Javier Contreras Sanz-

Universidad de Castilla-La ManchaJesús María López Lezama-

Universidad de Antioquia

Antonio Padilha-Feltrin-

Universidade Estadual PaulistaJose Ignacio Muñoz-Universidad de Castilla-La Mancha

IntroductionDistributed generationGeneral considerationsPower flow approximationsInner optimization problemOuter optimization problemBilevel modelTest and resultsFinal remarks

2

In the last decade the electric power industry has shown a renewed interest in distributed generation (DG). This new trend has been mainly motivated by advances in generation technologies that have made smaller generating units viable and feasible along with an increasing awareness of environmental issues.

3

DG can be broadly defined as the production of energy by typically small-size generators located near the consumers. There is a number of different technologies that can be used for small-scale electricity generation. Technologies that use conventional energy resources include gas turbines, fuel cells and microturbines.Technologies that use renewable energy resources include wind turbines, photovoltaic arrays, biomass systems and geothermal generation.

4

Motivations

Deregulation of electric utility industry

Significant advances in generation technologies

New environmental policies

Rapid increase in electric power demand

Main applications

Peak shaving

Combined Heat and Power

Isolated systems

5

Investment deferral in T&D

Increased security for critical loads

Relief of T&D congestion

Reduced emissions of pollutants

Distributed Generation

6

Two different agents are considered, namely, the distribution company (DisCo) and the owner of the DG.To attend the expected demand, the DisCo can purchase energy either form the DG units within its network, or from the wholesale energy market through its substations.Both agents have different objective functions, the DisCo procures the minimization of the payments incurred in attending the expected demand, while the owner of the DG procures the maximization of the profits obtained by selling energy to the DisCo.

7

The DisCo receives a contract price offer, and a declared capacity of the DG units located in its network. The DisCo must weigh the DG energy contract price offer (considering its location) with the potential benefits obtained from the dispatch of these units.If the power injected by a DG unit contributes to the enforcement of a voltage constraint and/or has a positive impact reducing power losses, then, even if the DG energy contract price offer is slightly higher than the wholesale market price, the DG unit is likely to be dispatched.

8

If the DG unit has a negative impact in the distribution network it might not be dispatched, even if its contract price is lower than the wholesale market price. On the other hand, in order to obtain maximum profits, the DG owner must consider the reasoning of the DisCo when deciding the contract price offer and location of its units.Regarding location, the DG owner is given a set of nodes in which he can allocate his units. This set of nodes is decided previously by the DisCo.The two-agent relationship described above can be modeled as a bilevel programming problem.

9

Bilevel programming scheme

10

Unlike transmission systems, in distribution systems, power flows are given mainly due to the difference in voltage magnitudes. Then, the following approximations are considered:

( ).n n mnm

nm

V V VP

Z−

≅

( )2n mloss

nmnm

V VP

Z−

=

lossnm nm mnP P P= +

Active power flow between nodes n,m

Active power losses in line connecting nodes n,m

11

The inner optimization problem corresponds to the DisCo who must minimize the payments incurred in attending the expected demand, subject to network constraints.

, ,( ) ( ) ( )

dgsegk ngj

se dgk gk j gj

P P V k K t T j J t T

Min t t P t tCp P tρ∈ ∈ ∈ ∈

Δ + Δ∑∑ ∑∑

Energy purchaded on the wholesale energy market through the substations.

Energy purchased from the DG units.

12

Subject to:Power balance

( ) ( )

( )2

( ) . ( ) ( ) ( ) . ( ) ( )

( ) ( )( ) ( ) 0 , : ( , )

n n m m m n

nm mnm n m nm n m n

n mgn dn

nmm n

V t V t V t V t V t V tZ Z

V t V tP t P t n N t T n t

Zπ

∈Ω ∈Ω> <

∈Ω

− −+

−− + − = ∀ ∈ ∀ ∈

∑ ∑

∑

Power flow limits

( )( ) . ( ) ( ); l , : (l , ); (l , )n n m

nm nm nm nm nmnm

V t V t V tP P L t T t t

Zφ φ

−≤ ≤ ∀ ∈ ∀ ∈

13

Voltage limits

( ) ; , : ( , ); ( , )n n nV V t V n N t T n t n tω ω≤ ≤ ∀ ∈ ∀ ∈

( ) ; , : ( , ); ( , )dg dg dggj gj gjP P t P j J t T j t j tβ β≤ ≤ ∀ ∈ ∀ ∈

DG active power limits

Substation active power limits

( ) ; , : ( , ); ( , )se se segk gk gkP P t P k K t T k t k tδ δ≤ ≤ ∀ ∈ ∀ ∈

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The outer optimization problem corresponds to the owner of the DG who must maximize profits.

( )

min max

( )

:( ) ; ,

; 0,1

j

dgj j gjCp

t T j J

dg dg dggj gj gj

i ii I

Max t Cp c P t

Subject toP P t P j J t T

B ndg B

∈ ∈

∈

Δ −

≤ ≤ ∀ ∈ ∀ ∈

= ∈

∑∑

∑

Contract price Energy cost

15

Both problems can be expressed as a bilevel programming problem:

( )

, ,

( )

:; 0,1

( ) ( ) ( )

:

j

dgsegk ngj

dgj j gjCp

t T j J

i ii I

se dgk gk j gj

P P V k K t T j J t T

Max t Cp c P t

Subject toB ndg B

Min t t P t tCp P t

Subject to

ρ

∈ ∈

∈

∈ ∈ ∈ ∈

Δ −

= ∈

Δ + Δ

∑∑

∑∑∑ ∑∑

Network constraints

16

A BLPP is a single-round Stackelberg game. In this game there are two types of agents, namely, the leader and the followers. The leader makes his move first anticipating the reaction of the followers, then the followers move sequentially knowing the move of the leader. In this case the leader is the owner of the DG units, and the follower is the Disco. Furthermore, the price and location of the DG units are parameters, and not decision variables, of the inner problem.Assuming convexity, the inner optimization problem can be substituted by its Karush-Kuhn-Tucker optimality conditions.

