eugene litvinov, senior director business … marginal pricing eugene litvinov, senior director...

Locational Marginal Pricing

Eugene Litvinov, Senior Director Business Architecture & Technology

Locational Marginal Pricing – WEM 301 © 2011 ISO New England Inc. 2

What We Will Cover

• Electrical Network and Its Model

• Contingency Analysis

• Sensitivities

• LMP Calculation

• Marginal Loss Pricing

• Market System Major Components


Electrical Network and Its Model

• One-line diagram and bus/branch model

• Ohm’s law

• Losses

• Kirchhoff's law

• Power flow calculations (different model idealizations)

• Reference bus

Locational Marginal Pricing – WEM 301 © 2011 ISO New England Inc.

One-Line Diagram and Bus/Branch Model

4

G1L1

Line1 Line2

Line4Line3

Sub1Sub1

G1

L1

Line1Line2

Line3Line4

G1L1

Line1 Line2

Line4Line3

Sub1Sub1-1

G1

Line1

Line3

+ Sub1-2

L1

Line2

Line4


Network Model

• Device – any electrical device like line, transformer, breaker, etc.

• Node – connection point of two or more devices in one-line model

• Bus – connection point of two or more branches in the network model

• Branch – physical or equivalent line connecting to buses

• Injection – flow of power into bus – generation

• Withdrawal – flow of power from bus – load

• Interface – a set of branches that, when opened, split network into two separate islands


Bus/Branch Model

6

1

2

0 3

P0

P2

P1

P4

Branch 1-3Branch 0-1

Bus 1

Withdrawal

Injection


Interface

7

Interface contains lines: 0-2, 1-2, 1-3

1

2

50 MW0 3

-120 MW

30 MW

P0

P2

P1

P4

Interface

Positive Direction

The flow through an interface is the algebraic sum of the flows in the lines comprising interface:

P = 30 + 50 - 120 = -40 MW

Negative sign means flow against positive direction.


External Interface

• Interface can be between two control areas (like NE and NY). It contains inter-ties only.

CA1 CA2

Positive Direction


Ohm’s Law

9

V

RI

RIV1 V2

V

V – Voltage

I – Current

R – Resistance

The current in the circuit: I = V/R

I = (V1 - V2) / R

V = V1 - V2


Power and Losses

Power: P = Vx I

1 2

V1 V2P121

P122

I12

1 2112 1 12 1 1 2 1 1 2( ) / ( )RP V I V V V R V V V= ⋅ = − = − ⋅2 21

12 2 12 2 2 1 2 1 2( ) / ( )RP V I V V V R V V V= − ⋅ = − = − ⋅1 2

12 12P P≠1 2 2

12 12 12 12lossP P P I R+ = = ⋅


Kirchhoff's Law

11

• All flows into the bus equal all flows out. In other words, the algebraic sum of all injections and withdrawals at the bus equal 0.

• Withdrawal is positive, injection is negative. 515 MW

i

55 M

W

75 MW100 MW

35 MW

80 MW

350 MW

100 + 80 + 75 + 350 - 35 - 55 - 515 = 0


Kirchhoff's Law (cont.)

This law is also true for any closed area of the network.

100 MW

50 MW

50 MW

100 MW

200 MW

150 MW

Area 1

100 MW

100 MW 150 MW

150 MW

350 MW 150 MW

100 + 50 + 150 - 150 - 150 ______ 0

12


Kirchhoff's Law (cont.)

This law is also true for the whole control area: sum of all generation, load and inter-tie flows equals 0.

CA1 CA2

CA3

13


Powerflow Calculation

• Given: injections and withdrawals at every bus, branch parameters, network topology

• Find: power flows and currents in each branch and voltage at each bus

• High voltage electrical networks are three-phase alternate current circuits.

• The theory of power systems provides ways to perform calculations with one-line models for symmetric conditions.

• System losses is a sum of all branch losses in the system.


Bus/Branch Model in PowerWorld™

15


Simple Two-Bus System

V1 V2

I2

I1

I12

R12 1 2

2 12 2 12

2 1 2

1 12 1 12

1 1 2

1 2 12

1 ( )

01 ( )

01 ( )

I V VR

I I I I

I V VR

I I I I

I V VR

I I I

= −

− = ⇒ =

= −

− + = ⇒ =

= −

= =

1 2 2

1 2 2

1 ( )

1 ( )

V V IR

V V IR

− = − =

The equations in the system are identical – the system has infinite number of solutions.

16


Simple Two-Bus System (cont.)

• This means that we can arbitrarily choose voltage at one bus and calculate another voltage using only one of the equations.

• The bus where we specify voltage is called a Reference Bus.

• Power flow model always solves n-1 equations, where n-number of buses in the network.


AC Power Flow

• The model described above is a direct current (DC) model.

• The alternate current (AC) system is calculated using complex numbers.

• This means that any quantity is described by two components: real and imaginary or active and reactive. For example, power S=P+jQ, where S is MVA, P is active power in MW, and Q is reactive power in MVAR.


AC Power Flow (cont.)

• In addition to resistance R, each branch has reactance X. Instead of just resistance we use impedance Z = R + jX.

• Voltage has two components as well: V = V’ + jV’’.

• Both Ohm’s law and Kirchhoff's law hold true for AC case.


AC Power Flow – Phase Angle

|V|

V'

V"

δ

V

VRef

' "V jV V δ+ → ∠

Any complex number can be presented as a vector in Cartesian coordinates.

V – Voltage magnitude,

δ - Voltage phase angle.

Phase angle at the reference bus is usually set to 0, so all other phase angles use it as a reference. The flow in any branch depends only on the difference of the voltages at the ends of the branch, so no matter which bus is a reference bus, the power flow is the same.

20


AC Power Flow

• Losses in any branch also depend only on the difference of the voltages at the ends, so moving reference from one bus to another does not change system losses.

• As we saw, losses occur in each line that has a resistance – generators have to cover all the losses in the network to supply required load.

• Thus, generators have to produce more power than just required by loads to keep system in balance.

• This means that in the system with losses the algebraic sum of all injections and withdrawals must be equal to system losses – conservation of energy.


Reference, Slack and Swing Bus

• We introduced the concept of the angle reference bus as the reference for voltage vectors.

• In power flow calculations, besides the reference bus, we have to use slack or swing bus.

• Recall that when calculating power flow, one has to specify all nodal loads and generation. It is impossible to guess the total value of losses in the system before the power flows are calculated.

