Alexander Suzdalenko
email: aleksandrs.suzdalenko@rtu.lv

Ilya Galkin

Instantaneous, Short-term and Predictive Long-term Power Balancing Techniques in Intelligent Distribution Grids

Keywords: Intelligent Distribution Grid, Nanogrid, Energy Management, Instantaneous Power Balancing, Short-Term Energy Management, Pricing

An increased number of distributed small generators connected to the power grid allows higher total efficiency and higher stability of electrical power supply by exporting energy to the grid to be achieved during peak demand hours. On the other hand, it poses new challenges in structuring and developing the control approaches for these distributed energy resources. This paper proposes an improved method of real-time power balancing targeted to reaching long-term energy management objectives. The novel long-term energy management technique is proposed, that is based on load categorization and regulation of energy consumption by regulating electricity price function estimated with the proposed mathematical model. The method was evaluated by a LabVIEW model by simulating various types of loads. The price function for the defined energy generation pattern from renewable energy sources was obtained.

Introduction

The dynamic of worldwide installed power utilizing renewable energy sources (RES) is increasing year by year [START_REF] Masson | Global market outlook for photovoltaic until 2016[END_REF], showing global awareness of climate change and the footprint of human behavior on the nature: like mining activities that change the landscape and damages caused by oil plants. Another problem is related to the total effectiveness of energy from the primary fuel. It relates to the total system efficiency of energy delivery to the end-user from a mining site, which includes energy losses at middle stages, like use of energy at the mining stage, energy conversion losses, transmission (mechanical and electrical) losses, as well as end-device efficiency. Consequently, locally generated energy (especially from renewable energy sources) is preferable because it excludes most of the mentioned losses.

As more distributed small generators are connected to the power grid, higher total efficiency and higher stability of electrical power supply by exporting energy to the grid during peak demand hours. On the other hand, it poses new challenges in structuring and developing the control approaches for these distributed energy resources. Thus, it can be concluded that the centralized generation is moving towards distributed, plug-and-play type of generation that requires a special control approach. Implementation of centralized control for such a dispersed grid is unacceptable, thus some kind of distributed control is to be realized.

The advantages of Multi-Agent System (MAS) use in control applications described in [START_REF] Jimeno | Architecture of a microgrid energy management system[END_REF] demonstrate the effectiveness of this approach in the organization and management of the intelligent distribution grid in large-scale applications.

This paper discusses different energy management approaches for an intelligent grid that consists of RES, energy storage devices and controllable loads. An improved instantaneous energy management method is described targeted to obtaining longterm objectives and a novel long-term energy management approach is proposed based on load categorization and regulation of energy consumption by regulating electricity price function.

Relationship to Internet of Things

Concerning growing interest in intelligent systems that acquire data from network sensors, much higher predictive operation of household appliances is expected [START_REF] Vermesan | Internet of things strategic research roadmap[END_REF]. The concept named Internet of Things delivers the idea of connecting each smart appliance to the Internet, helping to communicate with the data center and providing advanced functionality (such as predictive connection of loads) that could help to solve the problem of peak demand, by scheduling of distributed generators and shedding the end-user's load or in other words, enabling demand-side management [START_REF] Smith | Internet of Things 2012 New Horizons[END_REF] [START_REF] Atzori | The Internet of Things: A survey[END_REF].

The idea of Internet of Things covers a large functionality of the smart nodes, but this article uses a simplified model to test the long-term scheduling approach by using a bidirectional communication between the sources and loads on the one side and the data center on the other side.

State of the Art in Energy Management

Currently a major effort is focused on the approaches that keep energy balance in small-scale networks, such as nanogrids [6][7], microgrids [START_REF] Piagi | Autonomous control of microgrids[END_REF]. The energy management approaches are divided into groups which differ by the period of management: instantaneous (real-time) energy management, keeping balance in the period of milliseconds or seconds, short-term energy management, with periods of seconds to hours and long-term energy management, which defines energy management strategy to fulfil objectives for weeks and months.

Instantaneous Energy Management

The operation in islanded mode requires special control techniques for energy generation units as they should regulate the output power with the demand of the local grid.

For this reason the droop control approach is mostly utilized both in AC and in DC power distribution grids, which defines the method of power sharing between various sources by means of definition of different droops according to their nominal power [START_REF] Guerrero | Hierarchical control of droop-controlled AC and DC microgrids-a general approach toward standardization[END_REF]. The droop control differs between AC and DC grids, because in a DC grid only supply bus voltage should be regulated, but in an AC grid -voltage and frequency. AC droop control is realized in the following way -the amount of active power is tight with the frequency, but reactive power with voltage reference. When a generator's output power becomes more capacitive, it increases the voltage reference, which increases the output power of the generator, at the same time decreasing the output frequency, thus sharing power with other generators.

0 nom P 1 nom P 2 V max V min V 0 max 2 Q E min E max 1 Q max 1 Q - max 2 Q - max E Q 0 nom P 1 nom P 2 f f f min max Fig. 1.
Droop control method -V-droop for DC grid, Eƒ-droop for AC grid From the point of view of energy management, DC distribution grids provide less problems, as only single parameter should be regulated within the defined levels. In contrast, energy management in an AC grid requires more attention due to inherent problems of reactive power fluctuations, harmonics, unbalances, as well as certain difficulties for generators to operate in no load conditions. These problems are compared in the table below.

Table 1. Comparison of instantaneous power balancing control challenges in AC and DC grids

Problem

AC grid DC grid P -active power balance

• • Q -reactive power balance • F -frequency control; synchronization; • Harmonics, 3-phase unbalance • R=∞ -no load condition • Imax -Inrush currents • •

Short-term Energy Management

Instantaneous power balancing provides power sharing approach between various of sources. However it does not provide information about the resources of the grid, for instance, islanded DC grid consisted of 1 kW source and 500 W load will operate at the same voltage level as second islanded DC grid, which consists of 10 kW source and 5 kW load. It means, that in one case there is no possibility to supply additional 1 kW load, but in other grid -there is possibility. Thus, special control approach is needed to overcome this issue.

In [START_REF] Schonberger | Autonomous Load Shedding in a Nanogrid using DC Bus Signalling[END_REF] the DC bus signalling (DBS) approach is applied to this problem. The control technique is based on different disconnection delays for different priority loads when DC bus undervoltage condition is occurred, thus enabling to lowest priority loads with shorter delays to disconnect faster, releasing power for higher-priority loads. The drawbacks of described approach are the following: the delays are set constantly, meaning that the controllability of such approach is limited; there is always trade-off between the delay periods (with longer periods it is possible to define more steps of load prioritization) and the quality of DC bus voltage, as the longer are delays, the bigger are voltage dips.

In our opinion, the communication hardware should be used to provide better short-term energy management that would eliminate unwanted voltage dips.

Long-term Energy Management

According to literature [START_REF] Kanchev | Energy Management and Operational Planning of a Microgrid With a PV-Based Active Generator for Smart Grid Applications[END_REF], with proper control of microgrids it is possible to realize long-term objectives, such as the reduction of CO2 emissions by a certain percentage or export a defined amount of energy to the grid during some period (week, month or even year), which could be required by contract statements or transmission network operator's tasks.

It is recommended to solve that problem with artificial neural networks [START_REF] Reddy | Distributed ANNs in a Layered Architecture for Energy Management and Maintenance Scheduling of Renewable Energy HPS Microgrids[END_REF], which will successfully predict the amount of generated power by RESs. Another solution is to use genetic algorithms [START_REF] Logenthiran | Short term generation scheduling of a Microgrid[END_REF] that help to choose an optimized operation schedule for energy storage components providing reliable power supply and keeping lifetime of energy storages longer due to full charge and discharge cycles. An additional solution is provided by a pricing approach [START_REF] Ogimoto | Optimum Operation Scheduling Model of Domestic Electric Appliances for Balancing Power Supply and Demand[END_REF] based on changing the price per electrical energy. Different loads and sources behave according to an internal logic -how to operate under certain price limits. This solution results in additional load when the price is low and vice versa, load shedding when the price is high. However, it is not possible to predict amount of switched on or off load with this solution.

Contribution of Research

Instantaneous energy management approach based on the droop control method can be used to obtain long-term objectives in energy management by regulating the reference values of DC bus voltage for a bidirectional AC/DC converter that will influence the consumed energy in the local grid.

The proposed method of long-term energy management is based on the mathematical model of price function estimation. It can be interpreted as a graphical approach of predictive energy generation surface intersection with acquired operation schedule of categorized loads, which can also be drawn as a 3D surface, thus enabling the control of the intelligent grid by providing only price function information.

Improved Instantaneous Power Balancing

Our proposal is to use the well known droop control method for keeping power balance in the grid, but with variable reference values of a common point interface (CPI) converter. It is especially suitable for DC supply grids that always operate under the droop control method, as these are not connected to the utility grid directly, but through an AC/DC interface converter, which imports or exports energy to or from the grid, depending on the DC bus voltage. We propose to use variable reference value of the DC voltage grid for an interface converter that will influence the operation of local grid elements, for instance by decreasing the DC bus voltage reference value to few per-cents below the nominal value, the unnecessary loads will not operate or will operate at the highest energy saving mode and as a result, more energy will be exported to the grid. In conclusion, it is possible to fulfil long-term objectives with a CPI converter which regulates its reference value for the regulation of a local grid operation mode, as other grid elements choose their operation mode in accordance with the grid voltage -the parameter available for all elements of the grid.

