None Reading_14

Table of Contents

Challenge

In today's reading, your challenge is to return to our "toy" power grid system, and try to understand, given non-negiglible power line resistance, how much voltage will be available to your cabin at any given time. In general, our goal is to see whether the first law of thermodynamics offers us a "shortcut" to relating the voltages in this system to one another, the way it offered us an easy way to relate the currents in our system when we introduced the concept of a node.

We will consider the small, cabin-in-the-woods DC (direct current) power grid (including a storage system) given by the following schematic. You've set up this system to power small electronics in your cabin, because you didn't feel that a large, AC power system was necessary.

image-3.png

As before, the voltage source could be an intermittent power generation station like a wind turbine or solar farm. This source is connected to power lines with resistance $R_1$, and then to a storage bank, which we will model as an ideal capacitor (This may not be a very good model of a real storage bank, which likely consists instead of batteries that don't behave quite like capacitors). From the storage, bank, the power runs to your cabin through more power lines with an effective resistance $R_2$. The electrical load in your cabin can be modeled by an approximately constant resistance $R_h$.

Challenge: System Scoping

In scoping your model of this system, you choose to represent the intermittent renewable energy source as an external input. Given your newfound knowledge about "nodes," which represent idealized points between elements where power is transmitted "perfectly," making the first law of thermodynamics for a node reduce to $\dot{W}_{in} = \dot{W}_{out}$, you annotate the circuit diagram to mark the distinct voltages in the schematic like so:

image-3.png

Then, you make a list of independent and dependent variables (yes, this is often done as part of model construction, but there's no reason we can't do it now!). Because you're interested in the voltage at the house, your list reads:

independent

  • $R_1$
  • $R_2$
  • $R_h$
  • $C$

Dependent

  • $V_3,\dot{V}_3$

Your model scope's physical boundary is given by the dotted green line, since you'd like to avoid having to account for the potentially "infinite" power of the constant voltage source (supposedly, it provides a constant voltage no matter what current is required).

image-3.png

Representing this as an energetic diagram, you find the following simplified form of the time-derivative of the first law:

image-2.png

Which can be written as the following equation: $$\require{cancel}$$ $$\dot{E} = \dot{W}_{in} - \dot{Q}_{out}$$

The input work $\dot{W}_{in}$ is expected to come from the voltage source. While you don't expect any heat added to the system to increase its stored energy, you know that the resistors will transfer heat to the environment. you also know that because of the capacitor in the system, we know that the system will accumulate energy over time if $\dot{W}_{in}>0$. Now, as usual, you write the first law for each of the components in your system, with the appropriate assumptions for each "idealized" component:

R1

$$\dot{W}_{in} - \dot{W}_{out} = V_1i_1 - V_2i_1 = i_1^2 R_1$$$$\therefore V_{1}-V_{2} = i_1R_1$$

R2

$$\dot{W}_{in} - \dot{W}_{out} = V_2i_3 - V_3i_3 = i_3^2 R_2$$$$\therefore V_{2} - {V}_3 = i_3R_2$$

Rh

$$\dot{W}_{in} - \dot{W}_{out} = V_3i_3 - \cancel{V_g}i_3 = i_3^2 R_h$$$$\therefore V_{3} - \cancel{V}_g = i_3R_h$$

C

$$\dot{W}_{in} - \dot{W}_{out} = V_2i_2 - \cancel{V_g}i_3 = CV_2 \dot{V}_2$$$$\therefore i_2 = C\dot{V}_2$$

You also know that because of the node 2 that causes a split in current, you can write an equation for the "node" where voltage is equal to $V_2$ using KCL, which we derived using the first law in Reading 13.

$$i_1 = i_2+i_3$$

This collection of representations of the first law of thermodynamics for the elements in your system allows you to re-write the first law for the entire system, taking all of the individual components' contributions into account. Because we are looking for a relationship between the voltages in our system, we will leave all of the equations in the form that exposes the voltages $V_1,V_2,V_3$.

