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**Statistical Analysis**

**Introduction**

Overloading of an IPT power supply as mentioned in Chapter 1 happens when the aggregate power demand exceeds the maximum capacity in the power supply, and can have disastrous system-wide effects. Apart from the costly solution currently used – to install enough capacity to meet the maximal possible demand – the problem of overloading can also be solved if some extra energy is available in the power supply to meet the power demand during an overloading event without requiring extra capacity, or if the power demand during an overloading period can be reduced or deferred to a later time. The fact that IPT systems have energy storage elements in both the power supply and pickups means that they potentially already have the tools to deal with overloading: the DC capacitor and the track inductor in the power supply could temporarily provide the extra energy needed if the overloading magnitude is small and the duration is short, or some appropriate control methods could stop power flow to some of the pickups during an overloading when the output DC capacitors of these pickups are temporarily used to power the loads instead. However having the energy storage elements and/or appropriate control methods alone is not enough, one also needs to understand how reducing the power supply capacity affects the overloading characteristics (e.g. frequency and duration) and in turn how it affects the pickups and the system operation. Then an informed decision on the optimal power supply capacity which balances both costs and impacts can be made. Unfortunately, to date, such understanding is lacking.

The purpose of this chapter is to fill this knowledge gap by statistically describing the aggregate power profile of a multiple slow-switching pickup IPT system. In this study, the analytical expressions for two important parameters will be derived. The reason for choosing the slow-switching control is because this control is very simple and popular in many applications especially materials handling applications. In fact slow-switching controllers are used in more than 70% of the materials handling systems designed by the world leading materials handling company Daifuku corp.

The chapter starts with an overview of similar systems, in this case telephony systems and existing traffic models to identify the parameters which are currently missing but are very important in describing the overloading characteristics of the IPT system under considerations. This is then followed by the development of a statistical analysis for a multiple slow-switching pickup IPT system. The analytical expressions for the two parameters are then applied to a numerical example IPT system to illustrate how they can be used to evaluate the impact of reducing the power supply capacity on the system. Lastly the analysis result is compared against MATLAB simulations for verification.

**Relevance of telephony systems and existing traffic models**

A multiple slow-switching pickup IPT system is in many ways analogous to a telephony system where a single service provider provides telephone services to many end users. To have enough channels such that all users can make calls at the same time is both unrealistic and unnecessary. However, not to have the maximum capacity possible means there is a finite chance, however small, that some users will occasionally be denied service and would correspondingly reduce the system QoS (Quality of Service). To properly balance both the cost and QoS of the telephony system, many traffic models have been developed and they differentiate from each other by the following parameters.

- call arrival patterns
- blocked call disposition types
- call holding time
- number of loads

A typical multiple slow-switching pickup IPT system would have the corresponding parameters - pickup turn-ON patterns – which is essentially random
- blocked power transfer disposition types – which varies depending on the control mechanism
- switch “ON time” – which is nearly constant (average power demand is the same)
- number of pickups – which varies

The models that best suit an IPT system are the Erlang distribution and the Engset distribution between which the major difference is the number of loads. Both models enable calculations of the blocking probability and the probability of the system supplying the exact number of loads. These models are sufficient for telephony systems because in the event that all channels are busy, the system simply stops additional calls from being made. Regardless of whether those calls are lost forever, retried or put in a queue, this does not affect other calls or compromise the reliability of the whole system. This inherent reliability of the telephony system means no extra control or knowledge about overloading is required. The loss of QoS represents the sacrifice that the system designers are willing to make in order to reduce the number of channels and therefore the cost of the system.

In an IPT system however, the fact that overloading could affect all the pickups and cripple the system means that proper control for overload mitigation is a must and to do so, knowing the overloading probability alone is not sufficient. To illustrate this, imagine that a particular 10kW IPT system with an 8kW power supply has a probability of being overloaded 1% of the time. However without additional information, one cannot know if this 1% probability of overloading means a continuous 1s overloading period in every 100s or a continuous 10µs overloading period in every 1ms. In the former case, the overloading is clearly disastrous as almost certainly no energy storage elements in the system can provide enough extra energy for the whole overloading duration, whereas in the latter case system shut down is more likely to be prevented with possibly small impacts on the coupled loads (pickups) and the system.

Therefore in order to provide the extra information necessary, two parameters describing the overloading characteristics will be derived in this chapter and they are: the average number of times per second that the power demand exceeds the available power in the system ( ), i.e. the frequency of overloading, and the average duration of each time the power demand exceeds the available power in the system ( ), i.e. the duration of overloading.