Novel Long-term Energy Management Approach

Our proposal is to define categories of household appliances which switch and operate at different conditions:

1. Time-triggered load (see Fig. 2a) -pumps, general illumination etc. -the load which basically switches on at a defined time; 2. Price-triggered load (see Fig. 2b) -energy storage element, washing machines etc.

-the load which turns on when the price is within an acceptable level; 3. Hybrid-triggered load -heater, boiler, refrigerators etc.-the load which operates according to the daily schedule (depending on consumption), but it can also switch on a little earlier when the price per electricity is economically profitable; 4. Price-responsive load (see Fig. 2c) -controllable illumination -the load which changes the consumed power depending on the price per electricity. The proposed algorithm is realized in the following way. Each load based on the statistics of use and on the changes of electrical price provides an energy management system (EMS) with a schedule of operation (for time-triggered and hybrid-triggered loads), condition of operation (for price-triggered and hybrid-triggered loads):

( ) ( ) ( ) ( ) ( ) ( )
∑ ∑ - - + = Ψ i j j resp C i trig T load t C P t C P t C , , , 1 , (1) ( ) ( ) ( )
∑ = Ψ i i source source t C P t C , , . (2)
After having aggregated all data from loads ( The scheduling of price-triggered loads is also used for the determination of the price function. EMS combines the price-triggered loads in accordance with energy excess, which is calculated using (5) (see Fig. 4a):

( ) ( ) ( ) ( ) t C t C t C delta load source , , , 1 Ψ ≥ Ψ - Ψ , (3)
( ) ( ) ( ) ( ) ( ) ( ) t C t C t C t C delta load source , , , 1 ) 1 (
Ψ + Ψ ∩ Ψ = . ( 4
Generation surface Consumption surface ( ) ( ) ( ) ( ) ( ) ( ) t C t C t C t C delta load source excess , , , , 1 Ψ + Ψ - Ψ = Ψ , ( 5
) ( ) ( )      + ≤ ≤ + ≥ < ≥ Ψ - - - ∑ i i start i end i i start i start i satrt i i i t t i i i trig C excess t t t t t t t C C t C P t C end start max min 1 1 , , , . (6)
As a result, the price function

( ) t C ) 1 (
is changed in accordance with the switching schedule of price-triggered loads, keeping the price at a desirable level, to regulate the amount of switched price-triggered loads and the power balance (see Fig. 3b).

Conclusions and Future Work

This paper proposes an improved real-time power balancing approach for DC distribution grids. It is based on the possibility of changing the reference parameter of the DC voltage for a bidirectional AC/DC interface converter influencing the DC bus voltage. When the voltage is decreased, it will lead to the change of operation of grid elements controlled by the droop control method, unnecessary loads will be switched off during the period of low DC voltage. Consequently, a larger amount of the energy will be exported to the utility grid. Thus, it is possible to fulfil long-term objectives just by controlling the voltage reference value of a bidirectional interface converter.

The novel approach of long-term energy management is based on pricing technique. It uses the load categorization that defines different conditions of operation for different loads. Each grid element can communicate with the energy management system (EMS), which aggregates time scheduling and condition of operation of all elements and finds the interconnection line between the 3D surface of the load and the 3D surface of generation. As a result, a certain price function is defined that can be used to realize predictive energy management of an intelligent distribution grid.

Our future plan is to conduct experiments to test the proposed methods on a smallscale network to acquire real evaluation of long-term and short-term energy management approaches taking into account typical summer, winter and spring day consumption patterns, as well as appropriate forecast generation patterns.

Fig. 2 .

 2 Fig. 2. Load pattern graphs

-

  for price-responsive loads), EMS subtracts the 3D surface of de-. 3a) to extract the interconnection line, the projection of which to Time-Cost axes is used to find the price function

  3b

  ):

Fig. 3 .

 3 Fig. 3. Primary equalization of price function

Fig. 4 .

 4 Fig. 4. Proposed algorithm of finding the price function

Acknowledgements. This research is co-financed by the European Regional Development Fund within the project "Intellectual Hybrid Uninterruptible Power Systems and Component Development and Research to Improve Energy Efficiency". Project agreement No. 2010/0225/ 2DP/2.1.1.1.0/10/APIA/VIAA/160.


A F H Mendes

Reis

R Laia

H M I Pousinho
email: hpousinho@gmail.com

R Melício
email: ruimelicio@uevora.pt

V M F Mendes
email: vfmendes@deea.isel.pt

A H Reis

Schedule of Thermal Units with Emissions in a Spot Electricity Market

Keywords: Short-term schedule, thermal units, emissions, dynamic programming, Pareto-optimal solution

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Over the last years, ambitious policy targets and concrete actions have been proposed in order to encourage mitigation of greenhouse anthropogenic gases emissions and ensure environmental sustainability worldwide [START_REF] Yamashita | Pareto-optimal solutions for trade-off analysis of CO 2 vs. cost based on DP unit commitment[END_REF]. A significant share on emission of greenhouse gases into the atmosphere is through the burning fossil fuel on thermal power plants [START_REF] Catalão | Influence of environmental constraints on profit-based short-term thermal scheduling[END_REF]. In 2011, thermal power plants played a dominant role in the mixed power generation, accounting for 54.2 % of total generation in the EU-27 and 57.1 % in Portugal [START_REF]Archive agriculture, environment, energy and transport statistics[END_REF]. The expressive share of thermal power plants in a competitive framework forces as a fundamental tackling the schedule problem of thermal units during a short-term horizon in order to conveniently appraise favorable economic conditions for the Generation Companies (GENCO). Within the past 40 years, mathematical programming techniques have been used aiming at the optimization of thermal unit commitment. In early studies, still without the availability of enough computational power to support a full modeling of the problem, the unit commitment was based on priority lists [START_REF] Xie | Environmental-economic unit commitment using mixedinteger linear programming[END_REF]. The development of computational power allowed to model the complexity of the problem, allowing over the years the inclusion of new constraints not only related to operational costs, technical boundaries, but also reserve, minimum up/down time, ramp rate power constraints and supply limits. Furthermore, in nowadays the electricity market framework provides a trading mechanism based on bilateral contracts suitable for thermal power producers. This kind of mechanism is used in order to allow hedging price volatility. Bilateral contracts are agreements between power producers and consumers to provide a given amount of energy at a predefined price along with a delivering period [START_REF] Heredia | A stochastic programming model for the optimal electricity market bid problem with bilateral contracts for thermal and combined cycle units[END_REF]. Hence, additional modeling must be included in the schedule problem in order to conveniently model electricity market framework influence on the schedule. This paper proposes a bi-objective optimization approach, requiring the use of a method based on Dynamic Programming (DP) and nonlinear programming. A case study is presented for a schedule over a time horizon of 168 hours with hourly periods.

Relationship to Internet of Things

Electric power systems are tagged by evolution of computing technologies, allowing the development of powerful optimization approaches able to process decisions under the operational planning. Consequently, every GENCO in the new competitive market paradigm are envisaged in order to ensure convenient management conditions to be a thing in the Internet of Things.

The connection to the Internet of Things conveys adequate decisions for system operators. The interfacing with the internet allows bidirectional communication between the power producers and the remaining decision makers, for instance, regarding real time market information and bilateral contracts. This paper deals with an application contributing to take decision in real time on market information and bilateral contracts.

State of the Art

Technical literature presents several optimization methods for solving the unit commitment and the economic dispatch problem: methods have been reported since the old priorities list method [START_REF] Burns | Optimization of priority lists for a unit commitment program[END_REF] to the classical mathematical programming methods until the more recently reported artificial intelligence methods [START_REF] Mantawy | A simulated annealing algorithm for unit commitment[END_REF]. Although, easy to implement and requiring a small computation time, the priority list method does not ensures an economic convenient solution near a global optimal one, implying a higher operation cost [START_REF] Senjyu | A fast technique for unit commitment problem by extended priority list[END_REF]. Within the classical methods are included DP, linear programming, nonlinear programming and Lagrangian relaxation-based techniques [START_REF] Chandrasekaran | Thermal unit commitment using binary/real coded artificial bee colony algorithm[END_REF]. DP methods are flexible but suffer from the "curse of dimensionality", due to the increase in the problem size related with the number of thermal units to be committed and the number of states considered for modeling the thermal behavior of each unit during the time horizon, implying an eventually huge use of computation memory and processing time. Although the Lagrangian relaxation [START_REF] Zhuang | Towards a more rigorous and practical unit commitment by Lagrangian relaxation[END_REF] can overcome the previous limitation, does not always lead to a conveniently feasible solution, requiring in order to set a feasible solution the satisfaction of some violated constraints using heuristics, undermining the optimality. Artificial intelligence (AI) methods based on artificial neural networks [START_REF] Ouyang | A hybrid artificial neural network-dynamic programming approach to unit commitment[END_REF], genetic algorithms [START_REF] Kazarlis | A genetic algorithm solution to the unit commitment problem[END_REF], evolutionary algorithms [START_REF] Dhillon | Economic-emission load dispatch using binary successive approximation-based evolutionary search[END_REF] and simulating annealing [START_REF] Wong | An enhanced simulated annealing approach to unit commitment[END_REF] have also been applied. However, the major limitation of the AI methods is the likelihood to obtain a convenient solution near global optimum, especially with a few thermal units. These problems are the ones faced by some GENCO due to the new paradigm, admitting companies with a small number of units to go into business.

