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EE 137B, MAY 2016

1

Electricity Markets, Optimal Power Flow, & LMP
Ryan Mann1
1

Energy Engineering, University of California, Berkeley

Abstract – This paper aims to summarize the core concepts underlying deregulated electricity markets, with a specific focus
on generator bids, optimal power flow calculation, and locational marginal prices. Subsequently, a simple three-bus power system
model is used to demonstrate the impact of transmission constraints on total cost and LMPs. This model is then expanded to create
a one-day simulation with hourly time resolution that demonstrates the impact of solar PV on conventional generator ramping and
nodal prices. Finally, this model is augmented by converting the PV generator to a solar thermal facility with molten salt thermal
energy storage. Optimal economic dispatch is again run, and generation schedules and LMPs are compared between the two cases.

I. I NTRODUCTION
Although the core components that make up electric power
systems have not changed significantly over the past 134 years,
models for ownership and management of these resources have
seen an important restructuring in recent decades: the creation
of deregulated markets.
Traditionally, and in regions still governed by so-called
“cost-of-service” or “rate-of-return” regulations, electric utilities owned and managed generation, transmission, and distribution equipment, and sold power to ratepayers.
However, many regions have partially eroded this regulatedmonopoly structure. State regulators required utilities to sell
off the majority of their generation assets to independent
power producers, while allowing them to retain control of
all transmission and distribution infrastructure. These Independent Power Producers would then bid into newly created
markets run by Independent System Operators or Regional
Transmission Operators. After aggregating offers from all
generators and bids from utilities (buying wholesale power
to be sold to consumers), the ISO/RTO could find the clearing
prices of the market, and the dispatch signal to be sent to each
generator.
This restructured ownership model is nicely visualized in
Figure 1 [1] below:

The decision-making process occurring within ISO/RTO
markets, and the impacts it might see as a result of an
increasing penetration of renewable generation, is the primary
subject of this report.
II. G ENERATOR B IDDING
In a perfectly competitive market, where no one entity can
affect the clearing price of electricity, it is optimal for each
generator to submit a bid equal to its marginal cost in order to
maximize profit. This concept holds true in electricity markets,
as the presence of a single clearing price offers little incentive
to bid above marginal cost for fear of not being dispatched.
“Marginal cost” refers to the cost of producing each additional unit of electricity, or dC/dQ. In the case of generators,
this cost is dominated by the cost of fuel, with other variable
costs (such as maintenance) playing a far smaller role. As a
result, renewables (which do not consume fuel) can bid into the
market at low, zero, or even negative (due to policy incentives
such as the Production Tax Credit for wind) prices.
Nuclear power is typically next in the loading order –
uranium is relatively inexpensive and highly energy-dense, so
fuel requirements are relatively low. Next are the most-efficient
gas and coal facilities, followed by the less-efficient ones. Last
in the loading order are “peaker plants”, typically burning
natural gas, diesel, or fuel oil. These plants are relatively
inefficient, and are rarely run, so they must submit high bids
to recover their costs.
This loading order can be seen in Figure 2 [2] below:

Figure 1. Deregulated Electricity Markets.
Figure 2. Generator Dispatch Order.

EE 137B, MAY 2016

However, deregulated electricity markets are not perfectly
competitive - large players can exercise market power and
manipulate the clearing price in order to maximize their profits
if not properly regulated. Most famously, this occurred in
California during the early days of deregulation in 20002001. Energy traders realized that withdrawing some of their
generators from the market (often making spurious claims of
a need for maintenance) would, when combined with high
demand from air conditioning demand during the hot summer
months, force high-priced peaker plants to be dispatched,
raising the clearing price and boosting profits for the firm’s
remaining generators.
Average monthly wholesale prices during this time period
are shown in Figure 3 [3]:

2

Figure 4. Supply and Demand Curves for a Single-Clearing-Price Market.

IV. L OCATIONAL M ARGINAL P RICES

Figure 3. Wholesale Prices During CA Energy Crisis.

