is specified as a Bayesian network [31], which consists of a qualitative dependency structure and associated quantitative parameters. The qualitative dependency structure is defined by associating with each attribute X.A a set of parents Pa(X.A). Each parent of X.A has the form X.τ.B where B (X.τ) and τ is either empty, a single slot ρ or a sequence of slots ρ1, . . . , ρk such that for all i, Range[ρi] = Dom[ρi+1]. For example, the attribute reliability of class Function may have as parent Function.AssignedTo.reliability, thus indicating that the reliability of the class function depends on the reliability of the class which it is assigned to. Considering the quantitative part of the PRM, given a set of parents for an attribute, we can define a local probability model by associating a conditional probability distribution (CPD) with the attribute, P(X.A|Pa(X.A)). We can now define a probabilistic relational model (PRM) Π for a metamodel as follows. For each class X and each descriptive attribute A (X), we have a set of parents Pa(X.A), and a conditional probability distribution (CPD) that represents PΠ(X.A|Pa(X.A)). Given a relational skeleton σr (i.e. a metamodel instantiated to all but the attribute values), a PRM Π specifies a probability distribution over a set of instantiations consistent with σr: | ,Π

P . |Pa .

where σr(X) are the objects of each class as specified by the relational skeleton σr. A PRM thus constitutes a formal machinery for calculating the probabilities of various architecture instantiations. This allows us to infer the probability that a certain attribute, e.g. reliability, assumes a specific value, given some (possibly incomplete) evidence of the rest of the architecture instantiation. IV.

PRM FOR RELIABILITY ANALYSIS

The PRM presented in this section contains a set of classes and attributes suitable for reliability analysis of control and automation systems together with primary equipment. An extensive analysis framework is outside the scope of this paper; however is part of ongoing research efforts. The PRM is constructed in line with the PRM presented in [32] with an extension of primary equipment e.g. transformers and circuit breakers. A. Probabilistic Relational Model A total of six classes have been defined for the PRM Function, Equipment, Application component, Communication function, Device and Network, which constitutes a representation of both primary equipment and ICT components together with the functions they implement. The last four classes are strictly related to control and automation systems while Equipment represents primary power system components. The class Functions, however, may either be assigned to primary components (e.g. circuit breaker trip) or software dependent systems (e.g. control switch) or both; here represented by multiple AssignedTo reference slots. Both equipment’s and application component’s reliability are parents to the reliability of the class Function. Recall from the previous section that parents to an attribute X.A are defined in the form X.τ.B, where B (X.τ). The class here is Function with reference slot AssignedTo, i.e. Dom[AssignedTo] = Function, whose range type is either ApplicationComponent or Equipment, i.e. Range[AssignedTo] = {ApplicationComponent, Equipment}. With the attribute B here being reliability, the parent relation is defined as Function.AssignedTo.reliability. Moreover, a Function has an interrelation to itself, Trigger, with parental relation Function.Trigger-1.reliability; which represents the ability for functions to be reliant on each other’s execution. Another class is Communication function which represents the logical functionality provided by the ICT infrastructure, e.g. receiving measurements or sending control signals. The reference slot UsedBy has the domain type Communication Function and range type Function. With the attribute being reliability, the parental relation holds Function.UsedBy-1 .reliability. The Communication function is the only class, in addition to Function, that only holds the attribute of reliability. A Communication function is realized by class Device connected via class Network to other Devices. The Communication function’s reliability has as parents CommunicationFunction.RealizedBy.reliability and CommunicationFunction.RealizedBy.Connects-1.reliability.

The structural classes have in addition to the logical classes (functions) additional attributes, here represented as attribute i, i = 2,…,n (i =1 being reliability). These are attributes that causally influence the reliability of its own class. Equipment may, for example, hold the attribute of age which causally relates to its reliability. For Application components (software) an attribute may, for example, be complexity e.g. measured by the number of lines of code or the measure of cyclomatic complexity. Similar attributes can be defined for Network and Device. If regarding these undefined attributes, the reliability of class Equipment have the parent Equipment.attributei where i = 2,…,n. Equivalent parental relation does also apply for class Network and Device. The Application component’s reliability, in addition to the prior defined relation, also has as parent ApplicationComponent.AssociatedWith.reliability. In Fig. 1 the complete PRM is presented including classes, reference slots between classes, the attributes of each class and the attribute relations.

