# Control Volume Switchboard

This is how I use a control-volume approach to practically monitor circuits. It’s half maths and half “it’s easy that way”. The area below is a switchboard, monitored with multiple CTs. See phisaver.com.

Referring to the above:

**N**et- Import is positive, export is negative

**P**roduction is positive**L**oads are positive and monitored**U**nmonitored loads are same as Loads, just without a CT**B**attery is a positive when providing energy (i.e. like Net and Production) and negative when charging. (We are assuming Production === Solar. See alternative: P=B+S, further down)~~Battery is a special case of Load. Positive when changing, negative when discharging. It is not considered in the following.~~

We can then work out some useful equations. I use them in iotawatts.

**N + P – L – U + B = 0****U = N + P – L**+**B**

If **U**=0 (i.e. we monitor all loads):

**B = – N – P + L**

## Derived Measurements

We can further derive some measurements with cannot be directly sensed. There can be found on the Iotawtt:

- Import : Net max 0
- Export: (Net max 0) x -1
- Unmonitored: Net + Production – (monitored) Loads
- Consumption: Net + Production
*(since Consumption = U+L and U+L = N+P)*

Some further notes on these measures.

- Production and consumption are
*not*at the meter, as production can be self-consumed. - Production is of any kind (e.g. solar, battery, whatever). Battery storage within the building is consider production, it’s just shifted in time.
- Net is “at the meter”.
- It’s help to record Import and Export, because we can easily sum them. You can’t sum Net, then get Export: you have to get Export at each time then sum. This is because the positives and negatives of Net can cancel.

Given that:

`Net = Import - Export `

`Net = Consumption - Production`

.. then putting this together:

`Production - Export + Import - Consumption = 0`

## Import Consumption v Free Consumption

We want to state (e.g.) “your solar system covers *x *% of your energy”. This is a bit tricky. Imagine over two days:

- Day 1: 10 kWh consumption, no production (broken)
- Day 2: 10 kWh consumption, 20kWh production (exactly balanced)

So, do we mean:

- Option A: Fraction of consumption
*directly (*i.e. behind the meter) provided by solar. If so, then that’s 10kWh / 20 kWh or 50%. This is, perhaps, a more accurate measure of solar contribution. - Option B: Fraction of consumption provided by solar (either directly, or exported-then-imported). If so, then that’s 20kWh / 20kWh or 100%. The general definition of “Net Zero” is 100%.

Let’s break consumption up…

Consumption = Consumption_{free}(i.e. behind meter, from onsite production ) + Consumption_{imported}(i.e. imported from Grid)

… and then work out ImportConsumption and FreeFraction for the two options

#### Option A: “Direct”

ImportFraction_{direct}= Import / Consumption FreeFraction_{direct}= 1 - ImportFraction_{direct}...and by the way... FreeFraction_{direct}= Consumption_{free}/ Consumption FreeFraction_{direct}= (Production - Export) / Consumption

#### Option 2: “Net”

FreeFraction_{net}= Production / Consumption FreeFraction_{net}= 1 - FreeFraction

### Self-Produced Consumption versus Grid Consumption

Recall we split total consumption, C, into “Consumption from onsite production (free)” and “Consumption from imported grid energy(imported)”, or:

`(1) `**C** = C_{free} + C_{imported}

We also know that C_{free}, or “from onsite production” is the onsite Production, less anything Exported:

`(2) C`_{free} = P - E

and so we can re-arrange and substitute to find:

`C`_{imported} = C - C_{free}
C_{imported} = C - P + E

This is useful for PhiSaver as we monitor Production and Export directly, so can calculate C_{free} “free energy / onsite consumption / wadeveryoucalit ” and compare to C_{imported}

We see some sample data below. When solar is active, it’s all C_{free} (solar used on site). At night when production (integrated battery) expires at midnight, we import. The second chart just shows for a given hour (e.g. 00:00 to 01:00 and 07:00 to 08:00) we have both “produced” and “imported” consumption.

## Consumption, Unmonitored and ConsumptionNet

Imagine we’ve got a big switchboard with 40 poles. It’s impractical to monitor each circuit – we tong/guess to monitor the big users. Let’s say we cover 20 poles, with 90% of the load. Great! But what about the 10%? How do we report that?

**ConsumptionNet** is the real consumption. It’s Net + Production at the board. It’s 100% of the load.

**ConsumptionMonitored **is the sum of monitored circuits. It’s 90% of the load in our case.

**Unmonitored** is the difference (10% in our case)

Ideally we’d report this from an iotawatt, but the calculation is difficult and error prone. Instead we calculate it:

`Consumption(Net) = Net + Production`

`Umonitiored = ConsumptionNet - ConsumptionMonitored`

For convenience we do this:

- Iotawatt and Iotawatt bucket: Iotawatt Consumption is ConsumptionMonitored
- Influx (power, energy) buckets: Consumption calculated (as ConsumptionNet). ConsumptionMonitored is removed and Unmonitored takes it’s place.

