Markov chain models for power system loads

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Markov chain models for power system loads

• Presented by
• Dinesh Kumar Salem-Natarajan
• Arizona State University

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Contents

• Introduction
• A Markov chain process
• Terminology
• Using Markov chain process to model a load
• Tests/ Results of tests
• Analysis of test results
• Scope of future work
• Conclusion

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Introduction

• Need for a load model - A load model is needed in
  order to model the system and account for changes
  in system demand as system dynamics occur.
• Various techniques used - Of the many methods of
  modeling an electric load, the two most important
  are i) Component based and ii) Dynamic state
  based.
• Technique used - The second technique is used in
  this work.

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A Markov process

• A Markov chain process - This is a process which
  uses the history of the system to predict the
  future response of the system.
• Markov chain models - This is a mathematical
  device that uses discrete states and trajectories
  between states to model a process. A key element
  of this model is that the future trajectory is
  determined entirely by the present state.

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The Markov process
Markov chain model
Probabilities assigned to branches of a Markov
chain
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Terminology

• A system state
• System state space
• State transition
• State transition matrix
• State transition probability
• The Markov model

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• A system state - The values of the system
  parameters used to describe the system at any
  instant is the state of the system at that
  instant.
• System state space - The multi-dimensional space
  that is defined by the various states of the
  parameters that characterize the system, is the
  state space of the system.

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• State transition - The change in the system
  parameters between two sampling instants,
  represents a change in the state of the
  parameters and hence a change in the system
  state. This is referred to as the system
  undergoing a transition in state between the two
  sampling instants.
• State transition matrix - The matrix formed with
  the states of the system along its rows and
  columns, with the rows/columns representing the
  starting states or ending state for each sampling
  instant.

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• State transition probability - This is the
  probability that a particular system state be
  visited when transition takes place from another
  system state.
• The Markov model - This is the state transition
  matrix that has the memory of the states visited
  by the original system, in terms of the
  probabilities of transition from a given system
  state, at a sampling instant, to any other system
  state.

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Using Markov chain process to model a load

• 1. Use the input data of the system to find the
  number of hits/ visits to different states of the
  system
• 2. Form the Markov model for the above system.
• 3. Regenerate the response of the original system
  using the Markov model.

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Concept of this technique of load modeling
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Probabilities assigned to branches of a Markov
chain
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The state transition matrix
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Regeneration using Markov model

• Starting state - Select a start point for
  regenerating the original load.
• Random probability - Select a value of
  probability using a uniformly distributed random
  number between 0 and 1.
• End state - This is compared to the values stored
  in the matrix. The state that has a probability
  greater than that of the randomly selected number
  is the end state.

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• Determination of system state - The system state
  is the end state as found in the previous slide.
• Estimation of the current and voltage states -
  The states of the current and voltage are found
  from the system state.
• The output of the model - The output values of
  the current and voltage values are found from
  their respective state information.
• The next step - The end state of the previous
  step (sampling) is the new start state. And all
  the steps are repeated for this start state.

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Tests/ Results of tests

• Tests - To determine how well the Markov model
  represents the original system, tests were
  conducted on the model and the results are
  compared to the original system response.
• Data types - Two types of data are used in the
  testing i) Synthetic data and ii) Actual load
  data (Arc furnace data).

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• Test model - In all tests, the model that had a
  system sample space of 400 X 400 states were
  used. The voltage and current signals had 20
  states each.
• Note - The tests were conducted with noise and
  harmonics added to synthetic data.

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Test results
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Test 1
Test 4
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• Test results Synthetic data
• 1. As an example, in test 1, the model is tested
  using synthetic data that contains a pure
  sinusoidal voltage and a cosine current signal.
• 2. The model is tested using noise and harmonics
  and with both. An example, where both are
  included, is shown in test 4.

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Analysis of test results

• The analysis of the test results point to the
  following significant details.
• 1. Coarseness of response
• 2. Forbidden states
• 3. Response in presence of harmonics
• 4. Initialization

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Coarseness of response

• The regenerated waveform is coarse and hence the
  output approximates the input.
• Increasing resolution further results in an
  increase of the system state-space by a huge
  factor.
• To illustrate this fact, when the resolution is
  increased by a factor of 2, the system state
  space increases by a factor equal to the fourth
  power of the increment ratio.

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Forbidden states

• The forbidden state is a state, which is not
  encountered by the original signal
• When the model encounters that state, it tends to
  remain at that state.
• It is undesirable that the system enter these
  states.
• Sampling of the original data over a prolonged
  period of time is believed to mitigate this
  problem.

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Response in presence of harmonics

• The response of the model is shown in test 4.
  This signal has a THD of 37 percent, contributed
  by third and fifth harmonics.
• This signal is also contaminated with noise and
  the signal has a S/N ratio of 4.
• The response shows that the model holds good for
  signal contaminated with harmonics and noise.

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Initialization

• The start point/ state for regeneration must be a
  valid system state.
• In case this state is a forbidden state, the
  regeneration of the model may be incorrect, as
  the model is fed an invalid start state.
• Hence it is important to input a valid system
  state to the model for regeneration.

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Scope of future work

• This model is yet to be tested on the actual arc
  furnace data.
• To calculate the various measures of error
  involved in this method.
• The THD of the original and the regenerated
  signal could be compared as a measure of
  correctness of the model.
• As the EAF data is non-periodic, mean of the
  original and the model output signal could be
  compared, rather than RMS value.

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Conclusion

• The following conclusions can be drawn from the
  work done thus far.
• Once this model is tested for the actual load
  data ( in this case, the EAF data ) and
  validated, the model could be extended to
  regenerating other categories of loads.
• EAF - Electric arc furnace

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Conclusion
contd

• The response of various categories of load data
  could be used to train/ form the model and this
  model could possibly be used to predict the
  system dynamics, given a particular system state
  in terms of power or voltage or current, is made
  available.
• This technique could be expanded to include
  symbolic dynamics in generating a load model.