Emissions concerns in the literature are mostly addressed for the economic dispatch problem [START_REF] Abido | Multiobjective particle swarm optimization for environmental/economic dispatch problem[END_REF], stating only the power output level of each unit but not the on/off status and availability for generation at each hour. The short-term thermal unit schedule problem has to take into account two assessments, i) the thermal units online at each hour and ii) the power output level for each online unit at each hour, so as to suit economic and/or environmental targets on time horizon of one day to one week [START_REF] Chandrasekaran | Thermal unit commitment using binary/real coded artificial bee colony algorithm[END_REF]. The effects of the short-term thermal unit schedule are significant, in sense that a better operation achieves not only a reduction in the fuel consumption [START_REF] Wood | Power Generation, Operation and Control[END_REF], but meets environmental concerns: once electric power generation is obtained from fuel-fossil power plants, the anthropogenic emission cannot be overlooked in nowadays. The emission modeling in the short-term schedule problem [START_REF] Gjengedal | Emission constrained unit-commitment (ECUC)[END_REF] has not been deeply tackled as in the case of the economic dispatch problem. Hence, in this paper the goodness of being schedule and the emission concern are included in the short-term schedule of thermal units.

Problem Formulation

The short-term thermal unit schedule problem can be stated as to find the schedule on status and the power generated for each thermal unit i at each time period t that optimizes performance criterions involving costs, emissions and market trading subject to a set of constraints on the operation of the units.

Objective Function

The proposed framework considers two different objectives in which the first one expresses the total fuel cost of thermal units and the second one expresses the total emissions of pollutants into the atmosphere. Mathematically, the bi-objective vector to be minimized is given by: . ) ( ), (

1 1 1 1 } p , u E p , u C { it it T t I i it it it T t I i it ∑∑ ∑∑ = = = = ω (1)
I is the number of thermal units; T is number of periods in the time horizon; for unit i in t period, ) (

it it it p , u C , it u , it p , ) ( i it i p , u E
are respectively the fuel cost function, the commitment state (on/off) , the power generated and is the emission function. Since the total fuel cost and total emissions are expected to be conflicting objectives, implying the impossibility to find out a single optimal solution that simultaneously satisfies both objectives. Thus, the set of best compromise solution, known as the Pareto front, can be determine using the weight-sum method, converting the bi-objective vector (1) into a family of single objective function given by a convex combination of the objectives, i.e., given by:

. ) p , u ( E ) p , u ( C ) 1 ( it it T 1 t I 1 i it it it T 1 t I 1 i it ∑∑ ∑∑ = = = = + - ω λ λ (2)
ω is a unit price penalty factor associated with the emission and λ is a weighting factor, which should obey 1 0 ≤ ≤ λ

. If 0 = λ , the solution correspond to minimum cost, the usual unit commitment, and if 1 = λ , the solution is minimum emissions commitment. The fuel cost and emissions functions for the operation of each thermal unit depend on the power generated by that unit and can be modeled as a second order quadratic functions respectively given by:

. u ) p c 2 1 p b a ( ) p , u ( C it 2 it it it it it it it itop + + = (3) . u ) p 2 1 p ( ) p , u ( E it 2 it it it it it it it itop γ β α + + = ( 4
) i a , i b , i c and i α , i β , i
γ are respectively the cost coefficients and the emission coefficients for thermal unit i . A GENCO can trade energy via bilateral agreements.

Hence, an augmenting term must be included into (2). The augmented objective function is given by:

. ) d p ( ) p , u ( E ) p , u ( C ) 1 ( T 1 t I 1 i t it t it it T 1 t I 1 i it it it T 1 t I 1 i it ∑∑ ∑∑ ∑∑ = = = = = = - - + - π ω λ λ (5)
t π is electricity price at period t and t d is the power contracted with bilateral agreements associated with energy that have to be delivered at period t . This objective function admits the possibility of buying energy in the market if production is not enough to satisfy agreements. The objective function ( 5) can be seen as the application of the weighted-sum method for a bi-objective optimization problem with the objective vector given by: .

) ( ) ( ) ( ) ( 1 1 1 1 } d p p , u E , d p p , u C { t it t it it T t I i it t it t it it T t I i it - - - - ∑∑ ∑∑ = = = = π ω π (6)
In other words (5) can be seen as an objective vector with coordinates respectively given by the objective function associated with the minimum emission and minimum cost commitment both with bilateral contract agreements. The start-up cost of thermal units is a term to be added the fuel operation cost in (3) and depends upon the number of periods t

x the unit has been offline prior to startup. The start-up cost is given by:

. ) e 1 ( uccool SU i t utcool x 5 0 i it - - = (7)
0 i uccool is the cold start-up cost and i utcool is the cooling time constant. Also, the start-up emission can be included into (4) in the same way as given by [START_REF] Mantawy | A simulated annealing algorithm for unit commitment[END_REF].

Constraints

The optimization problem is subject to a set of constraints due to operation conditions that can include the following: a) Availability of thermal units: The sum of online units cannot exceed the maximum number of thermal units allowed online in period t .

∑ ∈ ≤ I i t it NMV u . (8)
b) Power balance constraint: The power generated by the thermal units must meet at least a demand t D in each period t , ignoring transmission losses in the system.

∑ ∈ ≥ I i t it D p . (9)
c) Spinning reserve constraint: The spinning reserves t R in period t are necessary in order to ensure reliability. h) Bilateral agreement constraint: If necessary in order to ensure that a bilateral agreement is fulfilled the difference between generated power and contracted power with bilateral agreements can be imposed as a non-negative constraint.

∑ ⊂ ∈ + ≥ I J i t t it R D p . ( 10
0 ≥ - ∑ ∈I i t it d p for T t , . . . , 2 , 1 = . ( 15
)
The minimum up/down time constraints are used to avoid thermal stress, implying future augmenting maintenance. More constraints are possible to into the problem, but the complexity will be augmented, implying an augmented processing time.

Particularly, instead of ( 15) is possible to have bilateral agreement only strong fulfilled at particularly periods, leaving to other periods the possibility of buying energy in the market to fulfill agreements.

Case Study

The proposed bi-objective optimization approach is tested on GENCO with three thermal units in a competitive electricity market with bilateral contracts. Simulation studies are carried out on hourly basis over a scheduling time horizon of 168 hours.

The fuel cost and emission coefficients are given in Table 1. Based on [START_REF]Sistema Electrónico de Negociação de Direitos de Emissão de CO 2[END_REF], the price penalty factor used for emissions is 3 . 12 = ω Eur/Mg. The numerical computing testing has been performed on a 1.9-GHz-based processor with 2 GB of RAM using as a computing language the VBA for Microsoft Excel platform. Nonlinear and DP methods was used in order to solve the proposed short-term thermal unit schedule problem, being the first step to determine the feasible solutions taking into account the minimum up/down time constraints and total cooling time of each unit. The generated power was computed for each thermal unit schedule. The total number of states is 1120 to be processed at each hour. The energy price profile and the energy contracted at each hour are shown in Fig. 1. The computed energy over the time horizon as function of λ is given in Table 2. The major surplus between the supplied and contracted energy happens for with 0 = λ . The power contracted, the power supplied and the hourly units committed are respectively shown for 0 = λ in Fig. 2 and Fig. 3. Fig. 2 shows that the power generated is greater than the contracted power at some hours. This is due to higher values of energy prices at these hours. The values of power contracted are closer to the ones obtain for lower energy prices, because is not profitable to generate more energy than the necessary to comply with bilateral agreement. Fig. 4 shows different behaviours for each thermal unit due to differences in startup costs. Both thermal units U1 and U2 are off when the energy price is lower, while U3 is on in order to avoid incur if shutdown occurred on the higher startup cost.

The power contracted, the power supplied and the hourly units committed are respectively shown for 1 = λ in Fig. 4 and Fig. 5. Fig. 6. shows that the objective function are conflicting ones due to the need to compromise between units with significant less cost, but with significant levels of emission, and others with greater cost but lesser emission. The first are committed regarding small costs, the second regarding small emissions.

Conclusions

A short-term generation scheduling approach for thermal units in small GENCO's is proposed. The approach considers the goodness of being schedule with the emission concern. Operation constraints of thermal units like ramp rate constraints and minimum up/down time constraints are taken into account, using DP and nonlinear programming. Numerical results allow concluding that the proposed approach is suitable to obtain the trade-off curve to assist on the generation scheduling decisions: according to the trade-off curve, a significant reduction on emissions can be achieved by rescheduling the thermal units, but implying an increase in operational cost. Although, DP suffers from the "curse of dimensionality", therefore impracticable to be used on large-scale systems, on small GENCO's the use of the DP is reasonable. Further work will be carried out including mixed integer programming in order to improve the computation performance and introduce new model considerations.

Fig. 1 .

 1 Fig. 1. Price profile (solid line) and energy contracted (dashed-dotted line).

Fig. 2 ..Fig. 3 .

 23 Fig. 2. Power generated and contracted, 0 = λ

Fig. 4 ..Fig. 5 ..Fig. 4

 454 Fig. 4. Power generated and contracted, 1 = λ

  .

Fig. 6 .

 6 Fig. 6. Pareto optimal solutions.

  have to be on by at least the minimum up time before shutdown. have to be down by at least the minimum down time before startup.

													i DR	it p ≤ +1	-	it p	≤	i UR	.	(11)
													it u	p	it	≤	it p	≤	it u	p	it	.	(12)
	it u	1 (	-+ it u	) 1 =	1	, if equal to 1, the shutdown occurs at period	t	+	1	.	(13)
		it u	+	1	1 (	-	it u	)	=	1	, if equal to 1, the start-up occurs at period	1 t . +	(14)

) d) Operating ramp rate constraints: The power generated over any two consecutive online periods is restricted by the ramp-down i DR and ramp-up i UR limits. e) Generator Capacity Constraints: Each thermal unit is restricted by its minimum and maximum limits on power generation. f) Minimum up time constraint: The minimum up time i UD imposes that unit i g) Minimum down time constraint: The minimum down time i DD imposes that unit i

Table 1 .

