12 Economic Dispatch

.title[Economic Dispatch] <br> .subtitle[BEE 4750/5750] <br> .subtitle[Environmental Systems Analysis, Fall 2022] <hr> .author[Vivek Srikrishnan] <br> .date[October 12, 2022]

---

# Outline

<hr>

1. Homework 3 Released
2. Regulatory Project Proposal Due Thursday
3. Questions?
4. Single-Period Economic Dispatch
5. Multi-Period Economic Dispatch
6. "Duck Curve"

---

# Poll

<hr>

URL: [https://pollev.com/vsrikrish](https://pollev.com/vsrikrish)

Text: **VSRIKRISH** to 22333, then message]

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***Any questions?***

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# Last Class

<hr>

* Generating Capacity Expansion Example

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# Power Systems Decision Problems

<hr>

.center[![Decision Problems for Power Systems](figures/elec-decision-problems.svg)] .center[.cite[Adapted from Perez-Arriaga, Ignacio J., Hugh Rudnick, and Michel Rivier (2009).]]

---

# Economic Dispatch

<hr>

**Decision Problem**: Given a fleet of (online) generators, how do we meet demand at lowest cost?

**New Constraints**: Power plants are generally subject to engineering constraints that we had previously neglected, including:

* Ramping limits
  * Minimum/Maximum power outputs
  * May include network constraints (we will ignore here)

---

# Simple, Single-Period Economic Dispatch

<hr>

What does this look like for a single period?

Let:

|        Variable        |                                    Meaning | Units |
|:----------------------:| ------------------------------------------:| -----:|
|          $d$           |                                     demand |    MW |
|         $y_g$          |                generation by generator $g$ |    MW |
|      $VarCost_g$       | variable generation cost for generator $g$ |  $/MW |
| $P^{\text{min/max}}_g$ |        generation limits for generator $g$ |    MW |

---

# Simple, Single-Period Economic Dispatch

<hr>

Then the economic dispatch problem becomes:

$$
\begin{alignedat}{3}
& \min\_{y_g} & \sum_g VarCost_g \times y_g \\\\
& \text{subject to:} & \\\\
& & \sum_g y_g = d \\\\
& & y_g \leq P^{\text{max}}_g \\\\
& & y_g \geq P^{\text{min}}_g
\end{alignedat}
$$

---

# Simple, Single-Period Economic Dispatch

<hr>

Let's say we have an online fleet of the following generators:

|        Type | $P^{\text{min}}$ (MW) | $P^{\text{max}}$ (MW) | $VarCost$ ($/MW) |
| -----------:| ---------------------:| ---------------------:| ----------------:|
|       Hydro |                     0 |                   150 |                0 |
|        Wind |                     0 |                   200 |                0 |
|     Nuclear |                   200 |                  1000 |                2 |
|        Coal |                   160 |                   700 |            21.50 |
| Natural Gas |                    40 |                   500 |               23 |

Also, demand in this hour is 2600 MW.

---

# Single-Period Dispatch: `JuMP` Implementation

<hr>

Let's load the generation data and demand:

```julia
generators = ["hydro", "wind", "nuclear", "coal", "natgas"]
p_min = [0, 0, 200, 160, 40]
p_max = [150, 200, 1000, 700, 500]
varom = [0, 0, 2, 21.50, 23]
demand = 1800
```

---

# Single-Period Dispatch: `JuMP` Implementation

<hr>

Setting up the optimization problem:

```julia
using JuMP
using HiGHS
using DataFrames
using Plots
gr()
using Measures
```

---

# Single-Period Dispatch: `JuMP` Implementation

<hr>

Setting up the optimization problem:

```julia
G = 1:length(generators)
dispatch = Model(HiGHS.Optimizer)
@variable(dispatch, p_min[g] <= y[g in G] <= p_max[g])
@objective(dispatch, Min, sum(varom .* y))
@constraint(dispatch, load, sum(y) == demand)
```

---

<hr>

```julia
print(dispatch)
```

```
Min 2 y[3] + 21.5 y[4] + 23 y[5]
Subject to
 load : y[1] + y[2] + y[3] + y[4] + y[5] = 1800.0
 y[1] ≥ 0.0
 y[2] ≥ 0.0
 y[3] ≥ 200.0
 y[4] ≥ 160.0
 y[5] ≥ 40.0
 y[1] ≤ 150.0
 y[2] ≤ 200.0
 y[3] ≤ 1000.0
 y[4] ≤ 700.0
 y[5] ≤ 500.0
```

---

# Single-Period Dispatch: Results

<hr>

```julia
set_silent(dispatch)
optimize!(dispatch)

gen = value.(y).data
results = DataFrame(
    "Name" => generators,
    "Dispatched Power (MW)" => gen
)
```

]

<div><div style = "float: left;"><span>5×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Name</th><th style = "text-align: left;">Dispatched Power (MW)</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">hydro</td><td style = "text-align: right;">150.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">wind</td><td style = "text-align: right;">200.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">nuclear</td><td style = "text-align: right;">1000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">coal</td><td style = "text-align: right;">410.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">natgas</td><td style = "text-align: right;">40.0</td></tr></tbody></table></div>

]

---

# Marginal Generators

<hr>

.left-column[ Once we hit the minimum for each generator, we then "fill up" capacity from lowest-cost generators first, then move on.