The scenario is based on an example for managing an increased penetration of Distributed Generation (DG) presented in [33]. Voltage rise effects may arise in local parts of the distribution network with high penetration; but can be reduced by active voltage control. By implementing a variable transformer at a feeder, local voltage management can be conducted at times with high generation output, hence reducing the need for generation curtailment. To apply the framework we first and foremost need to define the goal of the analysis. Without a clear goal it is difficult to delimit the components of interest, which could result in unnecessary extensive and complex models. Since the example here is based on active voltage management and the analysis framework on reliability, a reasonable scope would be to analyze the reliability of the automatic voltage control functionality. A. Automatic voltage control example The active voltage management is performed by a controllable transformer located at a feeder. The transformer holds an on-load tap changer (OLTC) controlling the demand side via a relay. Given the network’s state, the relay set-points can be adjusted to perform desired control actions for control of the voltage level. Additionally, a set of input parameters such as network constraints and generation capacity delimits the possible control behavior and thus also must be considered. For the analysis we, however, only regard one input measurement, provided by an IED located at the generator unit.

Figure 1. A PRM for reliability analysis

For each attribute in the PRM a conditional probability distribution is defined. However, defining a complete set of probabilities for each CPD is outside the scope of this paper. Instead a simple example is presented in Table I, here representing the reliability of the class Equipment, with domain of values, V(Equipment.reliability) = {X1, X2} and attributek with V(Equipment.attributek) = {Y1,Y2}. TABLE I.

EXAMPLE CPD FOR RELIABILITY OF EQUIPMENT

Equipment.attributek Equipment. reliability

X1 X2

Y1

Y2

P(Eq.rel=X1 | Eq.attrk=Y1)

…

P(Eq.rel=X2 | Eq. attrk =Y1)

…

The probability in bold font in the CPD is the probability of the Equipment’s reliability holding the value X1 if the value of attributek is Y1. V.

APPLICATION TO A TYPICAL RELIABILITY SCENARIO

This section provides a demonstration of the analysis framework applied to a generic active distribution network.

Figure 2. System setup - automatic voltage control

We begin by defining a relational skeleton with instantiated set of objects for all classes together with reference slots. Here, the first object being the actual Automatic voltage control (AVC), an instantiated object of class Function, which is the main target of the analysis. The AVC function is in turn dependent on the two functions Voltage control and Control scheduling; also instantiated objects from class Function reference slot Trigger. Hence, the parents to the AVC’s reliability are the reliabilities of the Voltage control and the Control scheduling, in PRM formalism represented as Pa(AVC.reliability) = AVC.Trigger-1.reliability. For the remainder of the example, attributes and their parents will not be fully described. Instead, for a complete description see Section IV.

The function Voltage control is assigned to both a Relay application and an OLTC transformer, where the prior is an object from class Application component and the latter from class Equipment. The Relay application is in turn associated with an object Relay (Device). The Control scheduling depends on a State estimation function (Function), where both functions are assigned to a DMS application associated with a Microcomputer (Device). The State estimation function is in turn using Measurements taken at the generator. The object Measurement – class Communication function - is realized by an IED (Device) located at the distributed generator. A wide area network, WAN, connects the IED and the Microcomputer which allows measurements to be sent between the two devices. The complete relational skeleton is presented in Fig. 3.

To asses the reliability of the AVC we further need to define the CPDs for reliability of the Functions. Here, the reliability of a function is defined to be in a true state if, and only if, all relating objects’ reliability is true. In Table III a CPD for the functions reliability is presented. If, we assume, all other components are 100% reliable, the reliability of the AVC will attain the reliability of the OLTC transformer, i.e. 99.1% if the temperature is high. TABLE III.