• Power system engineers resolve this problem by selecting a location in the network that would balance any difference between generation and load (generators have to supply losses in addition to the load)


Reference, Slack and Swing Bus (cont.)

• That location is called a slack or swing bus.

• This concept is very important in understanding sensitivities and LMP components later

• The reference bus and the slack bus do not have to be located at the same point of the network, however, in most cases, they are at the same location.

• That is why these names are being used interchangeably

• In the rest of this presentation, we will be using “Reference Bus” as a substitution for both reference and slack. Only when it is important, we will make a distinction between the two.


Slack Bus Total generation without slack: 600 + 30 + 110 + 193 = 933

24


Slack Bus (cont.)

Total generation without slack: 600 + 25 + 110 + 193 = 928

25


Slack Bus. Some Observations

• When slack bus location changes, all the flows change too

• Losses also change with the change of the location of the slack bus

• Therefore: AC Power Flow is dependent on the location of the slack bus

• The higher the imbalance between calculated and “guessed” losses is, the higher the difference

26


AC Power Flow

• Power flow equations are highly non-linear.

• There are different methods to solve power flow, the most popular one is the Newton-Raphson method.

• For real time calculations, very often we use different idealizations of the model to speed up time for solution.

• One of the most popular methods is using DC model.


DC Model

• DC model is based on the linearization of the power flow equations around certain base point to avoid iterations.

• This allows solving large series of power flows within reasonable time frame.

• This model is also being used in the economic dispatch to make it possible to use linear programming technique.


Linearization • Sinusoidal function of the flow is

replaced with the linear function.

• In the quite wide range of normal conditions, the error of linearization is reasonably small.

• When the loading grows close to the limit, the errors are getting high. This is usually far above the thermal limit of the line.

i jPij

Higher errors

Pij

δ ij

Linearized Flow

Non-Linear Flow

i iV δ∠ j JV δ∠

ij i jδ δ δ= −

sini jij ij

ij

VVP

Xδ=

29


DC Power Flow Model

• The following assumptions are made for DC idealization: – All branch resistances are equal to zero.

– All voltage magnitudes are constant. – The differences of phase angles between voltages at the ends of any

branch are within normal loading range (where the errors are not very high).

• Under these assumptions, there are no losses in the system (no resistance); active power solution can be obtained without solving simultaneously for reactive power.


DC Power Flow Model (cont.)

• For DC model, only active power injections and withdrawals are given. The result of calculation is just voltage phase angles.

• This is a system of linear equations and can be solved very quickly without iterations.

• Very often this model is used for rough estimates of the system conditions and calculating multitude of different cases in a very short period of time.


Questions

32


Contingency Analysis

• Contingency model

• Limiting elements

• Thermal limits

• Stability limits

• Contingency analysis


Contingency Analysis (cont.)

• Contingency Analysis is a process of identifying the consequences of potential component outages (contingencies) in the system.

• Contingency could be a line, transformer, breaker, generator, etc. outage or their combination.

• Each contingency is described by the set of outaged components.



• The main goal of contingency analysis is to determine conditions violating operating limits.

• These limits include: branch overloads, abnormal voltages, interfaces, and voltage angle differences.

• Contingency analysis is done both in real time and in a study mode.



• The components that could be violated are called limiting (or monitored) elements – they determine the constraints on system operating conditions.

• Transfer limits could be thermal and stability.

• A thermal limit is determined by the thermal rating of the limiting element – the maximum amount of power that can flow through the element without burning it.


Transmission Line Thermal Ratings

Normal Line Rating

Long-time Emergency

Short-time Emergency

Drastic Action Level

Must reduce loading to LTE within 15 minutes

Can operate in this range for several hours

Duration changes with season

Must reduce loading to LTE within five (5) minutes

Can operate in this range forever

37


Stability Limit

• Stability limit is determined by the consequences of dynamic (transient) processes in the system.

• Example is the stability of the synchronous generators that forces certain limitations on the power transfer due to the overload happening as a result of the short circuit at a substation.

• Stability limits are usually calculated in off-line studies that requires significant amount of time.

• There are new tools that may be used in the near future to calculate stability limits in real time.



• The following steps are performed by most of the contingency analysis tools: – Calculate base power flow (state estimator in real time).

– Check all limiting elements for violations

– Screen all the contingencies – this is a process of simulating each contingency from the given set one by one by DC model-based quick power flow analysis.

– Check each for potential violations.

– Run all suspicious contingencies through the full AC power flow analysis.

– Report violations in base case and under contingencies.


Contingency Analysis (branch R-S is open)

40


Contingency Analysis (branch P-Q is open)

41


Questions

42


Sensitivities

• Shift factors

• Loss factors


Sensitivities (cont.)

• Sensitivity is another way of linearization.

• It shows how a power flow variable (flow, voltage, phase angle, etc.) changes with the change of another value (injection, flow, etc.).

• Sensitivities are very widely being used in different industries for real-time control.


Sensitivities (cont.)

• The most-used sensitivities in electric network analysis are power transfer distribution factors (PTDF) and loss factors (LF).

• This is a fundamental security analysis tool. It can answer the questions: – How will solution change for variations in inputs?

– How must inputs be changed to control the output?


Power Flow and Linear Analysis

• How do the voltages change with increased load?

• How will branch flows change with the requested transfer?

• Which generators affect the limiting element?

• How will system losses change with the requested transfer?

Power Flow Linear Analysis

Inputs Outputs Answers

Questions


Power Transfer Distribution Factors

• PTDF determines a change in the power flow at each line when one (1) MW is transferred from one bus of the network to another.

• When one MW is transferred from one bus to another, it affects every single flow in the network!


PTDF’s. Transfer P -> T.

48


PTDF’s. Transfer P -> S.

49


Power Transfer Distribution Factors (cont.)

• In addition, depending on location of the two buses, the transfer causes different losses which are impossible to predict, so reference bus makes up for losses injecting additional MWs. That is why it is also called a slack (or swing) bus.

• This means that PTDFs are dependent on the selection of the reference bus. However, in the DC model, they are not dependent on the selection of the reference bus since there are no losses in DC network.


PTDF

ΔPij

ΔPij

i

j

Δπij

ΔPk Line k

kijα

ΔPmn

ΔPmn

m

n

kmnα

Δπmn

51


AC PTDF with Slack Bus at S

52


AC PTDF with Slack Bus at Q

53


AC PTDF with Slack Bus at P

54


Shift Factors

• Shift Factors (SF) are the PTDFs when one of the points is always a reference bus. In other words, shift factor is the sensitivity of the line flows to the change in injections at the buses.