 1 Fuel cost and emission coefficients for thermal units

				U1	U2	U3
	a	i	277	300	320
	i b	26.500	26.300	26.100
	i c	0.045	0.051	0.031
	α	i	40	43	38
	β		i	-1.6×10 -3	-1.3×10 -3	-1.9×10 -3
	γ	i	0.008	0.009	0.006
	i p (MW)	40	120	240
	i p (MW)	120	400	700
	i UR (MW)	40	120	240
	i DR (MW)	20	80	160
	i UD (h)	4	5	7
	i DD (h)	3	3	5
	i utcool (h)	4	5	7
	0 uccool (Eur) i	2200	2400	3000

Table 2 .

 2 Energy contracted and supplied, as function of the weighting factor λ

	λ	Total costs (Eur)	Energy contracted (MWh)	Energy supplied (MWh)
	0.0	4117780	119096	138084
	0.2	4092690	119096	120329
	0.4	3923691	119096	119135
	0.6	3729017	119096	119096
	0.8	3568313	119096	119096
	1.0	3408946	119096	119096


S K Pandey

S R Mohanty

N Kishor

J S Catalão
email: catalao@ubi.pt

J P S Catalão

An Advanced LMI-Based-LQR Design for Load Frequency Control of an Autonomous Hybrid Generation System

Keywords: Load frequency control, distribution generation, optimal control, linear matrix inequalities, Internet of Things

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In remote areas around the world an autonomous hybrid generation system is useful to supply the demand of electrical energy. A hybrid power system consists of a combination of two or more power generation technologies to enhance their operating characteristics and efficiencies than that could be obtained from a single power source [START_REF] Ahmad | Power fluctuation suppression of standalone hybrid generation combining solar photovoltaic/wind turbine and fuel cell systems[END_REF]. To meet increasing load demands for an isolated community, expansion of the distributed generation systems is required. The resulting power system may provide good quality service to the consumer load, which depends mostly on the type and action of the generation controller.

The aim of load frequency control (LFC) is to maintain the power balance in the system such that the frequency deviates from its nominal value within specified bounds and according to practically acceptable dynamic performance of the system.

Several controllers have been presented for LFC problems in order to achieve a better dynamic performance, where the most employed one is the conventional fixed gain controller like proportional-integral (PI) and proportional-integral-derivative (PID) controllers, because of their functional simplicity and simple structure, being widely used in industrial applications. However, their parameters are often tuned using experiences or trial and error methods and may no longer be suitable in all operating conditions. Different methods of tuning of PI/PID controllers are given in [START_REF] Cominos | PID controllers: Recent tuning methods and design to specification[END_REF]. The use of optimal control theory to power systems has shown that an optimal load frequency controller can improve the dynamic stability of a power system. Calovic [START_REF] Calovic | Automatic generation control: Decentralized area-wise optimal solution[END_REF] used output feedback techniques to design a decentralized controller.

Usually a linear model around a nominal operating point is used in the load frequency controller design. However, power system components are inherently nonlinear, so the operating conditions of power systems are continuously changing. Accordingly, the real plant usually differs from the model. Therefore robustness becomes a main issue in the attempt to design a controller to satisfy the basic requirements for zero steady state and acceptable transient frequency deviations.

Various robust control approaches have been used in LFC, such as Riccati equation approach [START_REF] Wang | Robust load-frequency controller design for power systems[END_REF], H∞-control [START_REF] Rerkpreedapong | Robust load frequency control using genetic algorithms and linear matrix inequalities[END_REF], µ-synthesis approach [START_REF] Wang | Robust control design for load frequency control using µsynthesis[END_REF], robust pole assignment approach [START_REF] Azzam | Robust automatic generation control[END_REF], Linear Matrix Inequality (LMI) based H∞-loop shaping control [START_REF] Patra | Design of static H∞loop shaping controller in four-block framework using LMI approach[END_REF],loop shaping control [START_REF] Majumder | LMI approach to normalized H ∞ loop-shaping design of power system damping controllers[END_REF] and inverse additive perturbation [START_REF] Supriyadi | Inverse additive perturbation-based optimization of robust PSS in an interconnected power system with wind farms[END_REF].The LMI approach based LFC has been mostly applied in conventional power systems, being very less used yetin hybrid energy systems for LFC.

In this paper, the formulation of LFC problem is based on LMI-Linear Quadratic Regulator (LQR). The basics of LMI-based-LQR control scheme are illustrated in [START_REF] Ko | An advanced LMI-based-LQR design for voltage control of grid-connected wind farm[END_REF]. The control optimization problem is obtained in terms of a system of LMI constraints and matrix equations that are simultaneously solved. The proposed LFC scheme with the advanced LMI-based-LQR (ALQR) is applied for an autonomous hybrid generation system. Results show that the robustness of the proposed controller is much superior to that of the optimal output state feedback controller, called as simple closed-LQR (CLQR), for different load and wind power perturbations.

Relationship to Internet of Things

The smart grids are expected to spread the intelligence of the energy distribution and control system from some central core to many peripheral nodes, thus enabling more precise control and adaptation. The concept and objectives of such intelligent nodes are similar to those of the Internet of Things (IoT).

IoT means extending the web paradigm to the connection, monitoring, and control of the objects of everyday life. IoT technologies can provide a viable solution toward a web-enabledsmart gridsystem [START_REF] Bui | The internet of energy: a web-enabled smart grid system[END_REF].

The future developments of the IoT related to energy-aware systems and distributed decision will be especially relevant for autonomous hybrid generation systems, making their control ubiquitous.Decision makers will be able to base their actions on real-world, real-time data [START_REF] Karnouskos | Towards an information infrastructure for the future Internet of energy[END_REF].IoT will make it possible for future energy systems to be self-managing, self-sustaining and robust. Therefore, the IoT based infrastructure will be tightly coupled with the energy domain [START_REF] Yu | The new frontier of smart grids[END_REF].

System Configuration and Description

The interconnected conventional power source and distributed generation (DG) system is shown in Fig. 1. The power-frequency balance is maintained by a proper control of the different power generation resources. In actual systems, WT, DEG, FC generator, AE and BESS are high order models and have nonlinearity. The controller design for higher order non-linear system would be complicated. Thus, the first order linearized model is taken into consideration for frequency control. The first order representation for the LFC is sufficient without sacrificing accuracy as the frequency control loop acts very fast. The generated power of the WTG depends upon the wind speed. The transfer function of the WTGis represented by a first-order lag [START_REF] Das | GA based frequency controller for solar thermal-dieselwind hybrid energy generation/energy storage system[END_REF], as:

( ) 1 w tg W T G w tg K G s sT = + (1)
Part of the wind power generation is to be used by the AE for hydrogen production. The transfer function model of the AE [START_REF] Das | GA based frequency controller for solar thermal-dieselwind hybrid energy generation/energy storage system[END_REF] can be expressed by:

1 a e A E a e K G sT = + (2)
The FC converts the chemical energy of the fuel (hydrogen) into electrical energy. The FC is represented by a first-order lag transfer function model [START_REF] Das | GA based frequency controller for solar thermal-dieselwind hybrid energy generation/energy storage system[END_REF], given as:

( ) 1 f c F C f c K G s s T = + (3)
The DEG works autonomously to supply the deficit power to the hybrid system in order to meet the supply-demand balance condition. The DEG is modeled by a firstorder transfer function [START_REF] Das | GA based frequency controller for solar thermal-dieselwind hybrid energy generation/energy storage system[END_REF], given by: d e g d e g ( ) 1

D E G K G s s T = + (4)
The BESS can be utilized to provide additional damping to power system swings, improving both transient and dynamic stability. The transfer function model of BESS is expressed by first-order [START_REF] Das | GA based frequency controller for solar thermal-dieselwind hybrid energy generation/energy storage system[END_REF], as:

( ) 1 bess BESS bess K G s sT = + (5)
The power balance is achieved by the equation: The transfer function for the system variation to p.u. deviation in power is expressed by: 1)

1 1 . (
sys e sys sys f G P K sT D sM ∆ = = = ∆ + + (7)
where s y s K is the system frequency characteristic constant of the system and s y s T is gain of the system. The parameters of proposed hybrid system [START_REF] Das | GA based frequency controller for solar thermal-dieselwind hybrid energy generation/energy storage system[END_REF] are given in Table 1.

Table 1.Parameters of the proposed hybrid system.

Components

LMI Representation of LQR

For a linear and robust control framework, the following linear time invariant multiple systems are considered:

x A x B y w = + + Γ & (8) y C x D u η = + + (9) where A is a n n × state matrix, B is a n p × control matrix, Γ is a m n × output matrix, x is a 1 n × state
vector, u is ap×1 control signal vector, y is a 1 m× output vector, w is a noisy signal vector, assumed to be white.