III. M ATCHING D EMAND AND S UPPLY
Wholesale electricity markets are cleared by first arranging
all generator supply offers in ascending order, and all utility
demand bids in descending order. The intersection point between these two curves sets the clearing price and quantity for
the market.
All infra-marginal generators are run at full rated capacity,
and the marginal generator is run so as to satisfy the remaining
demand. All dispatched generators are paid at the clearing
price of the market; this means that the marginal generator
does not make a profit (assuming it bid marginal cost), and all
other generators’ profits are equal to the difference between
the clearing price and their marginal cost.
As seen in Figure 4 [4], higher demand for electricity (here
shown as perfectly inelastic) results in a higher clearing price
for the market.

However, the “copper-plate” model presented above significantly oversimplifies the ISO/RTO market-clearing and
dispatch process, as it ignores the effects of power flow
through the transmission network, particularly the potential
for losses and congestion.
In reality, the clearing price of electricity can vary (often
dramatically) between grid nodes; these prices are known as
Locational Marginal Prices: “The marginal cost of supplying,
at least cost, the next increment of electric demand at a specific
location (node) on the electric power network.” [5]
LMPs can be deconstructed into three components:
• The marginal cost of supplying electricity, absent any
transmission constraints.
• A congestion cost, indicating the need to run more
expensive generators than would otherwise be used.
• The cost of electricity lost as heat during transmission.
Utilities effectively pay for the >1 MWh that was generated, not the 1 MWh that arrives at the load bus.
During periods of high grid congestion (typically summer
afternoons), LMPs in areas surrounding high-priced generators
can rise far higher than those at neighboring nodes. On
the other hand, LMPs can become negative during times
when renewable generators are producing more power than
transmission lines can easily transport away from the region.
As can be seen in Figure 5 [6], these congestion-related
spikes and dips can sometimes occur at the same time, causing
prices to vary widely even among nearby transmission buses.

Figure 5. High and Negative LMPs in the East Bay.

EE 137B, MAY 2016

3

V. O PTIMAL P OWER F LOW
With prices potentially varying at every point on the transmission network, dependent on the behavior of a complex
electrical system, this problem quickly becomes too difficult
to be solved by hand using the supply-demand-curve method
listed above. Instead, ISO/RTOs make use of an optimization
problem known as the Economic Dispatch Optimal Power
Flow problem.
The objective of OPF is to minimize the total cost of supplying electricity to consumers by choosing the optimal power
output for each generator. Some of the relevant constraints
considered by system operators include:
1) Total Generation = Total Demand (+ Losses)
2) 0 MW ≤ Generator Output ≤ Rated Capacity
3) Power Flows ≤ Rated Capacity of Lines
4) Generator Ramping Constraints – power output can’t
change significantly between timesteps.
5) Security-Constrained Optimal Power Flow, or (N-1)
Contingency Analysis – the system must be able to
survive any single line or generator failure.
Figure 6 [7] shows an optimal power flow problem running
in PowerWorld.

Figure 6. Optimal Power Flow in PowerWorld.

VI. AC O PTIMAL P OWER F LOW
The most accurate of these optimal power flow formulations
is AC OPF, which uses the full AC power flow equations to
describe power flow through the transmission lines, with no
simplifying assumptions made.

min

θ,|V |,P,Q

N
X

fi (PG,i , QG,i )

(1)

i=0

subject to:
Pij = |Vi ||Vj |(Gij cosθij + Bij sinθij )

(2)

Qij = |Vi ||Vj |(Gij sinθij + Bij cosθij )

(3)

PG,i − PD,i −

N
X

Pij = 0

(4)

j=0

QG,i − QD,i −

N
X

Qij = 0

(5)

PG,i,min ≤ PG,i ≤ PG,i,max

(6)

QG,i,min ≤ QG,i ≤ QG,i,max

(7)

Vmin ≤ |Vi | ≤ Vmax

(8)

θmin ≤ θij ≤ θmax
q
0 ≤ Pij2 + Q2ij ≤ Sij,max

(9)

j=0

(10)