• SF shows how the flow in the branch will change if the injection at the bus changes by one (1) MW.

• Because the reference bus always makes up for the change in the injection (to keep balance), shift factor values are dependent on the location of the reference bus. This is true even for the DC model.

• By definition, the shift factor at the reference bus equals to zero (0).

Shift Factors

1 MW

i

ΔPk Line k

1 MW

m

Xi MW Xm MW

ikSmkS

56 56


Shift Factors (cont.)

57


Linearization of the Line Flow

• Shift factors can be used to linearize flow in the line as a function of bus injections

• The shift factor will reflect the change of the line flow due to change in the injection

• In a linear model (DC model) we can use superposition to take into account the change in the line flow due to change in the injection at all buses


Linearization of the Line Flow (cont.)

• Assuming the injection change at node i being ΔPi the change in the line l flow would be:

• This linear form of the line flow will be used later in the Economic Dispatch formulation

• The constraints must be linear in order to be able to use linear programming

1

N

l li iP S P∆ = ∆∑


Loss Factors

• Loss factor (LF) is the sensitivity of system losses to a change in the injection at the bus. In other words, a loss factor at the bus shows how system losses will change if the injection at the bus is changed by one (1) MW.

• Because the reference bus always makes up for this additional MW, the values of the loss factors are dependent on the selection of the reference bus.

• Loss factors are often used in linear analysis to estimate the effect of different transfers or transactions on system losses.

• By definition, the loss factor at the reference bus equals to zero (0).


Loss Factors (cont.)

• The value DFi=1-LFi is called delivery factor.

• The delivery factor shows how much power is going to reach the reference bus if additional one (1) MW is injected at the bus i. This means that if one injects additional MW of power at the bus i, only 1-LFi MW is going to reach the reference bus, the rest is lost in the network.

• The inverse of the delivery factor is called loss penalty factor.


Examples of the Sensitivities

• Shift factor of the line i-j to the bus k SFk=20%. If we change the injection at k by 10 MW, the flow on the line i-j will increase by two (2) MW.

• Shift factor of the line i-j to the bus k SFk= -20%. If we change the injection at k by 10 MW, the flow on the line i-j will decrease by two (2) MW.

• Loss factors at the bus k LFk=2%. If we change an injection at the bus k by 10 MW, system losses will change by 0.2 MW.


Questions

63


LMP Calculation

• Commercial network model – Locations – Node, zone, hub

• Economic dispatch formulation

• Shadow prices

• Location-based marginal price

• LMP components


Commercial Network Model

• Unlike bus/branch network model that is being used in advanced network applications, the objective of the commercial network model is to provide pricing locations for trading.

• Locations provide points in the system where participants submit offers and bids, markets settle, and LMPs are calculated.

• Location is not necessarily a physical point in the electrical network model.


Locations

• Node – corresponds to a physical bus or collection of buses within the network

• Load zone – aggregation of nodes. Zonal price is the load-weighted average of the prices of all nodes in the zone

• Hub – representative selection of nodes to facilitate long term commercial energy trading. The hub price is a simple average of LMPs at all hub locations.

• External/proxy node – location that serves as a proxy for trading between ISO New England (ISO-NE) area and its neighbors


Network Model Hierarchy

One-line(Nodes)

AC Bus/Branch(Buses)

TP

DC Bus/Branch(Buses)

Linearization

AC Power Flow(Buses)

SE

Commercial(Private p-nodes)

SCADA

SCED

Commercial(Locations)

Agregation

Settlements(MD/Locations)

RQM

67


NEPOOL Control Area and Pricing Hub

Hub

68


Characteristics of a Hub

• Prices at the trading hub should move with prices in the target region.

• There should be very little intra-hub congestion.

• It should not be possible for the hub to be lost from service or disconnected from the rest of the system.

• Reasonable patterns of congestion should not cause the hub price to substantially diverge from the prices in the region.


Location-based Marginal Price (LMP)

• LMP is a cost of optimally supplying an increment of load at a particular location while satisfying all operational constraints.

• One can think of the LMP as a change of the total production cost to deliver additional increment of load to the location.

• LMPs are usually produced as a result of economic dispatch.

• LMPs can be calculated looking ahead – ex-ante LMPs, or after the fact – ex post LMPs. Ex-ante LMPs for generation locations are also called nodal dispatch rates (NDR).


Power System Normal Operation Control

• One of the most important power system control objectives is to keep the balance in the system.

• At any moment, the sum of all generation must meet all loads, losses and scheduled net interchange.

• There are three processes that achieve this goal under normal operations: automatic generation control (AGC), load following, and optimal/economic dispatch.


Load Following vs. AGC

Time

Load

Load Following

AGC

72


Balancing the System

• AGC is a fully automatic system that responds to comparatively small fluctuations of the load. Cycle – 4 sec.

• Load following is the look ahead process of making sure that the ISO has enough capacity online to meet the load. Due to characteristics of the units, they cannot instantaneously respond to the ISO instructions – they are limited by their response rates. If the load grows too fast, the operator may not have enough time to follow the load, so some units have to be started/committed in advance enough to be able to provide needed dispatch range. This process is also called resource adequacy analysis (RAA) in the ISO-NE.

• Response rate is the maximum speed at which the unit can move. It is measured in MW/min.


Economic Dispatch

• It is the least expensive way of supplying load in the system.

• Dispatching generators means changing their output to keep the system in balance.

• Economic dispatch is part of the load following control and is being run every five (5) min in ISO-NE to re-optimize the generation to meet load at minimum cost.

• The result of the dispatch is unit output levels – desired dispatch points (DDP) in MW and LMPs at each generator node – nodal dispatch rates.


Economic Dispatch Formulation

• As any optimization problem, economic dispatch is formulated by specifying objective function and a set of constraints.

• In case of economic dispatch, the objective function is the total cost of producing electricity that has to be minimized.

• Each unit submits offer that specifies the incremental cost of producing energy.


Economic Dispatch Formulation (cont.)

• In general, this problem is non-linear and has to be solved by using OPF – optimal power flow algorithm, but OPF software is not robust and quick enough to be used in real-time processes, so linearized version, utilizing linear programming (LP) technique, is used.