Using the H 2 representation of the LQR problem [START_REF] Anderson | Optimal Control: Linear Quadratic Methods[END_REF], [START_REF] Boyd | Linear Matrix Inequalities in system and control theory[END_REF], we obtain the statefeedback gain matrix k that minimizes the following cost function in terms of output y % as:

( ) m in { } T k J E y y   =   % % (10)
subject to:

( ) ( ) 0 T T A B k P P A B k Q k R k - + - + + < (11)
where 0

n n P × ∈ ℜ >
is a common Lyapunov matrix. The advanced LMI representation of H 2 LQR optimization problem can be obtained as the minimization of the cost function:

( , , ) m i n [ ( ) ( ) ] T T Y X J t r Q Y X t r N N Λ = + - Λ + Λ % (12)
subject to:

1 2 1 2 1 1 1 2 1 2 ( ) ( ) 0 0 T X R X R Y R R Y - -   Λ - Λ Λ > = >   Λ   (13) 1 1
(

0 0 0 T T T T T T A Y Y A B B Y R Y Q - -   + - Λ -Λ Λ   Λ - <     -   (14)

Control Formulation

Consider a linear time invariant model for the proposed hybrid power system with the following state space model:

x Ax Bu d = + + Γ & (15) y Cx = (16) u Kx = - (17)
where A is system matrix, B is the input distribution matrix, Γ is the disturbance distribution matrix, C is the control output distribution matrix, x is the state vector, u is the control vector and d is the disturbance vector consisting of load change and wind power change.

The state and other variables of the proposed hybrid power system are [ ] The design of the optimal controller based on output feedback control approach is discussed in detail in [START_REF] Calovic | Automatic generation control: Decentralized area-wise optimal solution[END_REF].

Results

Load Changes

The proposed controllers were applied to the hybrid power system shown in Fig. 1.

The parameters of the hybrid power system are considered to be the same as in [START_REF] Das | GA based frequency controller for solar thermal-dieselwind hybrid energy generation/energy storage system[END_REF]. In this section, the performance of the ALQR controller is compared with the CLQR controller for different load scenarios. The frequency deviation is shown in Fig. 2for ALQR and CLQR. It can be seen that using the ALQR controller the frequency deviation of the system is quickly driven back to zero with a smaller overshoot. It can be seen that the robustness of the ALQR controller is superior comparatively to the CLQR controller. In this case, it is considered that the load disturbance varies with time in step form, as shown in Fig. 4, and 0.5

wtg P pu =
is also applied to the system. The frequency deviation is shown in Fig. 5(a). The control effect of ALQR is better than CLQR. For clarity, Figs.

Random Wind Power Input

In this case, the random wind power input shown in Fig. 7

Robustness with Parameter Variations

Any controller designed for fixed plant parameter or tuned to nominal conditions may not perform satisfactory on variation in their values. Also, in some cases, the value of these parameters is not accurately determined, either experimentally or mathematically. Thus, they are said to belong in an interval range.

A controller design is desired to be robust enough against parameter variations of the system model.

The robustness of theCLQRand ALQR controllers is evaluated by computation of a quantitative performance index, integral square error (ISE) under variation in system parameters.

In order to be realistic, the simulations were done for a limited time. Moreover, the integral values are multiplied by some suitable constants, so that the outcomes of the performance indices attain comparable values.

Thisindexis modified as:

25 5 2 0 10 ( ) ISE f f dt ∆ = ∆ ∫ (18)
The values of ISE against parameter changes for theCLQRand ALQRcontrollers are shown in Fig. 8.

As can be observed with the parameters change over a large percentage range, the corresponding value of ISE varies significantly and increasingly in the case of the CLQR controller. However, the value of ISE with ALQR controller is much lower and change slightly.

These results confirm the high robustness of the ALQR controller against the random load change, and system parameter variations.

Conclusions

This paper presented a robust control design technique based on LMI-LQR approach. The proposed ALQR controller is applied for an autonomous hybrid generation system. A comparative study between the proposed ALQR controller and optimal output state feedback controller, called as CLQR controller, is provided in study. Simulation results using Matlab 2009b version demonstrated that the proposed controller is capable to guarantee robust stability and robust performance, such as precise reference frequency tracking and disturbance attenuation. The results suggest superior performance of ALQR against optimal output state feedback controller, in the case of load changes,parameter variationsand random wind power input. The incorporation of IoT technologies will foster the web-enabled smart grid concept, where all devices will be interconnected and able to interact through the internet, leading to the deployment of a ubiquitous energy controlsystem.

Fig. 1 .

 1 Fig. 1. Block diagram of the hybrid power system.

∆

  is the error in supply power and d P ∆ is the load change.

Case 1 :

 1 Step load changeIn this case, a load disturbance of 0

Fig. 2 .

 2 Fig. 2. Frequency deviation for case 1.

Fig. 3 (

 3 Fig. 3(a) and 3(b) show the frequency deviation with different load disturbances for ALQR and CLQR controllers, respectively.It can be seen that the robustness of the ALQR controller is superior comparatively to the CLQR controller.

Fig. 3 .

 3 Fig. 3. Frequency deviation for ALQR (a) and CLQR (b) with load disturbances. Case 2: Random step load change

Fig. 4 .

 4 Fig. 4. Random load change in step form.

Fig. 5 .Fig. 6 (Fig. 6 .

 566 Fig. 5. Frequency deviation for case 2 (a); frequency deviation for case 2from t=0s to 1s (b); frequency deviation for case 2 from t=9s to 11s (c); frequency deviation for case 2 from t=19s to 21s. Case 3: Random Load changeIn this case, the random load change shown in Fig.6(a)is applied to the system, also with 0.5 wtg P pu = .

Fig. 7 .

 7 Fig. 7.Random wind power input signal (a); frequency deviation with random wind powerfrom t=0s to 2s (b).

Fig. 8 .

 8 Fig.8. Robustness evaluation by computing performance index ISE for all parameters variation.

Acknowledgment. This work was supported by FEDER funds (European Union) through the Operational Program for Competitiveness Factors-COMPETE, and by Portuguese funds through the Fundação para a Ciência e a Tecnologia (FCT), under Project FCOMP-01-0124-FEDER-020282 (Ref. FCT PTDC/EEAEEL/118519/2010).


R Pereira
email: rpereira@deea.isel.pt

A Fagundes

R Melício
email: ruimelicio@uevora.pt

V F Mendes

J Figueiredo

J Martins

J C Quadrado

V M F Mendes

Demand Response Analysis in Smart Grids Using Fuzzy Clustering Model

Keywords: Demand response, smart grid, fuzzy subtractive clustering

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The smart grid engages consumer active role. This is crucial for grid management in order to efficiently ensure electric energy generation and usage. For instance, timescheduling or shedding of load requirements is more favorable when consumer has an active role. Demand Side Management (DSM) such as energy efficiency, energy conservation and Demand Response (DR) programs [START_REF] Boshell | Review of developed demand side management programs including different concepts and their results[END_REF] are required in order to adequate the profile of energy load diagram to generation. DR aims at shaping energy usage in some specified time periods with opportunely economic advantage due to adequate balance between load and generation. The shaping is attained by a modus operandi efficiently taking into account the need to apply time-scheduling or shedding of the load. The modus operandi is intended not only at lowering expenditures, such as the ones resulting from the need to call utilities expensive peaking power plants in short-time or to avoid building new power plant in order to satisfy future forecasted augmented energy consumption needs; but also in case of impossibility to avoid new power plants building, then chiefly discarding the ones with anthropogenic greenhouse gases emission [START_REF] Johnson | The fundamentals of liking demand side management strategies with program implementation tactics[END_REF]. Already, a European Union program started in 2008 for decreasing the end-user consumption is on effect, aiming to achieve 1 % reduction in energy consumption in the following nine years [START_REF] Feenstra | How to change consumers' energy-related behaviour[END_REF]. Consumers have to be persuaded to adhere to a smart grid environment and be an active element in the smart grid management, implying the necessity to furnish adequate conveniences to take advantage of the capabilities given by smart grid. DR techniques allows the accomplishment of shifting or decreasing the energy usage in several economic activity sectors, bringing benefits not only to consumers, but also to grid operators due to a reduction in system operation costs [START_REF] Ferreira | Demand Side Management, Igenium[END_REF].

In this paper the consumer´s active role in grid management is supported by a DR method developed using a fuzzy clustering model. The developed methodology provides the consumer an intended efficient tool which allows to support consumer decisions on load management regarding their consumption needs and consumption priorities along one day, taking into consideration the associated costs and possible economical benefits. The desirable massive consumer adherence to DR actions justifies a consumption pattern definition for the model implementation. This consumption pattern justifies the fuzzy clustering techniques applied.

This paper focuses on DR actions and modeling, giving details and results analysis. Domestic consumers are the target of the paper and are divided accordingly to their consumption profiles into three groups. Each group has an equipment priority list of Controllable Loads (CL) to be accomplished, considering for the decision on load satisfaction the available generation capacity and the energy price. Also, consumer non-controllable loads are considered.

The rest of the paper is organized as follows: Section 2 presents the contribution to innovation on DR involving research and customization of programming techniques. Section 3 presents aspects about DSM and DR. Section 4 presents DR model case study main assumptions. Section 5 presents the fuzzy subtractive clustering method to be applied to consumption scenarios. Section 6 presents results analysis based on the scenarios. Finally, concluding remarks are given in Section 7.

Relationship to Internet of Things

Internet of Things (IoT) is a name conceived to accommodate an extension of the web paradigm in order to consider the connection, monitoring and control of devices: things of everyday life [START_REF] Bui | The internet of energy: a web-enabled smart grid system[END_REF].

The smart grid characteristics [START_REF] Yun | Research on the architecture and key technology of internet of things (loT)[END_REF], the expectation to spread intelligence and control from some central core over peripheral nodes on electric energy usage entails the need to research, test and customize computing techniques for DR analysis, enabling opportune interfacing management of energy usage [START_REF] Bui | The internet of energy: a web-enabled smart grid system[END_REF].