**Development of Statistical Analysis**

For the purpose of this statistical analysis, the following assumptions are initially made:

- All pickups are slow-switching controlled (either fully ON, or fully OFF), more details in section 3.4.1
- All pickups switch independently of each other
- Steady state system operation is assumed (system start up is ignored)
- All pickups are identical and have the same average loading condition
- Transients and overshoots of internal voltages are ignored initially (some more realistic transient effects will be incorporated later on)

In many real life multiple pickup IPT systems, these assumptions are often realistic. Special attention will be given to assumptions 4 and 5. Assumption 4 is valid because in many applications such as overhead monorails, AGVs, and lightings systems, multiple identical pickups perform a single or a series of tasks. While at any instance in time each pickup may perform a different task from another (for example, due to the tasks being distributed at different positions along a monorail in a factory) representing different instantaneous loading conditions, over a long period of time (for example, one or more rounds of a monorail track), these differences average out and it is expected that the average loading condition for all the pickups are the same. Assumption 5 is valid because although transients and overshoots are unavoidable in real life systems, when the system is properly designed, their magnitudes should be relatively small compared with the actual voltage and current waveforms.

**Significance of the analytical results**

This section focuses on how the analytical results derived above (3.17) and (3.20) can be used to evaluate the effects of a reduced power supply capacity on the pickups and the system and is therefore able to help choose the most suitable power supply capacity. To facilitate easy interpretation, a numerical example of an IPT system will also be used.

**Slow-switching controller**

Fig. 3-6 below shows a typical parallel tuned IPT pickup circuit, its operation was described in chapter 2 and therefore will not be repeated here. Fig. 3-7 shows how the pickup’s input power and output voltage change during normal operation. Note that the slow-switching controller maintains the pickup output voltage to be within the hysteresis band of ± and for the purpose of this thesis, the upper and lower voltage boundary of this hysteresis band will be referred to as and respectively.

**CHAPTER 1. INTRODUCTION **

1.1. INTRODUCTION

1.2. FUNDAMENTAL THEORY OF IPT

1.3. TYPICAL IPT SYSTEMS

1.4. BENEFITS OF IPT

1.5. APPLICATIONS OF IPT

1.6. THESIS MOTIVATION

1.7. CONTRIBUTIONS

1.8. OUTLINE OF THE THESIS

**CHAPTER 2. IPT FUNDAMENTALS **

2.1. INTRODUCTION

2.2. OVERVIEW OF IPT SYSTEMS

2.3. PRIMARY CIRCUIT

2.4. COUPLING

2.5. SECONDARY CIRCUIT

2.6. COMMUNICATION

2.7. CONCLUSIONS

**CHAPTER 3. STATISTICAL ANALYSIS **

3.1. INTRODUCTION

3.2. RELEVANCE OF TELEPHONY SYSTEMS AND EXISTING TRAFFIC MODELS

3.3. DEVELOPMENT OF STATISTICAL ANALYSIS

3.4. SIGNIFICANCE OF THE ANALYTICAL RESULTS

3.5. SIMULATION RESULTS

3.6. CONCLUSION

**CHAPTER 4. POWER MANAGEMENT FOR MULTIPLE-PICKUP IPT SYSTEMS IN MATERIALS**

**HANDLING APPLICATIONS**

4.1. INTRODUCTION

4.2. TOTAL POWER DEMAND OF THE PS WITH MULTIPLE PICKUPS

4.3. POTENTIAL STRATEGIES AND COMMUNICATIONS

4.4. PROPOSED CONTROL – FAST SWITCHING CONTROL

4.5. PROPOSED CONTROL – SLOW SWITCHING CONTROL

4.6. EXPERIMENTAL RESULTS

4.7. DISCUSSIONS

4.8. CONCLUSIONS

**CHAPTER 5. DOUBLE-COUPLED SYSTEMS FOR IPT ROADWAY APPLICATIONS **

5.1. INTRODUCTION

5.2. FEATURES OF ROADWAY IPT SYSTEMS

5.3. CURRENT ROADWAY IPT SYSTEMS

5.4. DOUBLE-COUPLED SYSTEM

5.5. EXPERIMENTAL RESULTS

5.6. CONCLUSION

**CHAPTER 6. POWER MANAGEMENT FOR DYNAMIC ROADWAY EV POWERING APPLICATIONS **

6.1. INTRODUCTION

6.2. CHARACTERISTICS OF ROADWAY IPT SYSTEMS

6.3. COMPARISONS WITH MATERIALS HANDLING APPLICATIONS

6.4. ICC WITH LOW STORAGE (A SMALL ??)

6.5. ICC WITH LARGE STORAGE (A SIGNIFICANTLY LARGE ??)

6.6. SIMULATION STUDY

6.7. COST COMPARISONS

6.9. CONCLUSIONS

**CHAPTER 7. CONCLUSIONS **

7.1. GENERAL CONCLUSIONS

7.2. FUTURE WORK APPENDIX A. DERIVATION OF THE AUTOCORRELATION FUNCTION OF A REAL LIFE SWITCHING PATTERN

REFERENCES

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Power Management for Multiple Pickup IPT Systems