DYNA

2.6.1. Mathematical model

The location of Devices of different types in the network determines different effects on the loads connected to it. This is according to which device is closer (upstream) to these loads. The interrupter devices segment the network and have two functionalities: reaction to failures and maneuver under load. Within a segment, that is, in the part of the network between two Interrupters, there can be many combinations of lines, loads (users), fuses and disconnectors.

The different combinations determine which of the users are left without supply and for how long (see Table 1).

Table 1: Times without supply and quantity of interruptions per case.

Source: The authors.

The following four cases summarize all the configuration possibilities of a segment for these three types of device, illustrated in Fig. 1. All other combinations are redundant with respect to these four types.

Figure 1: Device combination cases

Source: The authors.

Case 1: Interrupter (I)

In this case, which is the most basic, there is only a combination of lines and loads between two Interrupters. If there is a failure in any of the sections, the entire network downstream of the switch is left without supply and cannot be restored to any of them until the fault is repaired. It is assumed that in the root node of the network there is an interrupter properly located.

Case 2: Interrupter Disconnector (ID)

In this case, before a fault in a section between I and D, proceed as in case 1. For a fault after D, I is actuated and left without supply downstream. However, the supply for the sections between I and D can be restored by opening D, thereby disconnecting the entire network downstream from D and by actuating I to restore the service to users between I and D. Once a fail occurs, D must be actuated to restore the downstream service, but first I must be pressed to interrupt the supply again. Once the cooling time of Iis over, the service is restored throughout the network.

Case 3: Interrupter Disconnector Fuse (I D F)

In this case, for a fault in a section between I and F, proceed as in case 2 or 1, depending on whether D is located. For a fault after F, this is triggered and left without supply downstream. Users between I and F are not affected by the cut.

Note that there may be devices of typeD between I and F which are of no use in this case. Once the fault has been repaired, F must be replaced, for which the supply downstream of I must be interrupted. Assuming that the cooling time of I is greater than the fuse replacement time, and after t CD is over, the network service is restored.

Case 4: Interrupter Disconnector Fuse Disconnector (I D FD).

In this case, when a fault occurs before D, proceed as in the previous case. For a fault after D, F is operated leaving no downstream supply. The fault does not affect the upstream of F. However, considering t r > t CD , the supply to users can be restored between F and D, opening D and replacing the corresponding fuse.

For the latter, it is necessary to cut off the supply from I, replace F and wait for the cooling time to re-start I, taking into account that D must be opened before completing the process. Once this first maneuver is performed, the fault is repaired and I must be restarted, cutting the supply, pressing D and waiting t CD to restore the supply downstream of D.

2.6.2. PSO Metaheuristic

The use of Metaheuristics makes it possible to obtain algorithms of general purpose, offering solutions which are generally satisfactory for a vast number of problems. This method is easy to implement and can be used with parallel processing. Within the set of metaheuristics available in the state of the art, the PSO metaheuristic is used due to its wide versatility, ease of implementation and good performance. The set of variants of the PSO metaheuristic is known as X-PSO forms.

In the Classic PSO model there is a set of alternatives called population or swarm (whose feasibility exhibits degrees of aptitude), which are called particles. From one iteration to the next, each particle moves in the search space, according to a certain rule of motion that depends on three factors, which are explained below. Vector b i k represents the best positions that the particles have reached individually in the previous iterations (an aspect referred to in the model as past life of the particle); Then it will correspond to the individual optimum of the particle i in its past life. In the same way, the global optimum achieved by the particle system will be indicated by b G k until the present iteration, according to the rule of motion in eq. (6) [7]:

The velocity vector for the i-th particle is expressed in eq. (7):

This rule of movement was represented in Fig. 2.

Figure 2: Movement Rule of PSO in two dimensions.

Source: The authors.

The term V i k represents the inertia or habit of the particle i: tends to maintain its motion in the direction in which it moved in the previous iteration.

The term w C ∙ r 1 k ∙ b i k − X i k represents the memory or cognitive ability of the particle: it is attracted to the best point in the search space achieved individually in its past life.

And the third term w S ∙ r 2 k ∙ b G k − X i k represents the cooperation between the whole swarm, or social capacity of the particle with respect to the swarm: the particles share information on the best overall position reached by the swarm.

The incidence of these factors on each particle is given by the constants or parameters of the model, w c and w s . The parameter w c is called cognitive constant and the parameter w s is called the social constant of the swarm.

The random numbers r 1 k and r 2 k are uniformly distributed in [0,1], U[0,1], and their objective is to emulate the stochastic behavior exhibited by the population or swarm in each iteration (somewhat unpredictable).

2.6.3. Particle swarm optimization metaheuristic with constriction factor

With this modification of the Classic PSO model a better convergence of the particles in time is promoted, by the constriction or reduction of the particle oscillation amplitude as they are focused on a certain focal point. In this case, the classical speed operator is modified as follows:

Where χ is called the Constriction Factor, and is obtained by the following expression:

With: ; and

A commonly used parameter configuration is φ=4.1, φ M = φ C =2.05. This results in a value of χ≈0.73.

When κ=1, the convergence is slow enough to allow detailed exploration before the search stabilizes at the point of maximum fitness found [7].

2.6.4. Extension to Multiobjective Domain according to decision theory in Fuzzy environment

Bellman Zadeh's principle:

Although the approach originally adopted by the PSO metaheuristic was Mono-Objective, in reference [7] an extension to the Multi-Objective domain of the EPSO is proposed. This method is referred to as FPSO (Fuzzy PSO), for the use of Fuzzy Sets. As detailed in the above mentioned reference, they are able to capture non-stochastic value uncertainties.