1

1 1

max

1min max

,

. .

0,

, 1,2,..., ,

, 1,2,..., ,

N

i gii

N L

gi lji j

N

ki gi k

gi gi gi

Min C P

S T

P P Loss

S P T k K

P P P i N

=

= =

⋅

− − =

⋅ ≤ =

≤ ≤ =

∑

∑ ∑

∑

Objective function – total cost

System Balance

Transmission Constraints

Capacity Constraints

Where kiS is a shift factor of branch k to the generator i, iC is an incremental price of energy at the generator i.

77



• Unlike other constraints, the first constraint is an equality. It means that the balance in the system must be maintained at all times.

• Any optimal solution must satisfy this condition.

• All other constraints are limits on branch or interface flows that reflect reliability criteria in the system.

• This is the simplest form of presenting the formulation – in real life it looks significantly more complicated.


Economic Dispatch Solution

• As a result of solving LP, the dispatch algorithm determines desired dispatch points for every dispatchable generator.

• These values are called primal variables. The prices are obtained using dual variables – shadow prices.

• Each constraint has a corresponding shadow price, even the system balance one.

• Each shadow price reflects the effect of relaxing corresponding constraint by one (1) unit on the value of the objective function, which means the change of total cost.


Economic Dispatch Solution (cont.)

• The shadow price λ of the system balance constraint is one for the whole system (only one system balance equation).

• Each transmission constraint k has its own shadow price μk.

• Some constraints may be binding.

• Binding constraint is the constraint that turns into equality for the optimal solution. For example, a particular branch has to be operated at its limit.



• The shadow price of the binding constraint is non-zero, while the shadow price of the constraint that does not bind is zero (0).

• The system balance constraint always binds, so its shadow price is never zero. This means that there is always a price to support system balance.

• If there are no binding transmission constraints, there is no congestion in the system.


150

100 70

180

250

250

1 2 250P P+ =

1 180P ≤

0

2 200P ≤

P2 [MW] @$10/MWh

P1 [MW] @$5/MWh

Non-binding Constraint

Binding Constraint $1600

Load: P1 + P2 = 250 MW Cost: 5 x P1 + 10 x P2

$1750 $1675

82 82



83




84



Optimal Solution with Non-Binding Constraint

85



One of the Constraints Binds

86



Optimal Solution with Binding Constraint

Binding Constraint

87



Binding Constraint

88



Binding Constraint – Mechanical Analogy

Non binding constraint Binding constraint

89


Economic Dispatch – Two Unit Example

1 2

P1

P2

PL

P12

Transfer Limit = 150 MW

P1max = 250 MW

C1 = $5/MWhSF1 = 1

P2max = 250 MW

C1 = $10/MWhSF2 = 0

1 1 2 2 1 2: * * 5* 10* minTotalCost C P C P P P+ = + −s.t.

1 2 LP P P+ = - System Balance

1 2 11* 0* 150P P P+ = ≤ MW - Generic Constraint

1 250P ≤ ,2 250P ≤ - Capacity Constraints

90

Economic Dispatch – Two Unit Example (cont.)

50

100

150

200

250

0 250 50

1 150P ≤

1 250P ≤

2 250P ≤

P1

P2

Total Cost = 5 x 150 + 10 x 50 = $1250

Total Cost = 5 x 200 = $1000

91 91


Economic Dispatch – Two Unit Example (cont.)

P1=150 MW 200 MW

150 MW

5 $/MWh 10 $/MWh

P2=50 MW

• Let Transfer Limit increase by 1 MW. We can load Gen 1 up to 151 MW. We will need only 49 MW from Gen 2. Total Cost in this case will be (5 x 151) + (10 x 49) = $1245.

• The change in cost μ = $1245 - $1250 = -$5 is a shadow price of the transmission constraint.

Total Cost: (5 x 150) + (10 x 50) = $1250

92


LMP Calculation

• LMP at any location is calculated based on the shadow prices out of LP solution.

• The following fundamental formula is used to calculate LMPs. For any node i:

1

,K

i i ik kk

LF Sλ λ λ µ=

= − ⋅ + ⋅∑

where λ is a shadow price of the system balance constraint.

93


LMP Calculation (cont.)

• While dispatched, all units will end up in one of three groups: – At the maximum limit

– At the minimum limit – Between minimum and maximum

• The maximum and minimum can be ramp rate constrained limit, regulation limit, etc.

• The third group of units is called Marginal Units – these are the units that determine LMPs at ALL locations.


LMP Calculation – Fundamental Properties

• The price at the location of each marginal unit is always equal to its offer price.

• n+1 Rule: for n binding constraints, there is at least n+1 marginal units. This does not include equality constraint.

• In the case of no congestion, there is only one marginal unit.


LMP Calculation – Fundamental Properties (cont.)

• Any increment of load at a particular location will be delivered from the marginal units.

• An LMP at any location will be a linear combination of the LMPs (offer prices) at marginal locations.

• If there is no congestion (all μk equal to zero) and no losses, the LMP will be the same at each location.

• LMPs at some locations can be higher than the highest offer price.

• Opening a branch can lower LMPs.

• LMP can be negative at some locations.


Base Case

97


The Price Can be Higher than the Highest Bid

98


The Difference Case

99


Prices Before Opening the Line

100


After Opening the Line

101


The Difference

102


LMP Components

Energy

Congestion

Losses

Each LMP can be split into three components.

103


LMP Components (cont.)

1

,K

i kik

i kSLF µλλ λ=

= − ⋅⋅ +∑

Congestion Component

Loss Component

Energy Component

• The energy component is the same for all locations and equals to the system balance shadow price.

• Congestion components equal zero for all locations if there are no binding constraints – all μk=0.

• The loss component is the marginal cost of additional losses caused by supplying an increment of load at the location.

104


LMP Components (cont.)

• At the reference bus, loss factor LFref = 0 and all shift factors Sref = 0.

• This means that both loss and congestion components are always zero at the reference bus.

• As the result, the price at the reference bus always equals to the energy component: λref=λ.


LMP and the Reference Bus

• LMPs will not change if we move the reference bus from one location to another.

• However, all three components are dependent on the selection of the reference bus (due to the dependency of the sensitivities on the location of the reference bus).


LMP Components

• The dependency of components on the selection of the reference bus proves that the value of each component by itself does not mean much – only the differences have a meaning and are not dependent on the selection of the reference bus.