One question: we can calculate **ConsumptionNet** in two ways; **are they equilivant:**

- ConsumptionNet = Sum (Net) + Sum(Production) : sum first, then add
- ConsumptionNet = Sum(Net + Production): sum each interval, then add results

And yes, I did a numerical test and it seems these are equivalent. I thought the -ve of Net would stuff up #1. But the maths…. dunno: F(x) + F(y) = F(x+y)?

## Proportions of Averages versus Averages of Proportions

TLDR> When doing proportions (solar fraction, etc) take the fraction of the averages. **That is, first take the sum or mean, then divide.** An example is below.

Time | Consumption | Import | ImportConsumptionFraction |

time1 | 0.05 | 0.00 | 0.00 |

time2 | 1.76 | 1.50 | 0.85 |

time3 | 1.66 | 1.65 | 0.99 |

time4 | 1.82 | 1.82 | 1.00 |

Average of columns: | 1.32 | 1.24 | |

0.94 | 0.71 | ||

Fraction of Average | Average of Fractions | ||

CORRECT | INCORRECT |

# Import, Export, Net

For single phase:

`Export = Net [-oo,0] * -1`

`Import = Net [0,+oo] * `

No worries, but for three phase there are two options for import/export

- Export = ((NetR min 0) + (NetW min 0) + (NetB min 0)) x -1
- Export = ((NetR + NetW + NetB) min 0)) x -1

And also, `Net = NetR + NetW + NetB`

and I`mport`

is similar to Export, just with the x -1

What? Well with #1 we treat each phase separately. Maybe Red is exporting and White is importing. For #2 we sum them together. Imagine #1 is 3 pipes measured separately, and #2 is putting the 3 pipes into 1 pipe then measuring that.

For example at a moment in time:

- Red: -10 Blue: 5 White: 20

- Net is 15. Export is -10 and import is 25. This is really happening: simultaneous import and export is real.
- Net is 15. Export is 0 and Import is 15. This is a simplification.

I don’t know how utility meters handle this. #1 or #2 or something different?

# Batteries (in progress)

The equations above assume no battery. But look what happens when I try it on some sites *without* and *with* a battery:

*Without a battery*, below: the green (“free”) and blue (“imported”) add to the dashed (total) Consumption. All good.

*With a battery,* below, “free” and “imported” don’t add to “total”. We need to add in a battery storage element. On the left at 08:00, consumption (dashed) is higher than free+import. The different must be *changing* to battery. On the right (12:30), consumption is lower than “free” and “import” is negative! This ain’t exporting (we accounted for that), it’s changing the battery.

Initially, I thought we could deduce battery charge/discharge:

`B = -N + P + L`

But, in this site’s case (and it’s common), the Production is actually Battery+Solar. That is, the battery doesn’t directly feed into our switchboard (as in our original control volume). We actually have:

Given that

, we cannot differentiate between **P = B + S**

and **B**

. We don’t have any equations between them. We actually have:**S**

**N + P – L – U = 0**

**N + B + S – L = 0** # assume U=0 and use P=B+S

Bummer. Can we be clever? Maybe a little. We know solar works in the day, so S=0 at night. Also, L>0. So we could approximate B/P breakdown. This is the intuitive thinking from the graph above.

#### This is might work if Battery is directly connected to switchboard

So let’s whack in a Battery (B) term. Let definite +ve as discharging (like ‘importing’ is +ve for Net) and -ve as charging (like ‘exporting’ is -ve for Net). It’s perhaps kinda like another Export term. We already know

`(1) `

**B = - N - P + L**

`(1) B = - N - P + `

C

ps. Loads = Consumption.

We also know `C`

(direct sum of all loads)_{total}

`(2)`** **C = C_{free} + C_{imported}

We also how that C_{produced}, or “from onsite production” is the onsite Production, less anything Exported, *and* less anything to Battery.

`(?) C`_{free} = P - E - B ???

and so we can re-arrange and substitute to find:

`C`_{imported} = C - C_{free} - B ???
= C - (P-E-B) - B
= C - P + E

Arrggghhh, don’t know. I think we have knowns: C, P, E and unknown Cfree,Cimported and B and only two equations. But going back to (1), less take B=0 to start. If C > C_free+Max(0,C_import) then B_charge = C-C_import and if C<C_free+Max(M,C_import) then B_discharge = Cfree-abs(Cimport)

Yeah, I think we can use the knowledge that C_free and C_import must be both (a) positive and (b) less then C to “make up” for the missing equation (i.e. 3 unknowns, 2 equations).

Need maths help!