For this problem, the coal generator is the *marginal generator*: extra capacity is added from coal, and so this unit sets the price of electricity. ]

]

---

# Merit Order and the Dispatch Stack

<hr>

This is the concept of *merit order*: plants are scheduled to supply additional electricity based (**mostly**) on their variable costs.

```
"/private/var/folders/24/8k48jl6d249_n_qfxwsl6xvm0000gn/T/jl_Qz4oGM/build/simple-merit-order.svg"
```

---

***What might complicate this simple merit ordering based on variable costs?***

---

# Multiple Time Period Example

<hr>

Now, let's consider multiple time periods.

Not only do we need to meet demand at every time period, but we have additional **ramping** constraints: plants can only increase and decrease their output by so much from time to time, by $R_g$.

---

# Multiple Time Period Example

<hr>

$$
\begin{alignedat}{3}
& \min\_{y\_{g,t}} & \sum\_g VarCost\_g \times \sum\_t y\_{g,t} \\\\
& \text{subject to:} & \\\\
& & \sum\_g y\_{g,t} = d\_t \\\\
& & y\_{g,t} \leq P^{\text{max}}\_g \\\\
& & y\_{g,t} \geq P^{\text{min}}\_g \\\\
& & \color{red}y\_{g,t+1} - y\_{g, t} \leq R\_g \\\\
& & \color{red}y\_{g,t} - y\_{g, t+1} \leq R\_g
\end{alignedat}
$$

---

# Multiple Time Period Example

<hr>

|        Type | $P^{\text{min}}$ (MW) | $P^{\text{max}}$ (MW) | $VarCost$ ($/MW) | $R$ (MW) |
| -----------:| ---------------------:| ---------------------:| ----------------:| --------:|
|       Hydro |                     0 |                   150 |                0 |      150 |
|        Wind |                     0 |                   200 |                0 |      200 |
|     Nuclear |                   500 |                  1000 |                2 |      100 |
|        Coal |                   160 |                   700 |            21.50 |      250 |
| Natural Gas |                    40 |                   500 |               23 |      300 |

---

# Multiple Time Period Example: `JuMP` Implmenetation

<hr>

```julia
ramp = [150, 200, 100, 250, 300]
demand = [1725, 1596, 1476, 1408, 1530, 1714, 1820, 1973, 2081, 2202, 2105, 2065,
    2045, 2195, 2309, 2390, 2486, 2515, 2075, 2006, 1956, 1902, 1865, 1820]
h = length(demand)
T = 1:h

dispatch2 = Model(HiGHS.Optimizer)
@variable(dispatch2, p_min[g] <= y[g in G, t in T] <= p_max[g])
@objective(dispatch2, Min, sum(varom .* sum([y[:, t] for t in T])))
@constraint(dispatch2, load[t in T], sum(y[:, t]) == demand[t])
@constraint(dispatch2, rampup[g in G, t in 1:h-1], y[g, t+1] - y[g, t] <= ramp[g])
@constraint(dispatch2, rampdown[g in G, t in 1:h-1], y[g, t] - y[g, t+1] <= ramp[g])
set_silent(dispatch2)
optimize!(dispatch2)
```

---

# Multiple Time Period Example: Visualization

<hr>

```julia
gen = value.(y).data
p = plot(gen',
    label=permutedims(generators),
    xlabel = "Hour",
    ylabel ="Generated Electricity (MW)",
    color_palette=:tol_muted,
    thickness_scaling=1.45, linewidth=5,
    size=(700, 700), left_margin=5mm,
    bottom_margin=5mm,
    legendfontsize=14, ticksize=13,
    guidefontsize=14
)
```

```
"/private/var/folders/24/8k48jl6d249_n_qfxwsl6xvm0000gn/T/jl_Qz4oGM/build/multi-period-dispatch.svg"
```

]

---

# Multiple Time Period Example: Visualization

<hr>

```julia
areaplot(gen',
    label=permutedims(generators),
    xlabel = "Hour",
    ylabel ="Generated Electricity (MW)",
    color_palette=:tol_muted,
    size=(700, 700),
    left_margin=5mm,
    bottom_margin=5mm,
    thickness_scaling=1.45, grid=:false,
    legend = :topleft, ylim=(0, 2800),
    legendfontsize=14, ticksize=13,
    guidefontsize=14
)
plot!(demand, color=:red,
    label="demand", linestyle=:dash,
    linewidth=5)
```

```
"/private/var/folders/24/8k48jl6d249_n_qfxwsl6xvm0000gn/T/jl_Qz4oGM/build/multi-period-dispatch-stacked.svg"
```

]

---

# Duck Curve

<hr>

.left-column[Ramping and minimum generation play a major role in systems with high levels of renewable penetration.

For example, a prominent feature of California's grid, which has a lot of solar generation, is the "duck curve" (right).]

.right-column[ ![CAISO Duck Curve](figures/duck_curve.jpeg) .center[.cite[Source: [Power Magazine](https://www.powermag.com/duck-hunting-california-independent-system-operator/)]] ]