To formalize the multiobjective extension, the application of the principle of decision making in fuzzy environments formulated by Bellman and Zadeh is proposed [3-6]. This extension, applicable to any variant strategy of the classical PSO metaheuristic, is the one used in this work.

Consider a set of fuzzy objectives {O}={ O 1 , O 2 ,…, O n } whose membership functions are μ Oj , with j=1..n, and a set of fuzzy constraints {R}={ R 1 , R 2 ,…, R h } whose membership functions are μ Ri with i=1..h.

The Decision fuzzy set, results:

Where C is a fuzzy sets operator called “confluence”.

The most common confluence, is the intersection.

Then, the membership function of D is expressed as:

Where C is an operator (called, in general, t-norm) between values of membership functions.

For example, if the confluence for certain instance is C ≡∩ (intersection), then C is the t−norm≡min (minimum value of all membership function in eq. (11)).

If t x,y : 0,1 × 0,1 → 0,1 is a t-norm, then: a) t 0,0 =0;t x,1 =x; b) t x,y =t y,x ; c) if x≤α and y≤βt x,y ≤t α,β ; and d) t t x,y ,z =t x,t y,z . Notice that all fuzzy sets (Objectives and Constraints) are “mapping” in the same fuzzy set of decision, D, and are treated the same way.

The crisp set S of “optimal” solutions or alternatives x ∗ is:

Fuzzy membership functions used:

For each objective (or restriction), the fuzzy functions associated to their objective or restriction are defined as follows: consider an upper and a lower limit in the possible values of the variable corresponding to a certain objective or constraint m, v m . These limits are referred to as v m Sup y v m Inf , respectively.

When v m ≤ v m Inf the fuzzy function μ m = 1, since v m is required to be minimized. Farther, when v m ≤ v m Sup , μ m =0 for the same reason. The fuzzy bounds of v m in [ v m Inf , v m Sup ] are calculated by the equation of the line passing through the points ( v m Inf , 1) and ( v m Sup , 0). In this way, the degree of satisfaction of the objective decays as it moves away from the value v m Inf and is not accepted from the value v m Sup .

In addition, let EW m be its exponential weight, whose effect on the fuzzy function is its contraction ( EW m >1) or dilation ( EW m <1). In this way, there is greater or minor importance of the criteria analyzed in the total fitness function. For the case EW m =1, the assumed membership function is linear. The lower and upper bounds of the fuzzy functions were obtained by simulations for the extreme scenarios. The upper boundaries correspond to the network without inversion in devices (except investment cost). The lower boundaries correspond to the network with Interrupters in all the sections.

The aptitude function μ D was obtained with the t-norm called Einstein Product, whose arguments are the membership functions of fuzzy functions associated to each objective. All the values used are shown in the results section.

The fuzzy functions of CE and ENS were presented in eq. (13) and (14).

Definition of the t-norm used:

The confluence operator must be established between the fuzzy sets in order to obtain the maximizing decision that will define the value of the fuzzy fitness function at each iteration of the F-PSO. For this, a t-norm is used, which is a function t defined in the interval [0, 1]([0, 1] also applied in [0, 1]. In the state of the art the Einstein Product was adopted:

Where x and y are generic membership functions. Then the fuzzy maximizing fitness function for the reliability problem is:

Intrinsic Cost (IC):

In the present research line, the economic valuation of the optimized criteria was made through the 'Intrinsic Cost' index (IC). This was defined by the derivative of the variable related to the criterion to be optimized, with respect to the variable generally related to the cost:

Where ∂ μ CE ∂CE and ∂ μ ENS ∂ENS are the derivative of the criteria analyzed with respect to their fuzzy functions (see eq. (13) and (14)).

Then the expression of the IC is:

In these works, the mathematical expression to determinate ∂μ CE ∂ μ ENS was obtained from the aptitude function applying the clause 'Ceteris Paribus'. This clause, which is generally used in economy, stands for 'all other things being unchanged or constant'. Then, the other variables associated with the optimization criteria were considered constant. From this assumption, by expressing the membership functions of these criteria and the t-norm used, the cost criterion was expressed as a function of the variable associated with the desired variable. This approach leads to extensive mathematical expressions when the number of objectives to be maximized or minimized is large. Therefore, another expression of intrinsic cost is proposed, by changing the development of one of its components. It starts from the membership function obtained from the fuzzy relationship between the investment cost criterion and ENS criterion as follows:

The logarithmic function and the properties associated to the product and division was applied to the eq. (21).

Differentiating the eq. (22) with respect to the fuzzy function μ CE .

Where f μ ENS , μ CE is the derivative of the third term of eq. (22).

Given that ∂ logμ ∂ μ CE =0, then the following is obtained:

Economic Penalty:

From the intrinsic cost of the ENS, the economic penalty is defined (see eq. (20)), and this refers to the resulting cost because the optimal alternative has not been selected.

Where EN S ∗ refers to the optimal (most satisfactory) Energy Not supplied and ENS′ refers to the non-optimal alternative. IC ENS ∗ , is obtained from (17).

Multi-objective analysis for each node of the studied system:

An analysis of each node of the MVPDS was proposed through the application of the fuzzy multiobjective extension for the analyzed criteria: ENS and CE. Following this proposal, the eq. (20) can be expressed for each node in the system, obtaining the intrinsic cost and the economic penalty per node. This progress makes it possible to distinguish penalties according to the importance and impact of each node.

The ENS at each node was obtained through an extensive and deep analysis of the impact that each load has on the total Energy Not Supplied. Due to the depth and complexity of this analysis, the algorithm and computational model required has not been described in detail in this work. Device Acquisition Costs for each node were obtained simply by observing the type and cost of the device installed.