• The only reason we need LMP components is the need to use them for FTRs and split congestion cost from energy.


Two Unit Example – LMP Components

150 MW250 MW

150 MW Ref

1 $5/ MWhλ = 2 $10 / MWhλ =

100 MW

1 1SF =

2 0SF =

1 1 10 1 ( 5) $5/c MWhλ λ λ= + = + ⋅ − =

$5µ = −

2 2 10 0 ( 5) $10 /c MWhλ λ λ= + = + ⋅ − =

1 $5/c MWhλ = − 2 $0 /c MWhλ = $10 / MWhλ =

150 MW250 MW

150 MWRef

1 $5/ MWhλ = 2 $10 / MWhλ =

100 MW

1 0SF =

2 1SF = −

1 1 5 0 ( 5) $5/c MWhλ λ λ= + = + ⋅ − =

$5µ = −

2 2 5 ( 1) ( 5) $10 /c MWhλ λ λ= + = + − ⋅ − =

1 $0 /c MWhλ = 2 $5/c MWhλ = $5/ MWhλ =

108


LMP Components

1

1 ) ,(K

i kk

ii kSLF µλλ=

− ⋅= +∑

Congestion Component

Delivered EnergyComponent

• Grouping energy and loss components together can be considered as one component – delivered energy component.

• This component is the marginal price of delivering an increment of load from the reference bus.

• In fact, the settlement process never needs energy and loss components separately.

109


LMP Components – Settlement

System wide, generators are being paid: ( ) .i i i ik k i

i i kP LF S Pλ λ λ µ− ⋅ = − − ⋅ + ⋅ ⋅∑ ∑ ∑

System wide, loads pay: ( ) .i i i ik k i

i i kL LF S Lλ λ λ µ⋅ = − ⋅ + ⋅ ⋅∑ ∑ ∑

Total revenue:

maxarg

( ) ( ) ( )

.

i i i i i k i i iki i k i

m k kk

P L LF P L P L S

Loss Loss T

λ λ µ

λ λ µ

− ⋅ − + ⋅ − − − ⋅ =

= − ⋅ + ⋅ −

∑ ∑ ∑ ∑

∑

110


LMP Components – Settlement (cont.)

Loss revenue:

argRe ( ).mLoss v Loss Lossλ= ⋅ −

Congestion revenue: maxRe .k k

kCong v Tµ= −∑

Loss is a value based on the Revenue Quality Metering.

111


LMP Components – Settlement (cont.)

• Both Loss and Congestion revenue depends on the selection of the reference bus. When moving the reference, total revenue stays the same, however the split into congestion and loss fund changes.

• This requires correct and consistent modeling of losses in the system.

• Under this condition, the dispatch and, therefore, μk will be the same.

• This also means that no matter where you are located, both payment and credit will stay the same.


Revenue with the Slack at Bus T

113


Revenue with the Slack at Bus P

114


Revenue with the Slack at Bus S

115


Transmission Losses

• Transmission losses cause the power flow in the beginning of a transmission line to be different than the flow at the end.

• This is due to the dissipation of energy in the wires.

• In order to supply energy to the load, the generator has to supply more to cover load and transmission losses.

U1 U2

PL

PG

R

PG PL

PG = PL+ Loss


Transmission Losses (cont.)

• Based on the Ohm’s law, the losses in the line A-B, π, are approximately proportional to the square of the power flow through the line:

where a is a coefficient that depends on the voltage and resistance of the transmission line.

2 ,AB ABa Pπ ≈ ⋅


Transmission Losses – Two Bus Example

1 2110 MW

110 MW100 MW

100 MW

• Let a = .001. Then the losses in the line 1-2 will be 0.001 x 1002 = 10 MW.

• Generator has to generate 110 MW in order to supply 100 MW of load.

• 10 MW are physical system losses.

118


Average Losses

• Average losses can be defined as the amount of losses per MW of transfer (power flow in the line):

• For the above example, average losses will be 0.001 x 100 = 0.1 MW.

• This means that every MW of load will cause an average 0.1 MW of losses.

./avAB AB AB ABP a Pπ π= = ⋅


Marginal Losses

• Marginal losses are the rate system losses change with the change of flow. This is being expressed as a derivative:

• For the two bus example: π = 2 x 0.001 x 100 = 0.2

2 .m ABAB AB

AB

a PPππ ∂

= = ⋅∂


Average vs. Marginal Losses

• Comparing average and marginal losses, one can see that marginal losses are about twice as much as average.

• These two quantities describe different properties of the system: marginal effect on losses of increasing transmission loading vs. average amount of losses per MW of flow.

• Note that both average and marginal losses are dependent on the state of the system – flow in the line A-B.


Two Bus Example – Economic Dispatch

1 2

100 MW

110 MW100 MW


P1max = 250 MW

C1 = $20/MWh

P2max = 250 MW

C1 = $30/MWh

0 MW

• This example is very easy to optimize.

• Since the flow in the line 1-2 is not higher than the transfer limit, all load can be supplied by the least expensive generator, so the system will be dispatched as shown above: generator 1 will be loaded up to 110 MW (supplying load and losses), and generator 2 will stay at zero.

• Total cost of production: 110 x 20 = $2200

122


Two Bus Example – LMP

• Location-based Marginal Price (LMP) at a location is defined as a change in the total cost of production due to increment of load at this location.

• We can figure out the values of the LMPs in the two bus example by adding one MW of load at each bus and determining the corresponding change in the total production cost.


Two Bus Example – LMP at Bus 1

• If one (1) MW of load was added at bus 1, there would be no need to transfer any additional energy over transmission line – this one (1) MW can be supplied by the cheapest generator 1 at the price of $20/MWh.

• Additional cost of producing one (1) MW will be $20 – total cost will be $2220.

• The change in total cost of production will also be $20, so the LMP at bus 1 will be $20/MWh.


Two Bus Example – LMP at Bus 2

1 2

101 MW

111.2 MW101 MW


C1 = $20/MWh

C2 = $30/MWh

0 MW

• Let us add 1 MW of load at bus 2.

• The flow in the line will become 101 MW, so the losses will be 0.001 x 1012 = 10.2 MW.

• Generator 1 will have to produce 101 + 10.2 = 111.2 MW.

• Total cost will be 111.2 x 20 = $2224.

• LMP at bus 2 will be equal to the change in total cost: 2224 - 2200 = $24/MWh.