The DR analysis in this paper is on a line of technological contributions tied with the IoT aptitude to: (i) collect data on energy usage, (ii) provide this data to other participants on local utility grid, and (iii) get energy prices information [START_REF] Ciuciu | Social network of smart-metered homes and SMEs for grid-based renewable energy exchange[END_REF].

Demand Side Management and Demand Response

The oil shocks of the 1970s brought a new research concerning policies and measures targeting energy demand. Since then development of DR and energy efficiency policies are on the way, aiming to influence quantities or patterns of energy [START_REF] Haney | Demand-side management strategies and the residential sector: lessons from international experience[END_REF]. A discussion and analyze DSM appeared in 1985 with alternatives related to load shape benefits derived from techniques concerning, for instance: valley filling of utility's loads, clipping of peaks, shifting of loads to off peak hours, strategic conservation to reduce demand [START_REF] Delgado | Demand-side management alternatives[END_REF]. DSM became more ambitious with smart metering in domestic buildings and services and bidirectional communication provided in smart grids.

DR allows load adaptation to the generation; this adaptation is important due to the intermittent characteristic of renewable energy sources integrated into the electric grid. DR programs are mainly conditioned by grids requests and consumers have to adapt the consumption in order to take advantage of economical benefits. DR is defined as any program which communicates with the consumer and either enables or encourages the consumer to lower or shift energy consumption during periods of unfavorable energy prices to favorable periods [START_REF] Stromback | The development of demand response in Europe[END_REF]. DR programs can be divided into: Incentive-Based Programs (IBP) and Time-Based Programs (TBP) [START_REF] Aalami | Demand response model considering EDRP and TOU programs[END_REF].

Demand Response Model Case Study

The proposed model is a TBP one, aiming to support consumer's decision and actions on load management also referred as CL management, taking into consideration the available power capacity and energy price. This TBP model intends to give consumers flexibility in order to take advantage of economical benefits allowing the load management that best fits consumer's profiles or life-styles. Consumers load management is possible throughout methods of load time-scheduling and load shedding. The economical benefits mainly come from valley filling of utilities' loads and shifting of loads to off peak hours.

The model assumes that grid power dispatch provides hourly information to consumers about available power and energy price. The assumptions considered in the model are the following: available power is considered non-constant along the time horizon to emulate available distributed generation behavior and consequently allowing a convenient DR modeling; consumer can deselect priority on a CL in order to allow available power to the next priority CL; once the in progress power sum of CL reaches the total available power, no further CL can be connected. These assumptions are intended to give to the consumer an opportunity to adapt the consumption diagram to what the consumer is willing to pay for energy price.

The consumers' profiles and operation modes were obtained from analysis of consumption behavior that allowed a consumption pattern definition. Three consumer profiles were set to ensure the coverage of DR generalization on the model and help consumers cope with price changes over one day period. These three consumer profiles were designated as follows: economic, moderate and extravagant. Two operation modes designated by cleaning and comfort are set for each consumer profile. An example of a priority list is shown in Table 1. In table 1 the CL priority list is the same for the three profiles because it guarantees that consumer decision is obeyed independently of the selected profile, regarding the energy price. For example, if economical profile is selected but for one time period the energy price is superior to the one allowed for the economical profile, consumer can change his profile to moderate, keeping the same CL priority list but with increased costs. Also, a common CL priority list to the three profiles allows a DR model comparison analysis for the scenarios considered in this study. The demand response model is shown in Fig. 1.

Start Finish Input Data

Fuzzy Controller

Consumer

Profile Save Data Fig. 1. Demand response model.

The input data was set considering one residence with 6.9 kVA of installed power and the CL are home appliances and air conditioner with nominal reference power given in [START_REF] Erse | [END_REF]. The model input data are chosen considering CL consumption stages, available power and energy price. The reactive power is neglected.

The consumption stages assumed are the null power or the reference nominal power, i.e., for the air conditioner the consumption stages are off power or full power, which is 1.8 kW. Nevertheless, the model is valid for any other consumption stages introduced as input data. The model is applied considering a power range between 1.8 kW and 6.9 kW, respectively given by the CL lower value and the installed electric power value.

Fuzzy Clustering Model

The required DR behavior is standardized in an Excel sheet by creating relations between the input and output parameters desired for the DR model. In order to represent the DR behavior in a fuzzy controller, data clustering technique was used. Given an input, available in the Excel data sheet, data clustering apprehends the pattern in it and gives the outputs. The outputs are cluster centers corresponding to the pattern, which can be utilized in a fuzzy controller. Each cluster center is composed by twelve elements. A total of 7 inputs are considered. These inputs are given by: 5 CL consumption states required by the consumer, the energy price and the available power. And 5 outputs given by the 5 CL states.

Data clustering techniques are necessary due to the fact that otherwise the required number of rules is a large number in order to acceptably define the fuzzy controller. Fuzzy clustering consists on a strategic division of the data space into fuzzy clusters. Each fuzzy cluster is introduced in order to stand for the convenient representation of one specific part of system behavior. After projecting the clusters onto the input space, the antecedent parts of the fuzzy rules can be found [START_REF] Priyono | Generation of fuzzy rules with subtractive clustering[END_REF].

The subtractive clustering technique is an extension of the mountain clustering method based on data sets evaluated using mountain function resulting in cluster centers. Accordingly to this technique, given a collection of n data points in an mdimensional space, the technique starts by setting each data point x i as a potential cluster center with a defined potential in function of the Euclidian distances of all data points. This function, specifying the potential at x i , is given by . r a is the cluster radius, setting a hypersphere of data points with significant influence on the cluster center potential in order to be acknowledged as neighbors.

The r a value has a strong effect on the numbers of the generated cluster. Three observation about this value are in order: a data point outside a neighborhood of radius r a has little influence on the potential of the neighborhood center data point; a higher value for r a generally results in a generation of few clusters and therefore a model too generalized; while a low value can lead to an excessive generation of clusters and therefore a model that has not enough generalization [START_REF] Chiu | Fuzzy model identification based on cluster estimation[END_REF]. The r a values were chosen in order to introduce an adequate amount of clusters which are related to the resulting number of fuzzy rules. The r a parameter can be adjusted based on the required model complexity and generalization ability. In this paper the r a values determined resulted in a rules number which allowed an adequate control action with an adequate model time response. Hence, it was considered that no major advantages arrived with improved mathematical methods or optimization methods applied on the r a determination. The values for r a are shown in Table 2.

The subtractive clustering technique after computing all the data point potentials, using the expression (1), identifies the first cluster center as the data point with the highest potential.

In ( 4) η is the squash factor defining the neighborhood of data points that will have significant measurable reduction in the potential value. Normally, a value of η = 1.5 is a good choice. After the first cluster center has been obtained and all the potential of the data points have been revised using the procedure given by assignment (3), the data point with the highest potential is selected as second cluster center. In general, after k th cluster center has been obtained, the potential of each data point is revised using the assignment given by .

2 || || * * k i x x k i i e P P P - - - ⇐ β (5)
In (5) P k * and x k * are respectively the k th cluster center potential value and center location. A description of further details about the subtractive clustering technique employed in this paper can be seen in [START_REF] Chiu | Fuzzy model identification based on cluster estimation[END_REF]. Fuzzy rules are defined based on valid cluster centers. The assumptions considered in the model presented in section 4 and the data sets to be submitted to the subtractive clustering technique were implemented in Excel ® . Each profile and respective operation modes have a proper data set. A file with Matlab ® extension .m was created, where the function genfis2 perform the subtractive clustering technique on the data sets. Then the Matlab ® function fuzzy is applied, which allows articulating the created clusters into if-then rules, establishing the fuzzy control saved in a file with Matlab ® extension .fis.

Other approach, such as neural networks (NN) can be applied to the model controller. However, for the model implementation in NN the required data sets characteristics in association with the controllable loads power consumption initial value definition, revealed a controller design complexity which have leaded to considerably more time dispended in this model implementation then with the fuzzy clustering model developed.

Results Analysis

Two scenario analyses are performed with and without the DR model, considering a residence with a limited power of 6.9 kVA and with a base power consumption shown in Fig. 3. The base power consumption represents the consumption of all loads excluding the CL.

The first scenario, shown in Fig. 2, refers to a consumption diagram where no planning or restriction to CL connection exists, i.e., no DR model is on consideration. This scenario considers that, when consumers are at home and awake the consumption is high and limited by the 6.9 kVA, otherwise the consumption values are low and limited by the base power consumption. This scenario is used as a comparison basis between the consumption diagram where no planning or restriction to CL connection exists and a consumption diagram resulting from the DR model. The second scenario has data information, shown in Fig. 3, about power and energy prices available in one day period on an hourly basis and consumer selects operation mode accordingly withwhen at home and awake, comfort mode is selected, otherwise cleaning mode is selected, considering two hypotheses: i) consumer sets an economic profile; ii) consumer sets a moderate profile.