CPD FOR RELIABILITY OF FUNCTIONS

Function.Trigger-1 .reliability Function.AssignedTo .reliability Function. reliability

True

True

False

False

True

False

True

False

True

1

0

0

0

False

0

1

1

1

Until now the oil temperature has either been high or low with known state. However, it is also possible to define the probability of the temperature being in a certain state, e.g. the P(Equipment.oilTemp = High) = 20% resp. P(Equipment.oilTemp = Low) = 80%. If, using the probabilities instead of evidence on the state of the oil temperature, the probability of the AVC’s reliability being true is 99.7%. VI. Figure 3. Relational skeleton of the AVC system

As seen in the figure attributes of the structural elements has been defined. For example, a device has the attribute age as a parent to its reliability, application components has lines of code, network has packet loss and equipment has oil temperature. Depending on the values of the attributes and the CPDs defined to each of them, it is possible to estimate the reliability of the automatic voltage control function. To give a short example here focusing on the OLTC transformer and its relating functions, we define the oil temperature of the OLTC transformer with value {high, low} and reliability with value {true, false}. In the example, the definition of high and low temperature is arbitrary; however for a real framework clear definitions are necessary in order to set the CPDs. The values defined for reliability represent the probability that the object is in a true or a false state, i.e. if it is working or not. In Table II the CPD for the reliability of the equipment – here being the OLTC transformer - is given. The probability in bold writing is read P(Equipment.reliability = true|Equipment.oilTemp = High) = 99.1%. If we instead were to have low oil temperature, the probability of the state of the transformer being true, i.e. in a working state, is 99.8%. TABLE II.

CPD FOR RELIABILITY OF EQUIPMENT

Equipment.oilTemp

High

Low

True

0.991

0.998

False

0.009

0.002

Equipment. reliability

CONCLUSIND REMARKS AND FUTURE WORK

The contribution of the present paper is two-fold. First, we have argued for and demonstrated the general feasibility of performing reliability analysis using probabilistic relational models. Second, we have shown how this analysis can be adapted to fit the specific context of the electric power industry when designing control and automation systems for active distribution networks together with primary equipment. A. Enhancment of PRM Of interest for future work, and parts of ongoing research, is the enhancement of the PRM; focusing on a more detailed description of classes and together with their attributes. The enhancement of the PRM includes a domain specific model based on the IEC61850 standard with the extension of defining attributes causally influencing objects reliability and their relations. B. Continuous variables Another interesting enhancement for the analysis framework is the integration of Hybrid Bayesian networks. These networks combine the ability to use both discrete states and continuous states, i.e. the use of probability distributions. REFERENCES [1] [2] [3]

[4] [5]

Distributed Management Task Force, Inc. (DMTF), “Common Information Model (CIM) Specification”, version 2.2, June 14, 1999. IEC 61850: “Communications Networks and Systems in Substations”, International Standard, 2003 K-P. Brand, V. Lohmann and W. Wimmer, Substation Automation Handbook, Utility Automation Consulting Lohmann 2003, ISBN 385758-951-5 R. Billinton and RN Allan, “Reliability Evaluation of Power Systems”, 2nd ed. New York: Plenum, 1996 R. Billinton, M. Fotuhi-Firuzabad, and L. Bertling, “Bibliography on the application of probability methods in power system reliability

[6] [7] [8]

[9] [10]

[11] [12] [13] [14] [15]

[16] [17]

[18]

[19]

[20]

