Transcript A SYSTEMS APPROACH TO CREATIVE WATER RESOURCES …

Higher order feedback loops
One stock – first order system
More stocks – higher order
Behavior
Second order negative feedback –
oscillating system
Second order positive feedback –
exponential growth and oscillations
(unstable)
Higher order feedback loops
Second order negative feedback
pendulum
Higher order feedback loops
Higher order feedback loops
Generalization
Higher order feedback loops
Higher order feedback loops
Second order positive feedback loop
Higher order feedback loops
Delays
Delays occur frequently in many systems.
When a pollutant is dumped into a river, it
takes time to dissipate.
Delays are divided into two types:
(a) Material delays - resulting from the time
involved in processing materials.
(b) Information delays - resulting from time in
perceiving and acting upon information.
Material delay example
Construction of apartments in a large city.
Builders construct apartments in response
to the gap between the total number of
apartments desired and available.
Start with 10,000 apartments.
Increase to 15,000.
Delay - completion time 2 years.
Material delay example
Graph 1
Table 1
Total Number of Apar tments
Apartments i n Constr uction
Completion Rate
Initi ati on Rate
Completion Time
Initi ati on Time
Differ ence
Desir ed Apartments
Material delay example
Initiation time = 1
Apartments in construction = 0
Completion rate=Apartments in
construction/Completion time
Total number of apartments =10,000
Desired apartments = 10,000 + STEP(5000,1)
Difference= Desired apartments – Total
number of apartments
Completion time = 2
Material delay example
Information delay example
Joe’s Cement Factory makes and sells
cement blocks Monday through
Friday.
Each morning Joe has to decide how
many blocks to make, and he relies on
the average sales over the past five
days in making his decision.
Information delay example
Cement bl ocks to make
Baking r ate
Sales
Graph 1
Information delay example
Cement blocks to make = 100 + STEP
(20,10)
Sales = 100 + STEP (20,5)
Making rate = SMTH (Sales, 5) –
Cement blocks to make
Information delay example
Solution interval DT
System Principle 11
Solution interval is in all level equations
and no others
Solution interval DT
DT
Solution interval
Period of measurement
Delta time
Time step
DT is the time period in which the
level is changed by the rate
DT in Vensim
The best size of TIME STEP is determined by the
following considerations:
TIME STEP should allow test inputs to be accessed
regularly.
TIME STEP should allow data to be accessed with
appropriate regularity.
TIME STEP should be smaller than 1/3 of the shortest
time constant in the model (not applicable with automatic
step size adjustment in Runge-Kutta integration).
TIME STEP should be smaller than the shortest period
for which a significant change in model behavior is at all
likely.
DT in Vensim
An inappropriately long TIME STEP leads to
incorrect behavior.
In general if you see oscillation with a frequency
that is near to twice TIME STEP you should test
TIME STEP to see if it has an appropriate value.
If you are using Runge-Kutta integration with
automatic step size adjustment, the third and
forth considerations do not apply.
Vensim will automatically determine how small it
needs to make TIME STEP in order to achieve the
desired accuracy or issue an error if it is unable to
do so.