The input data to simulate the DR model for moderate profile is shown in Fig. 3. A comparison between Fig. 2 and both DR model responses shown in Fig. 4 and Fig. 5, shows the influence of price on consumer consumption. On Fig. 2 the consumer is not influenced by the price because it is always the same, so the consumer consumes where is more convenient. On the other hand, accordingly to Fig. 4 and Fig. 5, the consumer is concerned with the price because it changes along one day, so the consumer reschedules the CL in a way to be connected on hours where the price is lower. A comparison of responses in the second scenario from hypotheses i), Fig. 4 and ii), Fig. 5, shows for the moderate profile that CL has more favorable conditions to be connected in one day period, than for the economic profile. Hence, the consumer is subjected to a lesser restricted DR due to the higher acceptance of the energy price in moderate profile than in economic profile. The extravagant profile allows the consumer a DR only depending on available power. It can be observed in Fig. 4 and Fig. 5 that DR model guarantees that the sum of base power consumption with CL consumption power is never superior to the available power. The individual CL power consumption for moderate profile in one day period is presented in Fig. 6. Together, both Fig. 3, Fig. 4, Fig. 5 and Fig. 6 allow to confirm that the energy price is satisfactory associated with CL power consumption. Both Fig. 3 and Fig. 6 allow to conclude that in the time period between 6 a.m and 8 a.m there is no CL available because the energy price is higher than the energy price assigned to moderate profile. The analysis of Fig. 5 and Fig. 6 allows to conclude that only at 8 a.m there is no sufficient power available to connect any CL. However, regarding the energy price shown in Fig. 3 and taking into account the defined CL priority list shown in Table 1, no CL is connected in the period between 6 a.m and 8 a.m as shown in Fig.

Conclusion

The proposed model ensures that power consumption is never superior to the available power and that no CL can be connected if energy price is superior to the correspondent energy price assigned to the selected consumption profile.

Consumer is subjected to a lesser restricted DR in moderate and extravagant profiles than in economic profile due to a higher acceptance of the energy price.

The consumption diagram can be shaped by the price changes along one day, which can lead to a more efficient grid because the consumption diagram can be conveniently adapted as it can be seen by comparison between the first and second scenarios, where a flatten diagram is desirable.

Consumer evidences a flexibility on consumption schedule when data on available power and on energy price are to be taken into account in due time. Hence, collecting and providing data on energy usage to participants on local grid, having energy prices information and model development, as the one proposed in this paper, are crucial for IoT to conveniently assist on a successful DR.

Fig. 2 .

 2 Fig. 2. Power consumption of consumer in first scenario.

Fig. 3 .

 3 Fig. 3. Base power consumption, available power and price evolution.

Fig. 4 .

 4 Fig. 4. Power evolution for economic profile.

6

 6

  .

Fig. 5 .Fig. 6 .

 56 Fig. 5. Power evolution for moderate profile. ´

Table 1 .

 1 Controllable load priority list for cleaning modeThe economic profile is set to allow only the use a CL when the energy price is low. The moderate profile is set to allow the use a CL when energy price is low or medium and in extravagant profile any CL could be used independently of energy price. The necessity of pattern recognition associated with a control which supports consumers' decisions for DR model design is fulfilled using fuzzy clustering method. The controller is implemented in the Matlab-Simulink ® software resorting to the Fuzzy Logic Toolbox.

	Economic profile	Moderate profile	Extravagant profile
		1 Dishwasher
		2 Washing machine
		3 Dry Machine
		4 Air Conditioner
		5 Thermo ventilation
	Low price	Low or medium price	Low, medium or high price

Table 2 .

 2 Fuzzy rules number and r a value

	Profile	Operation mode							Rules number
	Moderate		Cleaning Comfort									14 10	1.4 1.5
	Then, the potential of the data point is revised using the assignment given by
		i P	⇐	i P	-	* 1 P	e	-	β	||	x	i	-	* 1 x	||	2	.	(3)
	In (3) P 1 * and x 1 * are respectively the first cluster center potential value and center
	location, β is a parameter given by
				β =	4	/	( η	a r	) 2	.


Luís R Oliveira

A Marques Cardoso
email: ajmcardoso@ieee.org

Luís M R Oliveira

A J Marques Cardoso

Modeling and Characterization of Leakage Inductances for Transformer Winding Fault Studies

Keywords: Transformers, winding faults, leakage inductances, modeling

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Power transformers are key elements of the electric generation, transmission and distribution network and constitute one of the most capital-intensive investments made by power system utilities. The unexpected failure of a power transformer can generate substantial costs for repair and financial loss due to unscheduled electrical outage. Therefore, it is of crucial importance to detect internal defects in their incipient stage, so that the faulted unit can be immediately disconnected, avoiding the progression of the defective condition into a catastrophic failure, minimizing the damages in the transformer and other expensive neighboring equipment, and thus reducing downtime and total outage costs.

The development of new techniques for transformer condition monitoring and fault protection requires a detailed characterization of the transformer behavior during the occurrence of turn-to-turn winding short-circuits. The experimental study of these incipient internal faults presents some difficulties, mainly due to the high magnitudes of the faulty currents involved, which can damage the test transformer. Therefore, a detailed analysis of these phenomena can be better investigated by the use of a suitable digital simulation transformer model [START_REF] Oliveira | A Permeance-Based Transformer Model and its Application to Winding Interturn Arcing Fault Studies[END_REF].

Several circuit-based transformer models were presented in the last few years for winding fault studies [START_REF] Oliveira | A Permeance-Based Transformer Model and its Application to Winding Interturn Arcing Fault Studies[END_REF]- [START_REF] Avendaño | Transformer Internal Fault Modeling in ATP[END_REF]. One of the major difficulties in these approaches is to model the leakage inductances when the turn-to-turn short-circuit is present. The difficulty arises because the distribution of the magnetic flux is substantially modified when such a fault occurs [START_REF] Billig | Mechanical Stresses in Transformer Windings[END_REF]. In [START_REF] Bastard | A Transformer Model for Winding Fault Studies[END_REF] the method to determine the leakage inductances relies on correction factors, in order to take into account the radial component of the leakage flux. In [START_REF] Jablonski | Internal Faults in Power Transformers[END_REF] an analytical formula for the leakage inductance of the faulty winding is developed, but only for the simpler case of transformer no-load conditions. Nevertheless, the derivation of the models is not straightforward and difficult to be implemented. More recently a simplified method was proposed [START_REF] Avendaño | Transformer Internal Fault Modeling in ATP[END_REF], in which the leakage inductances are computed from the nameplate short-circuit reactance. Errors up to 68% are reported (as compared with experimental results) and only a model based on finite elements method would yield good results.

This work presents an analytical method to compute the leakage inductances of power transformers with a turn-to-turn winding fault. The influence of the fault position and fault progression is also characterized. Additionally, a leakage inductance equivalent circuit to represent the transformer with faulty turns is proposed. The results obtained from the application of the analytical method are validated by using data obtained from finite element analysis and experimental tests.

Relationship to Internet of Things

The smart grid can be seen as one of the Internet of Things application domains, in which intelligent electronic devices and their communication capabilities can be used to provide unprecedented reliability levels in the power network. However, for this to occur it is necessary the development of new fault detection methods, which must be integrated with the smart grid technologies. This in turn requires detailed and extensive modeling and simulation of the power system network. A power transformer model for winding fault studies is essential for these purposes.

Winding Fault Characterization

For the experimental investigation a three-phase, two winding, three limb transformer, of 10.3 kVA, 230/132 V, was used. The primary and the secondary windings have 152 and 90 turns, respectively. In each transformer winding there are five additional tappings connected to the coils, allowing for the introduction of shorted turns at several locations in the winding, as shown in Fig. 1(a), for one phase of the primary and the secondary windings. Only the coils of the center limb were used to perform the short-circuit single-phase tests, which results in a shell-type core design.

When a fault occurs in the primary-side, the short-circuited turns act as an autotransformer load on the winding, as shown in Fig. 1(b). The initial effect of the inter-turn short circuit is limited to a slight increase in the primary current. However, the insulation failure can lead to a high circulating current in the shorted turns, even if only a small number of turns is affected.

Fig. 2(a) presents the current waveforms in the transformer windings for the case of a load-test during the occurrence of a turn-to-turn fault in the primary winding. To protect the transformer from complete failure when the fault was introduced the current in the shorted turns is limited by an auxiliary resistor, which represents the fault contact resistance, R sh . It can be seen that the current in the shorted turns (i b ) and the current in the secondary winding (i s ) are in phase opposition with the primary-side current (i.e., they are opposing the primary magnetomotive force (MMF)).

The leakage inductance is usually obtained by performing the short-circuit test. It might be thought that the effect of the inter-turn short-circuit could be analyzed by performing a short-circuit test where both N s and N b turns were individually shorted, Fig. 1(c). However, as explained next, this test is far from representing the true leakage flux distribution. The resultant current waveforms obtained for this case are shown in Fig. 2(b). It can be seen that the current i b presents a completely different behavior than the one for the load test: it has a very small magnitude and is opposing the secondary MMF, instead of opposing the primary MMF. The main reason is that the mutual flux that links the three coils (N a , N b and N s ) is significantly different in these two tests, and, therefore, the induced currents do not follow the same pattern. As a result the flux distribution obtained by this short-circuit test is very different from the leakage flux component in the load test and it is not representative of this latter condition.

A better solution to study these phenomena can be obtained by performing the short-circuit test of Fig. 1(d), where the short is applied to N s and N b connected in series-addition. With this approach both currents in these coils are opposing the primary magnetomotive force, Fig. 2(c), which is much more similar to the results of the load-test of Fig. 2(a).