evaluation:1996–1999,” IEEE Trans. Power Syst., vol. 16, no. 4, pp. 595–602,Nov. 2001. M. Xie, Software Reliability Modeling. World Scientific, Singapore, 1991 H. Pham, Software Reliability. Springer-Verlag, Singapore, 2000 I. Eusgeld, F.C. Freiling and R. Reussner, Dependability metrics. Advanced lectures. Berlin: Springer (ISBN 978-3-540-68946-1/pbk). Lecture Notes in Computer Science 4909, 233-240 (2008). N.E. Fenton and S.L. Pfleeger, Software Metrics: A Rigorous and Practical Approach, 2nd ed., PWS Publishing Co., 1998. L.M. Laird and M.C. Brennan, Software Measurement and Estimation: A Practical Approach. John Wiley, New York, 2006. ISBN: 0471676225. B. Littlewood, “How to measure software reliability and how not to.” IEEE Trans. Reliability R-28 2, 1979, pp. 103–110. M. Rausand and A. Høyland, System Reliability Theory: models, statistical methods, and applications, Wiley, New York, 1994. W.E. Vesely, et al. Fault Tree Handbook. Washington, DC, USA:US Nuclear Regulatory Commission, 1981. B.W. Johnson, Design and Analysis of Fault-Tolerant Digital Systems, pp. 394-402. Reading, Mass.: Addison-Wesley, June 1989 W. Ching and M. Ng,. Markov Chains: models, algorithms and applications. International Series on Operations Research and Management Science, Springer, New York, 2006. F.V. Jensen. An Introduction to Bayesian Networks. Springer, New York, 1996 L. Pottonen, U. Pulkkinen and M. Koskinen, “A method for evaluating the reliability of protection”. Developments in Power System Protection, PAI Centre，Amsterdam，2004. C. L. Su and Y. C. Chang, “A SCADA system reliability evaluation considering performance requirement,” Proceedings of the International Conference on Power System Technology, Singapore, 21-24 November 2004. P.A. Anderson, B. Flemming, T.J. Lee, and E.O. Schweitzer, III, “Reliability Analysis of Transmission Protection Using Fault Tree Methods,” Proceedings of the 24th Annual Western Protective Relay Conference, Spokane, Washington, October, 1997 Y. He, Modelling and Evaluating Effect of Automation, Protection, and Control on Reliability of Power Distribution Systems. Ph.D. thesis, KTH, Stockholm, 2002. ISBN: 91-7283-345-9

[21] P. Zhang, L. Portillo, M. Kezunovic, "Reliability and Component Importance Analysis of All-Digital Protection Systems," IEEE 2006 PES Power Systems Conference and Exposition, Atlanta, GA, November 2006. [22] B. Hussain, C.E. Beck, T.E. Wieman, C.R. Sufana, “Transmission System Protection:A Reliability Study” IEEE Transactions on Power Delivery, Vol. 12, April 1997, pp 675-680. [23] L. Andersson, K.P. Brand, C. Brunner, W. Wimmer “Reliability investigations for SA communication architectures based on IEC 61850”, IEEE PowerTech 2005, St.Petersburg, Paper IEEE_PPT05-604 [24] Z.B. Tan,. “Reliability and availability analysis of two-unit warm standby microcomputer systems with self-reset function and repair facility”, Microelectronics and Reliability, 37, 1251-1253, 1997. [25] P.M. Anderson, R.F. Ghajar, G.M. Chintaluri, and SM Magbuhat, “An improved reliability model for redundant protective systems - Markov models,” IEEE Trans. Power Syst., vol. 12, no. 2, pp. 573–578, May 1997 [26] R. Neapolitan, Learning Bayesian Networks. Englewood Cliffs, NJ: Pearson Education (Prentice Hall), 2004. [27] B.A. Gran and A. Helminen, A Bayesian belief network for reliability assessment, Safecomp 2001 2187, 2001, pp. 35–45. [28] A. Helminen, “Case Study on Bayesian Reliability Estimation of Software Design of Motor Protection Relay”. SAFECOMP 2007: 384396. [29] A. Helminen and U. Pulkkinen, “Quantitative reliability estimation of a computer-based motor protection relay using Bayesian networks”, Safecomp 2003 2788, 2003, pp. 92–102. [30] L. Getoor and B. Taskar, Introduction to statistical relational learning. MIT Press, 2007 [31] F.V. Jensen, Bayesian Networks and Decision Graphs. Secaucus, NJ, USA: Springer-Verlag New York, Inc., 2001. [32] J. König, U. Franke and L. Nordström, ” Probabilistic Availability Analysis of Control and Automation Systems for Active Distribution Networks”, To appear, Transmission and Distribution Conference and Exposition, 2010 IEEE/PES. [33] G. Strbac, N. Jenkins, M. Hird, P. Djapic, and G. Nicholson, “Integration of operation of embedded generation and distribution networks,” Manchester Center for Electrical Engineering, Tech. Rep., 2002.