We know that the heat transfer out of the system is simply the balance of work into each resistor:

$$\dot{Q}_{out,system} = (V_1 - V_2) i_1 + (V_2 - V_3)i_3 + (V_3)i_3$$

We also know that the energy stored inside the system is due to energy stored in the capacitor. Using the first law equation for the capacitor, we can see that:

$$\dot{E}_{system} = \dot{E}_{cap} = V_2 i_2$$

Finally, we know that the work into the system comes from the voltage source.

$$\dot{W}_{in,system} = V_1 i_1$$

This means that the entire system's first law equation:

$$\dot{E} = \dot{W}_{in} - \dot{Q}_{out}$$

Can be re-written as:

$$V_2 i_2 = V_1 i_1 - ((V_1 - V_2) i_1 + (V_2 - V_3)i_3 + (V_3)i_3)$$

Using the node 2 equation, we can replace $i_2 = i_1 - i_3$ to get:

$$V_2 (i_1 - i_3) = V_1 i_1 - ((V_1 - V_2) i_1 + (V_2 - V_3)i_3 + (V_3)i_3)$$

Expanding this, rearranging it, and solving for the voltage $V_1$, we find that a curious identity emerges:

$$V_1 = (V_1 - V_2) + (V_2 - V_3) + (V_3)$$

This, as with many forms of the first law, doesn't tell us everything we need to know, but if we define:

$$V_1 - V_2 \equiv V_{12}$$

as the "voltage drop across resistor $R_1$," and we define:

$$V_2 - V_3 \equiv V_{23}$$

as the "voltage drop across resistor $R_2$," and:

$$V_3 - \cancel{V}_g = V_{3g} = V_3$$

as the "voltage drop across $R_h$, we can re-write this equation as:

$$V_1 = V_{12} + V_{23} + V_{3g}$$

This is what engineers call "the principle of compatibility," also known as "Kirchoff's Voltage Law" for electrical systems. We know that it is just a statement of the first law of thermodynamics, but it can actually be derived for any subsystem you could think of comprised of our system's components. If we "cut off" our system boundary to only include $C$, $R_2$, and $R_3$, with an input of $V_2i_2$, we could perform a similar analysis. The same is true if we were to "cut off" our system boundary to only include $R_1$ and $C$, with an input power of $V_1 i_1$ and an output power of $V_2 i_3$, we could do yet another similar analysis.

But this would take a long time! This is why system dynamicists use "Kirchoff's Voltage Law" to speed up the process.

Commonly, Kirchoff's Voltage Law is written:

"The sum of the voltage drops across the elements in any "closed loop" in a circuit must be zero.

Because in our system $V_1i_1 = \dot{W}_{in}$ represents a voltage gain (a negative voltage drop across the source), this is equivalent to what we've derived.

The Principle of Compatibility (Kirchoff's Voltage Law)

The principle of continuity states that the sum of the voltage drops around any closed "loop" in a circuit-like network must be zero. This is a direct consequence of the first law of thermodynamics. Consider the following circuit, for which 3 closed loops exist:

image-3.png

For Loop 1, the principle of compatibility states: $$\require{cancel}$$ $$ V_{41} + V_{12} + V_{23} + V_{34} = 0$$

Which, given the definition of $V_a - V_b = V_{ab}$, could be re-written as:

$$ \cancel{V}_{4} - V_1 + V_1 - V_2 + V_2 - V_3 + V_3 - \cancel{V}_{4} = 0$$

Similarly, for Loop 2, the principle of compatibility states:

$$V_{41} + V_{12} + V_{24} = 0$$

And for Loop 3, the principle of compatibility states:

$$V_{42} + V_{23} + V_{34} = 0$$

Compatibility can also be appled to fluid systems (replacing voltages with pressures) and to mechanical systems (replacing voltages with velocities or angular velocities).