where:
• PG,i and QG,i are the real and reactive power generated
at each bus i.
• PD,i and QD,i are the real and reactive power consumed
at each bus i.
• Pij and Qij are real and reactive power flows between
each bus i and neighboring bus j.
• Gij and Bij are conductance and susceptance for each
edge of the transmission network.
• |Vi | is the voltage magnitude at each bus.
• θij is the voltage angle between neighboring buses.
Here, generator cost is shown in Equation (1) as a generic
nonlinear function f of both real and reactive power production. As seen later in more simplified power flow formulations, system operators typically require that cost instead be
described as a quadratic or piecewise linear function of real
power alone.
Equations (2) and (3) describe power flow using the AC
power flow equations. Conductance Gij is typically far outweighed by susceptance Bij for transmission lines, yet both
are included in this formulation.
Equations (4) and (5) impose conservation-of-energy constraints on real and reactive power at each node: power
generated minus power consumed minus power exported must
equal 0.
Equations (6) and (7) require that generators’ real and
reactive power production be no more than their maximum
rated capacity and no less than their minimum limit (often 0
MW). These constraints could also be used to impose ramping
constraints, by only allowing a set amount of deviation from
the PG,i and QG,i generated in the last OPF solution for the
previous timestep.

EE 137B, MAY 2016

4

Equations (8) and (9) ensure power quality and system
stability by maintaining bus voltage near 1 p.u., and keeping
voltage angles relatively small.
Equation (10) describes transmission line constraints –
apparent power through each line must be less than the rated
capacity of that line.
The AC Optimal Power Flow Formulation describes power
system behavior in full detail, but at a significant computational cost: the formulation is both nonlinear and nonconvex,
so it cannot be easily computed using standard optimization
solver packages. Although many researchers are discovering
novel methods of solving the AC OPF problem, such as convex
relaxation, most ISO/RTOs currently use a simpler, linearized
formulation that can be solved quickly enough to clear the realtime market and send generators optimal dispatch commands
every five minutes.

VIII. S IMPLE O PTIMAL P OWER F LOW
The following simulations make use of a further-simplified
optimal power flow formulation that allows power flows to be
calculated without finding voltage angles for each bus.
This formulation was adapted from CE 295’s Homework
4, which tasked students with solving OPF for the three-bus
system shown in Figure 7 [8] below.

VII. DC O PTIMAL P OWER F LOW
The DC OPF formulation uses the simpler DC power flow
equations, which rely on a number of simplifying assumptions
that hold relatively well for transmission systems:






Reactance far exceeds resistance, so all Gij terms can be
omitted.
Per-unit voltage magnitude is close to unity, so those
terms can be removed from the power flow equations,
and no longer appear as decision variables.
Voltage angles are small, so sinθ ≈ θ.
Reactive power can be ignored.

Figure 7. CE 295 HW 4 Three-Bus System.

The Simple OPF equations are shown below:

This results in the following set of equations:

min cA PA + cB PB + cC PC + cD PD
P

min
θ,P

N
X

subject to:
ci,1 Pi +

ci,2 Pi2

0 ≤ Pi ≤ Pi,max

(11)

i=0

Pi − D i −

subject to:
Fij = Bij (θi − θj )
Pi − Di −

N
X




Fij = 0

x12 F12 + x23 F23 + x13 F13 = 0
Fij = 0

(13)

Pi,min ≤ Pi ≤ Pi,max

(14)



θmin ≤ θij ≤ θmax

(15)



Fij,min ≤ Fij ≤ Fij,max

(16)

where:



N
X

ci,1 and ci,2 are the quadratic cost parameters for each
generator.
Pi is the real power output of each generator.
Fij is the real power flow between each bus i and
neighboring buses j.
Di is the real power demanded at each bus.

This formulation’s objective function is quadratic in the
decision variables Pi , and its constraints are linear in the
decision variables Pi and θij , so it can be quickly solved using
any quadratic programming package.

(18)
(19)

j=0

(12)

j=0



(17)

(20)

where:





ci is the marginal cost for each generator.
Pi is the real power production of each generator.
Di is the real power demanded at each bus.
Fij is power flow between nodes, in the direction shown
in the figure above.
xij is the reactance between nodes i and j.

In this formulation, the cost function (17) is linear in Pi each generator has only a single marginal cost instead of an
output-dependent cost curve. This makes the simulation results
easier to understand, and allows the optimization problem to be
solved by a linear programming solver, instead of a quadradic
programming solver.
The power flow equations have been replaced by Equation
(20), which serves as an analogue to Kirchhoff’s Voltage Law,
except with real power instead of current. (Bus voltage is
assumed to be 1 p.u., so the two are equivalent.)