$20/MWh $24/MWh

125


Two Bus Example – LMP

• As we can see, even in the non-congested case, LMPs are different at different locations.

• This is due to the marginal effect of losses – an increment of load at any location causes additional losses that require more energy to be produced by the generators.

• Note that we have not mentioned any LMP components so far.

• Have we used loss factors?

• Have we used marginal loses?

• The market can be settled using calculated LMPs.


Two Bus Example – Settlement

• For the last example, generator will be credited: 20 x 110 = $2200.

• The load will pay 24 x 100 = $2400.

• ISO will be left with $200 surplus.

• The surplus comes from the marginal pricing – this is inherent to location based marginal pricing.


Marginal Pricing

• Loads implicitly pay for physical losses just due to the fact that generation is higher than load by the amount of physical losses, so this is coming out of the surplus.

• The surplus of money has nothing to do with payment for losses – it is the result of marginal pricing.

• If we accept the principles of marginal pricing, the surplus is inevitable, both with respect to congestion and losses.


Who Paid Extra Money?

• Looking at the LMPs, it is impossible to tell which part is payment for marginal losses and which is payment for energy – LMP is the price of energy at a location.

• As soon as we want to split LMPs into components in order to separate “energy” and marginal loss money, we have to arbitrarily define a reference (slack) bus.

• Depending on the location of the reference bus, the values of “energy” revenue and marginal losses revenue will change, even though the total will not.


Reference Bus

• Energy component of the LMP is a price of energy at the reference bus – it is the same for all locations in the system.

• Moving reference bus from one location to another will preserve all the LMPs.

• This means that the energy component will change and marginal loss component will have to change as well to preserve the value of LMP at each location.

• Loss factors are used to split LMP into energy and marginal loss components.


Loss Factors

• Loss factor is a sensitivity of system losses to the change in injection at a location.

• There are as many loss factors as locations in the network.

• The values of loss factors are dependent on the location of the slack bus because it is the slack bus that has to balance the increment of injection by consuming additional increment of load to keep the system balance.


Loss Factors – Example

1 2

100 MW

110 MW100 MW


0 MWRef

LF2=0LF1=0.167

• With the reference bus at bus 2, the loss factors will be 0 at 2, and 0.167 at 1.

• Note that loss factor at the reference bus is always zero – there is no change in power flow (and, therefore, transmission losses) if we balance the increment of injection at the same bus.

• The change in injection at bus 1 will cause the increase in the transmission flow and, therefore, system losses.

132


Loss Factors – Example (cont.)

1 2

100 MW

110 MW100 MW


0 MWRef

LF2=-0.2LF1=0

• With the reference bus at bus 1, the loss factors will be 0 at 1, and –0.2 at 2.

• The change in injection at bus 2 will decrease the flow in the transmission line and, therefore, system losses – this is why the value of the loss factor is negative.

• It can be interpreted this way: if 1 MW of generation is added at bus 2, the system losses will drop by 0.2 MW.

133


LMP Components

• In the system without congestion, an LMP can be split into two components: energy component and marginal loss component.

• As we discussed, the term “energy component” is not very reflective of the meaning – it is the price of energy at the reference bus.

LMP at bus i: .Li i iLFλ λ λ λ λ= − ⋅ = +

Loss component at bus i: .Li iLFλ λ= − ⋅

134


LMP Components – Two Bus Example

1. Reference Bus 2 • Energy component: $24/MWh

• LMP at 2: $24 + $0 = $24/MWh

• LMP at 1: $24 - $0.167 x 24 = $24 - $4 = $20/MWh

Was generator penalized for marginal losses?

2. Reference Bus 1 • Energy component: $20/MWh

• LMP at 2: $20 – (-0.2) x 20 = $20 + $4 = $24/MWh

• LMP at 1: $20 + $0 = $20/MWh

Did load pay for marginal losses?

Note that in both cases load paid the same amount and generator was credited the same amount.


LMP with Congestion and Losses

1 2

100 MW

110 MW120 MW


C1 = $20/MWh

C2 = $30/MWh

20 MW

• The binding transmission constraint causes price separation. Both generators become marginal and set the prices at their respective locations.

• The price at 1 will be $20/MWh.

• The price at 2 will be $30/MWh.

$20/MWh $30/MWh

136


LMP with Congestion – Settlement

• G1 is being credited: 110 x 20 = $2200.

• G2 is being credited: 20 x 30 = $600.

• The load is charged: 120 x 30= $3600.


LMP with Congestion – LMP Components

• Let us assume that the reference bus is at 2.

• Then the energy component of the price will be equal to the price at 2: $30/MWh.

• The loss and congestion components are both equal to zero at this location.

• The loss component at bus 1 will be: -0.167 x 30 = -$5/MWh.

• Then the congestion component will be: 20 + 5 – 30 = -$5/MWh.

• So the price at 1 can be decomposed as follows: $20/MWh = $30/MWh - $5/MWh - $5/MWh.


Modeling Losses in the Lossless System

• LP methodology uses DC network model to calculate LMPs.

• In the DC model, there are no losses in the transmission lines, but sum of all generation is greater than sum of all loads by the amount of losses.

• This brings up an issue: if there are no losses in the network, where to put losses to keep the balance?


Losses in the DC Model

• In a traditional approach, slack bus always makes up for losses, which means that all system losses are withdrawn at one bus.

• This may significantly distort the power flow in the network and, as a result, change LMPs.

• The slack bus that has been selected as a reference for shift factors determines the location of the losses in the network.


Two Bus LP Formulation

1 2

P1

P2

P12


L1=50MW L2=250MW$5/MWh

$10/MWh

LP1:

1 2 5 10Min P P+S.T.

1 2 1 2 0;P P L L Loss+ − − − =

1 1 1 2 2 2( ) ( );Loss lf P L lf P L= − + −

1 1 1 2 2 2( ) ( ) 150.sf P L sf P L− + − ≤

141


Two Bus LP Formulation with Distributed Losses

LP2:

1 2 5 10Min P P+S.T.

1 2 1 2 0;P P L L Loss+ − − − =

1 1 1 2 2 2( ) ( );Loss lf P L lf P L= − + −

1 1 1 1 2 2 2 2( ) ( ) 150.sf P L d Loss sf P L d Loss− − ⋅ + − − ⋅ ≤

142


Distributed Slack

• The reference for shift and loss factors does not have to be located at a particular physical bus.

