Systems Diagrams and Their Uses
My article posted on The Oil Drum generated a number of helpful hints that the complexity of the proposed model was a bit much for many readers (I got over twenty e-mails with friendly suggestions along these lines!) Admittedly the model of the interactions between economic and energy factors is complicated and hard to follow, even for those schooled in systems diagrams. The real life system is many times more complex than what I tried to represent but that is no consolation if the proposed model is confusing enough. So my apologies to all who got irritated with the confusion my model may have caused.
One long-time reader suggested I put up a tutorial on systems diagrams that might help orient readers for whom this kind of approach is new. As it turns out, I've been working on a series of explanations for my book on The Fundamentals of Systems Science targeted for newbies to systems science on just this subject. So here I have done a little extraction from that work to provide a short, hopefully helpful, tutorial on the use of these diagrams in thinking about complex systems.
In the below figure I show two kinds of representations of the 'same' phenomenon, one showing the flow of influences that are involved in a simple system, the other a 'process' control flow diagram. Let's look at how these work and their relationship.
Figure 1. Two perspectives on the same phenomenon, heat flow through a system and the need to maintain a relatively constant temperature. Round-edged rectangles are called mechanisms (or in math, these are the transfer functions that compute the dependent variable output based on the independent variable input). Variables and constants, which have names, are numeric values. Curved arrows represent the direction of influence. Plus and minus signs indicate the direction of influence (up or down). In the lower diagram the circle is a process and the thick arrows represent the flows, in this case of energy. The thin arrow represents the flow of information used to regulate the process.
The top diagram contains representations for variables (the values that change over time), mechanisms (the devices that either act upon the variables or are acted upon by the variables), and the directional arrows show the direction of influences. The plus signs indicate that an increase in the mechanism's output or the variable's value drives the receiving end in a particular direction.
For example, in the upper right corner of the diagram note that an increase in the ‘night sky’ (the sky gets darker at night becoming a radiation sink - no clouds) causes objects to increase their radiation of heat toward the sky, thus lowering the temperature. This takes some getting used to. And I should warn you that not all systems modelers use this convention. But you should be able to get the general idea that variables go up or down dependent on the influences of other variables (sometimes through explicit mechanisms). And the direction of influence should be made clear in some way.
On the right-hand side, insulation is noted as either a variable or constant. If you put on a shirt and still feel cool you might put on a sweater, thus increasing the insulation value. If that doesn't do the job you can put on a jacket. In this way insulation can change over time. An example of an essentially constant value of insulation would be that in the walls of your house. Until you decide to upgrade your insulation rating it is effectively constant over long periods of time. Below I will introduce the problem of feedback loops with multiple time constants by changing that constant in a step function, making it, for a bit anyway, a variable.
The general phenomenon represented here is that of heat loss from a building (or body) due to the fact that heat always flows from a warmer source (the inside of the building) to a cooler sink. It does so via the three represented mechanisms. The point being that the temperature will always tend to fall under normal earth conditions unless supplemented by a heat source, the heater. In this diagram I have explicitly included a control feedback loop, the thermostat, that activates the heater when the temperature is below a set threshold.
The lower diagram shows flows of heat and information (the feedback loop) in what I call a process model. To build a model of this process you need to know the details from the upper diagram. That is, you need to explicitly identify each mechanism and variable (or constant). The mechanisms are modeled as functions that change influence (variable) inputs into influence outputs. For example a warmer medium (right side) will cause a reduction in convection's influence, which, in turn, will reduce the rate of lowering of the temperature. The process model is convenient to show overall flows without the details of the influence diagram but the two go together to provide a complete visualization of what is going on in the phenomenon.
What isn't shown, but could be with a little more work and clutter(!) is the specific numbers representing rates of flow and influence. For example, in the lower diagram one could specify the BTU's per unit of time that the heater produces while it is on. The model would also need to know the rate of temperature rise within the building given that heat flow rate from the heater. These are the details that need to be established either empirically or from solid theoretical models. Once all of these details are added, you have a systems dynamics model that can be programmed into a computer and run.
You do need one last thing and that is setting initial conditions for all the variables, like the starting temperature, time of day, wind conditions, etc. Plug all of that into the model and run it. What you will get is a trace of the variables of interest (in this case just temperature) over time, called a time series. Figure 2 shows a continuous time series of temperature for a house, in, say, a temperate zone in early spring, from early morning through the middle of the day when the sun shines more brightly and the outside air temperature warms up. The heater comes on when the temperature falls more than two degrees below the desired set point (say, 70°F). It is designed to turn off when the temperature rises to two degrees above the set point. As the day warms up the heater comes on less often and, in fact, the internal temperature may rise quite a bit more than the turnoff point for the thermostat because heat loss hasn't just slowed, but possibly increased due to solar insolation through south-facing windows (in reality there are numerous other heat sources inside a building that can produce excess heat). The rate of heat loss from the building, given the conduction mechanisms shown in Fig. 1 is proportional to the temperature difference between the inside and outside air. With the right insulation, on a warmish spring day (say 60°F outside) this may drop to near zero heat transfer.
Figure 2. If we plot the measurement of temperature continuously over time for the above system, it would look something like this. Note that the wave form is not a perfect sinusoidal because events outside the walls of the house (sunshine, air temp, etc.) are varying as the day proceeds. The system still attempts to keep the temperature within a 'comfortable' band around the desired temperature set point.
The behavior of this kind of system with a control negative feedback loop is dynamically the same as the phenomenon of homeostasis in living systems. This is the situation when a system is attempting to maintain an ideal variable value against external disturbances (e.g. outside temperature fluctuations). Such systems are a main concern of the science of cybernetics, of which I have written a good deal.