EE 137B, MAY 2016

5

IX. S IMPLE OPF R ESULTS
Running the OPF solver both with and without transmission
constraints illustrates the impact that congestion can have on
the optimal optimal dispatch solution, and the LMPs at each
node. These results are documented in Table I below:
Table I
S IMPLE OPF R ESULTS
Total Cost
PA
PB
PC
PD
LM P1
LM P2
LM P3

w/o Transmission Constraints
$2,647.50
125 MW
285 MW
0 MW
0 MW
$7.50
$7.50
$7.50

w/ Transmission Constraints
$2,835.00
50 MW
285 MW
Figure 8. The CAISO “Duck Curve”.
0 MW
75 MW
$7.50
For this simulation, load data from March 31st were sourced
$11.25
$10.00
from CAISO OASIS [10], and scaled such that their peak

Without transmission constraints, Generator B (which has
the lowest marginal cost) is run at full capacity, and Generator
A (which has the second-lowest marginal cost) is run to satisfy
the remaining demand. LMP is the same at all three buses.
Adding the transmission constraints prevents Generators A
and B from fully satisfying demand. Instead, Generator D must
be run as well, despite its higher marginal cost. Total cost is
higher for this additionally-constrained optimization problem,
and LMPs differ between the three nodes.
X. M ULTI -T IMESTEP OPF WITH S OLAR G ENERATION &
R AMPING C ONSTRAINTS
This simple three-bus OPF model was then expanded on by
increasing the time horizon of the problem; instead of simply
finding the cost-minimizing dispatch command for a single
point in time (or equivalently, a single value for demand),
OPF was solved with a time horizon of 24 hours and a time
resolution of 1 hour.
The purpose of this multi-timestep OPF problem was to
illustrate the effects of high PV penetrations on conventional
generation, and observe the corresponding effects on LMPs at
each bus.
From the operator’s perspective, one of the most challenging
issues posed by PV is the steep ramp conventional generators
must perform during the evening hours, as decreases in PV
production and increases in demand occur simultaneously.
This potential problem was famously depicted by the California ISO in the form of the “duck curve” seen in Figure
8 [9], which shows projected net demand profiles (demand
minus renewable generation) in the face of increasing solar
penetration. The image specifies a March 31st load profile;
this ramping issue is most pronounced in the spring, because
solar irradiance is relatively high, yet mid-day demand from
air conditioners is not as prominent as in summer months.

value matched the demand values given in the original singletimestep OPF problem. Generator A was assigned to be the
PV generation facility, and became non-dispatchable. Instead,
its output values were given by hourly simulation outputs from
PVWatts [11] for a single-axis-tracking 110 MW PV facility
in Bakersfield, CA.
The remaining generators (B, C, and D) were assigned to be
a nuclear power plant, a simple-cycle gas peaker plant, and a
combined-cycle gas turbine respectively. Marginal cost values
for these plants came from the values listed for Diablo Canyon,
Kearney, and El Segundo 1 & 2 in the Electricity Strategy
Game generator infosheet [12], and are $11.50, $90.05, and
$44.84 respectively. (The solar PV generator has a marginal
cost of $0.)
The ramp rates of these three dispatchable generators were
set to be 0 MW/hour, 90 MW/hour, and 20 MW/hour. These
values were chosen purely for illustrative purposes. Although
their relative values matches the characteristics of their corresponding generators, their absolute values may be far below
such facilities’ actual ramping capabilities.
The formulation of this multi-timestep OPF is written as:

min
P

N
X

cA PA (k) + cB PB (k) + cC PC (k) + cD PD (k) (21)

k=0

subject to:
0 ≤ Pi (k) ≤ Pi,max
Pi (k) − Di (k) −

N
X

(22)
Fij (k) = 0

(23)

j=0

x12 F12 (k) + x23 F23 (k) + x13 F13 (k) = 0

(24)

Pi,ramp,min ≤ Pi (k) − Pi (k + 1) ≤ Pi,ramp,max

(25)

PA (k) = PP V (k)

(26)

As mentioned above, costs must be minimized for all
timesteps k, subject to constraints on the generators and
transmission lines for all hours of the day. Equation (25)
mandates that power outputs of each generator in successive
timesteps must not exceed the ramping limits described above,
and Equation (26) fixes the values of Generator A’s power
production equal to the datapoints from PVWatts.