• If we select a distributed slack, we assign participating factors for each bus to cover imbalance in the system.


Distributed Slack – Example

1 MW

i

ΔPk Line k

x

1-x

ikS

144


Distributed Slack – Example (cont.)

• Let us use distributed bus with the load-weighted participating factors.

• Let us distribute losses among all load buses as well.


Distributed Slack

Loss distribution factors:

1

2

50 1 ;50 250 6

250 5 .50 250 6

d

d

= =+

= =+

To convert loss sensitivities to refer to a distributed slack:

1

2

0.2 (1/ 6 0.2 5/ 6 0) 0.17241;1 (1/ 6 0.2 5/ 6 0)

0 (1/ 6 0.2 5/ 6 0) 0.03448.1 (1/ 6 0.2 5/ 6 0)

lf

lf

− ⋅ + ⋅= =

− ⋅ + ⋅− ⋅ + ⋅

= = −− ⋅ + ⋅

146


Distributed Slack (cont.)

To convert shift factors to refer to the same distributed slack:

1

2

0 1/ 6 0 5/ 6 ( 1) 5/ 6;1 1/ 6 0 5/ 6 ( 1) 1/ 6.

sfsf

= − ⋅ − ⋅ − == − − ⋅ − ⋅ − = −

The LP2 solution will be:

Case 5 SF @D LF @ D Generation Load LMP LMP_Energy LMP_Loss LMP_CongestionBus 1 0.8333 0.1724 205.1724 50 5 9.16666665 -1.580425 -2.586206916Bus 2 -0.167 -0.034 125.8621 250 10 9.16666665 0.3160667 0.517241383

Loss 31.034Lambda 9.1667Tau 9.1667Mu -3.103

147


Distributed Slack (cont.)

The results can be presented as follows:

126 MW

1 2

150 MW


L1=50 MW L2=250 MW

205MW

5 MW 26MW

Loss=31 MW

Distributed Slack

$5/MWh $10/MWh

$9.17/MWh

148


LMP with Distributed Slack

When distributed slack participating factors are selected the same as loss distribution factors:

1 1 2 2 .d dλ λ λ⋅ + ⋅ =

This means that the energy component will be the weighted average of all load locational prices.

For the two bus example:

1/ 6 5 5/ 6 10 $9.1667 / .MWhλ = ⋅ + ⋅ =

149


5 Bus Model

ETransfer Limit = 240 MW

300 MW

A

B C

D

Brighton

Alta

Park City

Solitude

Sundance

XED=2.97%

XAD=3.04%

XCD=2.97%XBC=1.08%XAB=2.81%

XEA =0.64%

300 MW

400 MW$20/MWh

$14/MWh$15/MWh

$30/MWh

$40/MWh

EcoMax=110EcoMax=100

EcoMax=500

EcoMax=200

EcoMax=600

150

5 Bus Model: Distributed Slack

Alta ParkCity Solitude Sundance Brighton LossGeneration 110.00 100.00 348.59 0.00 463.31 21.91

Bus Name Bus NoBus Gen

Bus Load

Bus Loss

Net Injectio D W Lf@W SF@W LMP LMPE

LMP Loss LMPC

A 1.00 210.00 0.00 0.00 210.00 0.00 0.00 0.0588 0.2554 23.07 31.12 -1.83 -6.22B 2.00 0.00 300.00 6.57 -306.57 0.30 0.30 -0.0002 0.1045 28.58 31.12 0.01 -2.55C 3.00 348.59 300.00 6.57 42.02 0.30 0.30 -0.0002 0.0464 30.00 31.12 0.01 -1.13D 4.00 0.00 400.00 8.76 -408.76 0.40 0.40 0.0003 -0.1131 33.87 31.12 -0.01 2.76E 5.00 463.31 0.00 0.00 463.31 0.00 0.00 0.0698 0.3674 20.00 31.12 -2.17 -8.95

LHS RHSShadow

PriceEnergy Balance 1021.91 1021.91 31.12Loss Balance 21.91 21.91 31.12Constraint ED 240.00 240.00 -24.36

obj 22764.13

Distributed slack with the following weights is selected: WB=0.3, WC=0.3, WD=0.4

151 151


5 Bus Model: Distributed Slack (cont.)

ETransfer Limit = 240 MW

300 MW

A

B C

D

Brighton

AltaPark City

Solitude

Sundance

240 MW

187 MW

19 MW61 MW246 MW

223 MW

300 MW

400 MW

$20/MWh

$23.07/MWh$28.58/MWh $30/MWh

$33.87/MWh

463 MW

349 MW

110 MW100 MW

0 MW

23 MW

9 MW

7 MW 7 M

W

DistributedSlack $31.12/MWh

0.3 28.58 0.3 30 0.4 33.87 $31.12 / MWhλ = ⋅ + ⋅ + ⋅ =152


Loss Model with Loss Distribution

• With the appropriate and consistent loss distribution, the selection of the slack bus/market reference is not important.

• Under this design, with the change of the slack bus, LMPs do not change. Moreover, the congestion component and the sum of energy and loss components stay the same.

• Loss component of the LMP is never used by itself in settlements. Even for analysis, only differences of components between locations make sense to look at.


External Transactions

• External transaction (ET) is a purchase by a participant of energy external to the control area or a sale of energy by a Participant that is external to the control area in the day-ahead energy market and/or real-time energy market or a through transaction scheduled by a non-Participant in the real-time energy market.

• Each ET is associated with two locations (not necessarily different).

• ET always uses proxy node as one of the locations.


External Transactions (cont.)

• Types: – Fixed and Dispatchable – Up-to congestion (in DAM only)

• Direction: – Import and export – Wheel-through (in RTM only)

• ETs imports and exports look to SPD just like generation or load respectively.


Proxy Node

• Proxy node is a node outside of the control area that provides a proxy pricing location for the entities willing to perform trades between different markets or control areas.

• The proxy node reflects the price of bringing in, moving energy out or through control area.

• The price at this node only reflects constraints internal to the control area of the market.


Proxy Node (cont.)

NeigboringCA

NetworkModel

Control Area

ExternalPricing

Location

Border

PT

157


LMP at the Proxy Node

• The LMP at the proxy node is calculated the same way as any other internal to the market location.

• The only difference is the calculation of the loss component – losses in the parts of tie lines outside of the control area must be excluded.

• This is achieved by modifying loss factors at the proxy nodes so that they do not include the effect of losses in the neighboring area.