Using Kirchoff's Voltage law, or KVL, we can further update our disciplined process for model construction using conservation of energy. This modification is reflected in the cell below:

Updated Disciplined Process for using Energy Conservation to Construct a Model inclusive of Kirchoff's Current Law (KCL) and Kirchoff's Voltage Law (KVL)

Now that we have a "new tools" (KCL/KVL) to help us with model construction, we can update our "step 3" in the general discplined process for Model Construction to include the use of nodes in our process. Our update of the disciplined process is as follows:

  1. Draw an energetic diagram for each element in your model scope, crossing off any terms you are assuming to be insignificant. When you do this, justify each assumption clearly in writing. Making an assumption is always a deliberate risk; the model evaluation step of our overall disciplined process for ES103 will guide you in determining the quality of the assumptions you make.
    1. You may reference the assumptions in the "standard" lumped element table in your justification, but you must still give a reason for assuming that an element acts like one of these "standard" element.
  2. Write the time-derivative of the first law of thermodynamics for each element to describe how the element transfers energy between itself and its surroundings. Take care to substitute what you know about the energy, work, and heat terms for each element into the equation.
    1. You may cross out any "extra" power variables in each final first-law equation for each element (e.g. $V_{12} i = i^2R \rightarrow V_{12} = iR$)
  3. Write equations that indicate how each element in your scope exchanges energy between itself and other elements in the scope, or between itself and the overall system's surroundings. Use physical connections between elements in your system to guide you here. Specifically, power linkges between elements can be written using Krichoff's Current Law (continuity) and Kirchoff's Voltage Law (compatibility):
    1. Identify "nodes" in your system and write the principle of continuity for these nodes (e.g. $i_1 - i_2 - i_3 = 0$).
    2. Identify "loops" in your system and write the principle of compatibility for each loop (e.g. $V_{12}+V_{23}+V_{34}=0$.
  4. Using algebraic manipulation, combine your equations from steps 2 and 3 to obtain a final system model.
  5. Check your final system model for internal validity, including the signs and units of each term, and the overall form of the equation.

Assignment:

Making explicit use of the first law for each component, as well as your "newfound" principles of continuity and compatibility, derive a differential equation describing how the voltage available at the house evolves over time, given a known input voltage $V_{in}$ from the power source.

I have provided "data" (it's fake... sorry!) to help with model validation. Your power source is a wind turbine that can be approximated as providing a constant 24V, but only for 60% of each day. The measured voltage vs. time for your turbine (system input) for one week is given in the data file turbinedata.txt and the measured voltage at your house is given in the data file housedata.txt. Each file has a column for time (seconds) and a column for voltage.

In your simulation, you will need to use the input voltage at the last time step to help compute the derivative of the house voltage because the input voltage is not constant.

What You Know

  • The electrical line from the turbine to node 2 has an effective resistance of $100\Omega$
  • The electrical line carrying power to the cabin has an effective resistance of $150 \Omega$
  • The "ideal capacitor" storage bank, when fully charged to $24V$, stores $450,000$ Joules of energy, or approximately the same energy as a typical deep-cycle marine battery.
  • The small appliances (bulbs, phone chargers, etc.) that are always on in your cabin have an average effective resistance of about $1000\Omega$

Deliverable: Model Scoping and Construction

Place your work below, and circle your final differential equation. Please make sure it is in symbolic form (no numbers!)

YOUR ANSWER HERE

Deliverable: Model Evaluation

Simulate your model and compare it to the data provided with this assignment

In [1]:
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error('No Answer Given!')

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Deliverable: Use your model for design

Using a copy of your simulation, which you can run in the cells below, and knowing that the devices in your cabin need at least $12V$ to operate properly:

  1. does your micro-power grid perform satisfactorily?
  2. What would happen if you increased the effective capacitance of your storage system by 10 times? Would this be beneficial? Why or why not?
  3. What would happen if you left a large appliance on in the house, decreasing the house's effective resistance to 100 ohms?

YOUR ANSWER HERE

In [2]:
% YOUR CODE HERE
error('No Answer Given!')

error: No Answer Given!