EE 137B, MAY 2016

6

XI. S OLAR OPF R ESULTS
As seen in Figure 9, net demand closely matches the “duck
curve” shape seen in CAISO’s report.

Figure 11. Power Transmission Through Three-Bus Network with Solar PV.

Figure 9. Demand and Net Demand with Solar PV.

Generator A follows the typical hourly output pattern for a
PV facility on a clear day. Generator B, as a nuclear power
plant, is run at full capacity all day, due to its low fuel cost
and inability to ramp. Generator D, the combined-cycle gas
turbine, is forced to follow the resulting net load profile.
However, at the end of the day, as solar generation falls
off sharply and demand increases, the CCGT can’t keep up
with the steep ramp required. This forces Generator C, the
expensive peaker, to turn on for several hour of the evening.
These generation patterns are shown below in Figure 10:

This mid-day congestion due to oversupply of solar, and the
resulting need for the peaker plant to generate, is reflected in
the LMPs seen in Figure 12. The price at Bus 1, where the PV
facility is, becomes highly negative, indicating that additional
demand at that node would actually reduce system cost by
mitigating congestion. At the same time, the LMP at Bus 3
is equal to $90.05/MWh, indicating that the peaker is acting
as the marginal unit there. At the end of the day, congestion
has eased, but ramping constraints again force the peaker to
generate power, bringing the LMPs at all three buses up to
match its marginal cost.

Figure 12. LMPs at Buses 1, 2, & 3 with Solar PV.

Figure 10. PV and Conventional Generator Power Output.

Figure 11 shows the role of transmission line flow limits
in affecting the optimal dispatch schedule. Line 1-2 reaches
its 126 MW limit in the middle of the day; however, the PV
generator’s output cannot be controlled, so the peaker plant
is again forced to turn on to “push” power in the opposite
direction and meet local demand.

EE 137B, MAY 2016

7

XII. M ULTI -T IMESTEP OPF WITH S OLAR T HERMAL
(S OLAR + S TORAGE )
The simulation was then re-run after transforming the solar
generator from a PV facility to a solar thermal power plant
with molten salt thermal energy storage, similar to the one
shown in Figure 13 [13] below:

Figure 13. Diagram of Solar Thermal with Molten Salt Energy Storage.

Generator A was given an energy storage capacity equal to
ten hours of full output. This value was taken from Crescent
Dunes Solar Molten Salt Thermal Storage, which also has a
rated power of 110 MW [14].
To account for the dispatchability provided by the presence
of on-site energy storage, the formulation of the multi-timestep
optimization problem has been altered as shown below:

min
P,E

N
X

XIII. S OLAR + S TORAGE OPF R ESULTS
With storage capacity added, the net demand profile (shown
below in Figure 14) is significantly flattened.

Figure 14. Demand and Net Demand with Solar Thermal.

With PV output nearly flat at 33 MW over the majority
of the day, the CCGT only needs to follow fluctuations in
demand, without any sharp ramp down at mid-day (the “belly”
of the duck) or ramp up in the evening (the “neck” of the
duck). As a result, the peaker plant only runs for three hours
at the end of the day, and only generates a total of 7 MWh.
These results can be seen in Figure 15:

cA PA (k) + cB PB (k) + cC PC (k) + cD PD (k) (27)

k=0

subject to:
0 ≤ Pi (k) ≤ Pi,max
Pi (k) − Di (k) −

N
X

(28)
Fij (k) = 0

(29)

j=0

x12 F12 (k) + x23 F23 (k) + x13 F13 (k) = 0

(30)

Pi,ramp,min ≤ Pi (k) − Pi (k + 1) ≤ Pi,ramp,max

(31)

PP V (k) − PA (k) = E(k + 1) − E(k)

(32)

E(0) = 0

(33)

0 ≤ E(k) ≤ Emax

(34)

Instead of requiring that Generator A’s output exactly match
the output from the solar thermal system, Equation (32) now
states that the difference between the energy produced and
the energy sent to the grid be stored in the molten salt. For
the sake of simplicity, the turnaround efficiency of the storage
system was assumed to be 100%.
The initial energy level E(0) of the storage system was
set at 0 MWh in Equation (33), and its energy level E(k)
was constrained to the range {0 MWh, 110 MW·10 hours} by
Equation (34). Interestingly, the optimal solution only made
use of 55.8% of the available energy capacity, suggesting that
a smaller storage system could have been built.