Alternatively, the leakage inductances can be obtained by performing the traditional short-circuit test were one winding is shorted at a time, Fig. 1(e). With this approach the three-winding transformer theory can be applied to obtain the leakage inductance transformer model, which must be consistent with the conditions of the non-standard short-circuit test of Fig. 1(d).

                 p N x i sh R          s N      b N b i p i            a N load R s i 2 turns
i p v      b N            a N          s N s i 1444424444 3 p i      b N          s N s i b i           

Leakage Inductance Analytical Computation

Several formulas for the analytical computation of the leakage inductances have been proposed in the past. For the case of concentric transformer windings, with the same height h, the leakage flux that flows due to the load current is virtually parallel to the axis (except in the ends of the windings). Under this assumption, Fig. 3(a) shows the per unit MMF distribution (m k ) obtained for the windings geometric structure of the test transformer. It is considered that the MMF's of the two windings are equal and opposite, which is valid for normal operating conditions (no fault). The leakage inductance can be computed by using (notation as per Fig. 3) [START_REF] Blume | Transformer Engineering. 2nd Ed[END_REF]:

( ) n 2 2 2 2 1 0 ( ) 1 1 1 2 3 σ σ - - = π µ   = ⋅ ⋅ + + + ⋅ ⋅ ⋅   ∑ ax k k gk k k k k k wk k N L K m g r m m m m w r h (1)
where N 1 is the number of turns of the excited winding, µ 0 is the permeability of free space, n is the number of vertical layers (7 in this case), r wk is the mean radio of the coil k, r gk is the mean radio of the gap between coils k and k+1, and m k is the per unit MMF acting on gap between coils k and k+1, see Fig. 3(a).The Rogowski correction factor, K σ , is used to take into account the flux fringing at the top and bottom of the windings and the effect of the iron core. For the case of a shell-type design: being E the height of the winding, T the wave length of the MMF wave, L the mean coil perimeter, L 1 that part of the perimeter which has iron on both sides, L 2 the rest of the coil perimeter, and C and C 1 the distances from iron to coil on the two sides [START_REF] Blume | Transformer Engineering. 2nd Ed[END_REF]. Equation ( 1) can also be used, with proper adaptations, for the computation of the leakage inductance of a disk-type winding configuration. Fig. 3(b) presents a generic disk-type winding arrangement. In this case it can be assumed that the leakage flux has only one component in the radial direction and the radial leakage inductance becomes:

( ) ( ) 2 2 2 2 2 2 1 1 4 2 1 1 1 1 1 1 1 2 2 π π π - π + + π - - - - σ   - -     = - - + + -       π         E E C C E C C T T T T C T L L L e e K
( )

n 2 2 2 2 1 0 ( ) 1 1 1 2 3 σ σ - - = π µ   = ⋅ + + + ⋅ ⋅   ∑ rad k k k k k k k k N r L K m g m m m m h w (3)
When there are irregularities in the concentric windings, such as a fault, the leakage flux is no longer parallel to the axis, but has significant components of radial flux, depending on the amount of asymmetry [START_REF] Stephens | Transformer Reactance and Losses with Nonuniform Windings[END_REF]. The formulas given by ( 1) and ( 3) are no longer valid under these asymmetrical conditions and a direct computation of the leakage inductance for these winding arrangements can be very complicated and extremely laborious. A very ingenious and useful way of dealing with these situations was proposed by Stephens [START_REF] Stephens | Transformer Reactance and Losses with Nonuniform Windings[END_REF] (and generalized in [START_REF] Blume | Transformer Engineering. 2nd Ed[END_REF]), in which the leakage inductance is divided into two components, one axial and the other radial. Each component can be computed separately and then added together to give, very nearly, the value of the total leakage inductance. Fig. 4 illustrates the basic principle of the method with a simplified diagram, assuming a fault in the middle of the primary winding. The axial component is obtained by dividing the affected winding into two coils, one referring to the healthy portion of the winding and the other to the faulty part. These two parts are uniformly distributed along the axis, resulting in a concentric design configuration, which can be computed by [START_REF] Oliveira | A Permeance-Based Transformer Model and its Application to Winding Interturn Arcing Fault Studies[END_REF].

The procedure to obtain the leakage inductance radial component is shown in Fig. 4(b). First, the secondary-side current is converted to the primary-side. Next, the secondary winding is divided into three segments, with the same dimensions of those in the primary winding. The MMF of each segment is then computed assuming an uniform distribution of the ampere⋅turns in the windings. Finally, the MMF's of the corresponding segments of each winding are summed up and the radial component of the leakage inductance can be computed by using [START_REF] Jablonski | Internal Faults in Power Transformers[END_REF].

The total leakage inductance is then obtained:

( ) ( ) σ σ σ = + ax rad L L L (4)
A finite elements method (FEM) based transformer model [START_REF] Meeker | Finite Element Method Magnetics, Version 4.2, User's Manual[END_REF] is also used to investigate the adequacy of the analytical calculations. The energy method was used to compute the leakage inductance from the FEM results [START_REF] Kulkarni | Transformer Engineering: Design and Practice[END_REF]. Fig. 5(a) presents the results obtained when the position of the faulty turns (N b ) are moved along the vertical axis of coil 1, from top to bottom, using the series-addition short-circuit test of Fig. 1(d). The leakage inductance takes greater values when the N b turns are located at the coil ends, because the radial component is higher in this situation. Obviously, the asymmetry increases with the number of the faulty turns, and, as a consequence, there is also an increase in the variation of the leakage inductance when the N b turns are moved along the winding. Only one experimental result can be obtained for this specific test conditions, due to the fixed tap positions in the coils.

Fig. 5(b) presents the variation of the leakage inductance when the fault progresses vertically, involving additional turns, first from the top to bottom in coil 1, and then affecting the neighboring vertical layer (coil 2). The leakage inductance initially grows, due the increasing values of the radial component. It is interesting to note that this behavior is opposing the fault progression, since it tends to limit the faulty current. The radial component reaches its highest level when the fault has extended to just about the vertical center of the coil and then decreases more or less symmetrically. This pattern is repeated in the other vertical layers (only shown for coils 1 and 2). The axial component of the leakage inductance drops, since the number of primary turns is effectively decreasing as the fault evolves. The results obtained by the analytical calculation, the FEM analysis, and the experimental tests are in relatively good agreement.

The aforementioned method is also valid for computing the leakage inductances between pairs of windings for the case of the standard short-circuit tests of Fig. 1(e). Fig. 6(a) presents the corresponding results as a function of the N b turns, which are also in good agreement with the FEM analysis and the measured values (L σ(ij) ≡ leakage inductance when the N i turns are excited and the N j turns are shorted).

Leakage Inductance Equivalent Circuit Model

The equivalent circuit for the leakage inductance of three-winding transformers proposed in [START_REF] León | A: Dual Three-Winding Transformer Equivalent Circuit Matching Leakage Measurements[END_REF] is adapted here to represent the transformer with faulty turns. The equivalent circuit is shown in Fig. 7 and its parameters are computed from the short--circuit inductances between the pairs of windings obtained by the tests of Fig. 1(e

): 1 ( ) σ σ = ab L L (5) ( ) 2 2 ( ) σ σ = bs a b L L N N (6) ( ) 2 ( ) ( ) ( ) 2 σ
σ σ σ   = - -   as ab bs a b M L L L N N . (7)
The leakage inductance equivalent circuit model can be used to simulate the short-circuit test of Fig. 1(d). By analyzing the circuit of Fig. 7, with a short applied to the series-connected N b and N s turns, the equivalent inductance becomes:

( ) ( ) 2 (a, b+s) 1 2 2 σ σ σ σ   = + + + +   s s b s s b L L L N N N M N N N (8)
Fig. 6(b) compares the results obtained by applying [START_REF] Meeker | Finite Element Method Magnetics, Version 4.2, User's Manual[END_REF] and the ones previously presented in Fig. 5(b). Globally, the proposed leakage inductance model yields good and consistent results. The FEM results are almost coincident. Minor differences can be detected in the analytical computed results, mainly due to approximations in the calculation of the mean radius and the Rogowski correction factors.

The leakage inductance network of Fig. 7 can be integrated with other transformer models [START_REF] Oliveira | A Permeance-Based Transformer Model and its Application to Winding Interturn Arcing Fault Studies[END_REF], [START_REF] León | A: Dual Three-Winding Transformer Equivalent Circuit Matching Leakage Measurements[END_REF], in order to take into account the core and loss components.

Conclusions

This paper has presented a method for the analytical determination of the leakage inductances of transformers with winding interturn short-circuits. An equivalent circuit for the representation of the leakage inductance of transformers with winding faults is also proposed. The experimental and FEM analysis results confirm the adequacy of the proposed analytical calculation method.

Work is currently in progress to further simplify the leakage inductances computation method, in order to allow their determination from nameplate data and core window dimensions.

Fig. 1 .Fig. 2 .

 12 Fig. 1. (a) Location of the tappings of the windings; (b) equivalent circuit for a fault occurring in the primary winding; (c), (d) and (e) short-circuit test schematics.

Fig. 3 .

 3 Fig. 3. Geometric structure and per unit leakage MMF distribution for the case of: (a) axial configuration; (b) radial configuration.

Fig. 4 .Fig. 5 .

 45 Fig. 4. (a) Equivalent representation of a fault in terms of two components, axial and radial; (b) diagram illustrating the procedure to obtain the radial component.

  Fig. 5. Leakage inductance as a function of the: (a) number of N b turns and their relative position along the winding (coil 1); (b) fault progression. (Short-circuit test of Fig. 1(d).)

Fig. 6 .Fig. 7 .

 67 Fig. 6. (a) Leakage inductance as a function of the number of N b turns for the case of the shortcircuit tests of: (a) Fig. 1(e); (b) Fig. 1(d).