Seams Issues with External Node Pricing

• Scheduling ETs and calculating prices at the proxy bus is one of the major sources of seams issues.

• While calculating price at external proxy node, SPD does not take into account offer and bid prices and constraints in the neighboring markets.

• This produces inconsistent prices across the border.


Seams Issues with External Node Pricing (cont.)

• The resolution to that seams issue could be: – Large RTO with a single dispatch, network model, and market

– Coordinated Markets, where LMPs are coordinated – Super RTO clearing inter-control area transactions

• There are different possible approaches to coordinate prices – from very simple approximations to a rigorous decomposition.


Questions

161


Market System Architecture and Major Components

• Unit Commitment (UC)

• Simultaneous feasibility test

• Security constrained UC

• Security constrained economic dispatch

• LMP calculator

• State estimator

• Settlements


Major Business Streams in the ISO

163


SMD System Architecture

Settlements

Operations

Publishing

MarketAssessment

andPerformanceMonitoring

Billing

Accounting

FTRs

MUIMUI

164


Market Operations

Day Ahead Scheduling

Hour AheadScheduling

ResourceAdequacy

Assessment

Real TimeScheduling& Dispatch

Unit Schedules

Supply Offers and Demand Bids

Unit Schedules

HourlyTargetsArchiving

(AnalysisSystem)

MOI

MOI

MO

I

MO

I

5-minTargets

165


Operations. Major Components

SCED

RT LMP Calculator

Real Time

EMS


State Estimator

SCADANetwork Model

SCUC

FTR

Unit Commitment

SCED

SFT

FTR - Financial TransmissionRights

SCED - Securuity ConstrainedEconomic Dispatch

SFT - Simultaneous FeasibilityTest

SCUC - Security ConstrainedUnit Commitment

EMS - Energy ManagementSystem

166


Unit Commitment

• Unit Commitment is the process of optimizing the total production cost over comparatively long period of time, for example 24 hours.

• The result of this process is units’ start/stop schedules.

• Unlike economic dispatch, this problem cannot be solved by one LP solution, even in the linearized form. In general, UC is a mixed integer programming problem.


Simultaneous Feasibility Test (SFT)

• SFT, in effect, is a contingency analysis process as described earlier.

• The objective of SFT is to determine violations in all post-contingency states and produce generic constraints to feed into economic dispatch or FTR auction.

• Generic constraint is a transmission constraint that is linearized for the unit outputs using shift factors in post contingent states.


Security Constrained Unit Commitment (SCUC)

• SCUC is a combination of UC, economic dispatch and SFT.

• For each hourly schedule produced by UC, economic dispatch determines unit dispatch points to meet hourly load.

• Each set of dispatch points is then tested by SFT for violations.

• If any violations are found, new generic constraints are generated and are fed into dispatch or UC.

• This process ensures that the set of schedules produced by UC satisfies reliability criteria.


Day-Ahead Scheduling and Pricing

OutageScheduler

Constraint Logger

DAPublishing

Unit SchedulesLMPs

Binding Constraints

Unit SchedulesExternal Transaction Schedules

Supply Offers Demand Bids

SecurityAnalysis

UC

SPD

Daily

S chedulesH

ourly Schedules

Generic

Constraints

Generic C

onstrraints


Virtual Offers & Bids

SCUC

170


Security Constrained Economic Dispatch (SCED)

• SCED is an economic dispatch combined with the contingency analysis.

• CA runs periodically finding violations in post- contingent states.

• If there are any violations, new generic constraints are generated. Operator can use constraint logger to activate (make available to the UDS) any constraints.

• SCED is the process used in the UDS software to produce desired dispatch points and nodal dispatch rates in real time.

• In the Day-Ahead Market, it is used to produce LMPs.


Real-Time Dispatch

VSTLF

AOL

UDS

CD-SPD

ED DEEMS

Unit AvailabilityReserve & AGC Requirements

Dispatch

LMPC Publishing

Unit Offers

SE

Security Constraints

TTCs

5-min Load Forecast

Hourly Targetsfrom Hour Ahead

DDPNDR

Unit Data

NDRConstraints

DDPNDR

Unit Data

LMPs

Uni

t Out

put

172


State Estimator

• State estimator (SE) is a component of the energy management system (EMS) that calculates the current state of the system based on raw telemetry from SCADA.

• SE is very similar to power flow except it finds the solution closest to the metered state.

• SE runs every three (3) min and produces bus/branch model (topology processing) and solves power flow in the system.


Ex Post LMP Calculation

• The main idea of ex-post LMP calculation is to determine LMPs that accurately reflect physical operation of the system.

• The LMP calculation is formulated as an incremental optimization problem around the operating point.

• Comparatively narrow limits are placed on unit outputs to keep system in the close neighborhood of the current state.

• Based on the performance, each unit and external transaction is evaluated for eligibility to set the price.


Ex Post LMP Calculation (cont.)

• If unit is not following dispatch instructions, it would not be able to set the price; moreover, it will be paid only for the amount, no more than 10% over its DDP.

• The resources that are eligible to set the price are called flexible resources, so if the unit is found inflexible and was marginal ex-ante, it will not be able to set the price ex post. This test is impossible under ex-ante approach.

• Active constraints honored by LMP calculator must be the same constraints that were used by operator to dispatch the system.


LMP Finalization

• If operator’s actions during dispatch are close to optimal, the LMP values at each flexible resource location will be consistent with its current NDR and offer price.

• If the dispatch selected by operator is not close to optimal, the LMPs will become inconsistent.

• A special consistency check is done during finalization process to verify that all marginal resources have their LMP value equal to its offer price.

• If the LMP value is outside of certain tolerance, the error is reported and the measures are taken to correct the price.


Settlements and Market Information Server

Settlements

Billing

MarketInformation

Server

Disp

atch

Dat

aSu

pply

Offe

rsDe

man

d Bi

ds

Internal ContractsExternal Transactions

Meter Reading

Invoice

SettlementData

Bill & Invoice

Data

Day

Ahea

dDa

ta

Real

Tim

e Da

ta

FTPServer

ExternalWeb

Server

InternalWeb

Server

MUI

PrintInvoices

Operations Gateway

Invoic

e

GeneralLedger

InternalDataMarts

External

Transaction


Gateway

SettlementsTask

Management

177


Questions

178


Evaluations

179

eugene litvinov, senior director business … marginal pricing eugene litvinov, senior director...

Documents