Figure 15. Solar Thermal and Conventional Generator Power Output.

Transmission Line 1-2 again reaches its 126 MW capacity
limit, and remains there for the majority of the day in
conjunction with the solar thermal facility’s output. Figure 16
shows much flatter power transfer values for all three lines.

EE 137B, MAY 2016

8

XIV. S UMMARY

Figure 16.
Thermal.

Power Transmission Through Three-Bus Network with Solar

Locational marginal prices also see significant changes from
the PV-only case. LMPs never go below zero (even at Bus
1, where the solar is located), and the peaker plant is only
marginal for a few hours at the end of the day (and only at
Bus 3, where it is located). Due to congestion on Line 1-2,
LMP values still vary between the three buses for the majority
of the day, as seen in Figure 17.

The goal of this project was to synthesize some of the core
concepts behind deregulated wholesale energy markets, with
an emphasis on the optimization-driven dispatch decision process performed by ISOs and RTOs. In addition, several simple
simulations were run to demonstrate this process at work, and
to illustrate the effects that increased solar penetration could
have on grid operations and wholesale prices. This modeling
work confirmed that grid-scale storage (here taking the form of
molten salt thermal energy storage for a solar thermal facility)
demonstrates clear value in reducing the ramping requirements
placed on conventional generators, in displacing expensive
and polluting peaker plants, and in reducing system cost and
stabilizing wholesale prices.
One interesting future avenue for study would be to shift
the location of the storage resource away from the solar
generation facility, or even distribute it among all locations
on the network, to illustrate the effects of adding distributed
storage, electric vehicles, and demand flexibility co-located
with demand centers.
XV. ACKNOWLEDGEMENTS
The author would like to thank Professor Meredith Fowlie,
Professor Raja Sengupta, Professor Scott Moura, and Professor
Javad Lavaei, whose courses in Energy Markets & Regulation,
Engineering Systems Optimization, Energy Systems & Control, and Power Systems Optimization inspired and directly
contributed to the contents of this report.
R EFERENCES
[1] National Academy of Sciences, “America’s Energy Future: Technology
and Transformation”, 2009, pg. 566, Available: nap.edu
[2] PJM Learning Center, “Economic Dispatch”, Available: learn.pjm.com
[3] Chronicle Graphic/California Independent System Operator, “The
California Energy Crisis”, Available: hdnux.com
[4] Meredith Fowlie, “Electricity Strategy Game Instructions”, 2015
[5] R.Treinen/California ISO, “Locational Marginal Pricing (LMP): Basics
of Nodal Price Calculation”, 2005, Available: caiso.com
[6] California ISO, “Market Price Maps”, Available: caiso.com
[7] PowerWorld, “WebHelp Images – OPF”, Available: powerworld.com
[8] Scott Moura, “CE 295 Homework 4”, 2016

Figure 17. LMPs at Buses 1, 2, & 3 with Solar Thermal.

Adding a storage resource significantly reduces total system
cost; the original solar PV system has a total daily operating
cost of $176,595.78, whereas the solar thermal system has a
cost of $125,184.59, representing a savings of 29%.
These results illustrate a clear economic case for grid-scale
energy storage. Solar thermal technology has waned in popularity as multi-crystalline PV prices have fallen, but its ability
to easily integrate efficient large-scale thermal storage may
improve its future outlook, if capacity planning is performed
from a holistic perspective that accounts for each resource’s
effect on total system cost and operations.

[9] California ISO, “What The Duck Curve Tells Us About Managing a
Green Grid”, Available: caiso.com
[10] California ISO, “CAISO Demand Forecast”, Available: caiso.com
[11] NREL, “PVWatts Calculator”, Available: pvwatts.nrel.gov
[12] Severin Borenstein and Jim Bushnell, “Electricity Strategy Game
Generator Info”, 2016
[13] BrightSource Energy, “Solar Thermal Storage Diagram”, Available:
cnet3.cbsistatic.com
[14] DOE/Sandia National Lab, “Global Energy Storage Database - Crescent
Dunes Solar Energy Project”, Available: energystorageexchange.org


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