A More Complicated Example
Many (possibly most) houses in the temperate latitudes in the USA have both heaters and air conditioners (active coolers). My house has the former but not the latter. Heaters are used to keep the temperature up during cold months and coolers are used to keep the temperature from getting too hot during hot months. Figure 3. shows an influence diagram of this situation. The mechanism in the upper right side encapsulates all three of the heat loss mechanisms shown in Fig. 1. As can be seen the natural heat loss mechanisms operate in parallel (though different times) with the mechanical cooler. The same is the case for the external heat sources and the heater. This example is particularly symmetrical making it easy to analyze. In more advanced versions of this problem we will start to see asymmetrical influences that start to change many variables, especially the temperature.
Figure 3. A process (home) that is subject to both heating and cooling influences from the environment and needs to maintain a relatively constant internal temperature. In this model the system has both an active heater and an active cooler to compensate for the outside influences. In the upper left corner there are two time series graphs, the top one showing a short time scale (3 years), the bottom one showing a longer time scale (20 years) in which we speculate a general upward trend in climate warming.
Referring to the graphs which show the difference between short-term and long-term behaviors of the external heat sources (read global warming), over twenty years we show the average temperature rising in a trend. This could well translate into hotter summers and subsequent need to run the cooler more frequently, drawing more electricity (demand) from the grid. Here we assume that the grid will respond with sufficient supply as required. Similarly, if more fuel were needed to heat the home in winter, we assume it would be provided.
Figure 4. This graph shows a time series of electricity usage for cooling as summers get hotter over time. The graph shows how a step function (one time increase in insulation (effectiveness) can hypothetically reduce the electricity usage in spite of the hotter summer.
Here we run into one of the fundamental problems with understanding the cause and effect issues in systems. The primary problem is that we have some, generally exogenous, variables that are changing systematically (trending) over longer time periods that can be difficult to compensate for (or even recognize) in the short run. In this particular example the addition of insulation, a one-shot operation, is relatively easy to predetermine the desired effect (lowering the electricity bill that had become a larger proportion of the household budget). There are systems where the time lag inherent in the change in short-term behavior can cause an over-compensation leading to undesired effects. We'll see examples sooner or later. For the moment let's consider what happens when the longer term heating trend causes a human (home owner) to make some decisions regarding the operation of the cooling plant based on comfort and the economic system.
Bringing Economics Into the Picture
Now we start to really complicate the model because we are going to introduce another exogenous variable (the above one was the long-term warming trend that led to warmer summers - more cooling hours), and two decision makers, the home owner and the electricity producer. The new exogenous variable is shown at the bottom of Fig. 5 — the cost to produce electricity. Figure 5 is a repeat of the cooling side of Fig. 4 slightly rearranged and somewhat simplified to accommodate the new mechanisms.
Figure 5. The home owner, a decision maker, and the electricity supply industry couple the mechanical cooling system to the economic system through costs and prices. This diagram includes both short-term feedback loops and those with longer-term changes that introduce time delays or have larger time constants. The oval with trend lines inside represent the loops that may change more slowly over longer time scales.
Lets run through a scenario starting with short run conditions as depicted in Fig. 3 (the 3-year time horizon graph). The home owner notices that the summers are hotter, and more cooling time is needed. He has set the thermostat to a comfortable temperature, but notices that his electricity bill is increased due to the increased demand for electricity to run the cooler. In the short run, he may decide to set the thermostat to a slightly higher setting (a little less comfortable perhaps, but saves on electricity). Another home owner might just accept the increased cost of the monthly electric bill and insist on keeping the temperature where they think it is comfortable (also businesses may feel it necessary to keep temperatures down for their customers' comfort).
In that latter case, the electricity producer is going to notice a trend in increasing demand for power and may have to either build more generating capacity or arrange to buy more (expensive) electricity from the grid. Either way, the cost of electricity may go up as a result and this will eventually be passed on to the home owner. However, notice that these effects do not happen in the same time scale as the daily or monthly demand/supply loop. It takes time to build new facilities (though the utility might start charging more in advance as a means to finance the project). Thus this model mixes feedback loops having inherently different time scales.
Suppose that a fair number of home owners see their bills going up even when they set their thermostats higher, say over a few more years as the summers get hotter. Further suppose that a fair number of them realize that installing higher valued insulation will reduce their cooling needs and thus reduce their short-term demand for electricity and save them money. Recalling the situation represented in Fig. 4, the installation of insulation reduces the electricity requirements, which pushes demand for more electricity down. Now put yourself in the shoes of the poor electricity producer. They built new capacity, or made contractual arrangements to buy more expensive electricity and somewhat suddenly they aren't selling all that they need to to pay off their investment (or contracts). This is where things will get very interesting as we continue to build economic models based on systems dynamics with feedback loops having inherently different (and often substantially different) time lags. Decision makers need to take actions that will have an impact on future performance of the system. But it isn't generally clear what the best decision would be without knowing what other decision makers will do (in the long run) or how exogenous factors may impact the system (for example if the electricity were generated by coal-fired plants and the price of coal goes up due to scarcity or increased costs to mine/deliver). All too often in these situations decision makers may “over-react” or over compensate for what they see as a trend, which then later causes some other problem to pop up.
This is the inherent problem with markets as mediators of supply and demand. Either of these factors can change over multiple time scales in such a way as to throw the short-term price system into erratic behavior. Price is supposed to be the ‘signal’ that provide information in the market such that decision makers can act, in theory, in a rational way. But some of these systems can become highly unstable. This is especially true for systems that include positive feedback loops that are inconsequential at low levels for some variables, but grow to dominate when those variables pass a certain threshold. Such positive feedback loops have a tendency to make matters worse and amplify the problems. This will be the subject of a future tutorial on systems diagrams and how they can help us understand the world we live in.