17

( )

:( ) 0( ) 0

xMin f x

Subject toh xg x

=≤

1 1

( ) ( ) ( ) 0

( ) 0 1,...,( ) 0 1,... ,

( ) 0 1,... ,

QP

p qp q

q

p q

p

q

q

Stationary condition of the Lagrangean

Primal feasibility condition

Complementarity condition

f x h x g x

h x p Pg x q Q

g x q Q

λ μ

μ

= =

∇ + ∇ + ∇ =

= =

≤ =

= =

∑ ∑64444444744444448

644474448

6444 7

1,...,0q

Dual feasibility condition

q Qμ =≥

4 44448

644474448

KKT optimalityconditions

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Substituting the inner optimization problem by its KKT optimality conditions, the following single-level optimization problem is obtained:

( )

, , ,( )

:; 0,1

dgsegk n jgj

dgj j gj

P P V Cp t T j J

i ii I

Max t Cp c P t

Subject toB ndg B

Primal feasibility conditions

∈ ∈

∈

Δ −

= ∈

∑∑

∑

19

Stationary condition of the Lagrangean:

( ) ( )

( )

( )

( ) ( ) ( ) ( )( )

( ) ( )( , ) ( , ) ( , )

( ) ( )( )( , ) ( , )

( )( , ) 0; ,

n m n mm

nm mn nmm n m n m nm n m n

n mnm n m

nm

n mmnm n m nm n m

nm nm

mnm n m

nm

V t V t V t V tV tZ Z Z

V t V tn t n t l t

ZV t V tV t

l t l tZ Z

V tl t n N t T

Z

ω ω φ

φ φ

φ

∈Ω ∈Ω ∈Ω> <

<

> <

>

− −+ +

−+ − +

−− −

+ = ∀ ∈ ∀ ∈

∑ ∑ ∑

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Complementarity and dual feasibility conditions:

( )( )( )

( , ) ( , ) ( , ) 0; ,( ) ( , ) ( , ) ( , ) 0; ,

( , ) ( ) 0; ( , ) 0; ,

( , ) ( ) 0; ( , ) 0; ,

( , ) ( ) 0; ( , ) 0; ,

( , ) (

j

k

n n

n ndg dg

gj gj

dggj

Cp n t j t j t j J t Tt n t k t k t k K t T

n t V t V n t n N t T

n t V t V n t n N t T

j t P t P j t j J t T

j t P t

π β βρ π δ δω ω

ω ω

β β

β

− + − = ∀ ∈ ∀ ∈

− + − = ∀ ∈ ∀ ∈− = ≥ ∀ ∈ ∀ ∈

− + = ≥ ∀ ∈ ∀ ∈

− = ≥ ∀ ∈ ∀ ∈

−( )( )( )

) 0; ( , ) 0; ,

( , ) ( ) 0; ( , ) 0; ,

( , ) ( ) 0; ( , ) 0; ,

dggj

se segk gk

se segk gk

P j t j J t T

k t P t P k t k K t T

k t P t P k t k K t

β

δ δ

δ δ

+ = ≥ ∀ ∈ ∀ ∈

− = ≥ ∀ ∈ ∀ ∈

− + = ≥ ∀ ∈ ∀ ∈

21

Several tests were carried out with a 10-bus distribution system. Load and price duration curves were consider for a one-year contract.

Example of a load duration curve and its approximation.

22

Load and price duration curves are related since higher prices on the wholesale market are expected to take place precisely during the peak hours, conversely, lower prices are expected during off peak hours.

Load duration curve Price duration curve

20 40 60 80 10025

30

35

40

45

50

55

Time (%)

Load

(MW

)

20 40 60 80 10035

40

45

50

55

60

65

Time (%)

Ene

rgy

mar

ket p

rice

(€/M

Wh)

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For the sake of simplicity and without lose of generality, we consider the loads to be equally divided among the 10 nodes. Furthermore, we consider an impedance of 0.0012 Ω

for all lines. However, any load distribution and impedance can be used.

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1 2 3 4 5 6 7 8 9 1060

65

70

75

Bus

Loca

tiona

l mar

gina

l pric

e (€

/MW

h)

Locational marginal prices for a peak load of 50 MW and a wholesale market price of €60/MWh are shown in the figure below. It can be observed that despite of the fact that the energy price at the substation is €60/MWh, providing an additional MW to bus 10 costs €74.4/MWh

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1 2 3 4 5 6 7 8 9 100.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Vol

tage

(p.u

)

Bus

The voltage profile of the system for the peak hour (50 MW) without DG is presented in the figure below. It can be observed that the further away from the substation, the lower the voltages are.

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Case 1:We assume that there is only one single DG unit of 4 MW to be allocated in any node from 6 to 10. Furthermore, we consider a production cost of €50/MWh for the DG unit.

Solution:Location: Bus 10Contract price: €63.38/MWh

Set of possible nodes where to allocate the DG unit.

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1 2 3 4 5 6 7 8 9 100.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Vol

tage

(p.u

)

Bus

Without DGWith DG

1 2 3 4 5 6 7 8 9 1060

65

70

75

Bus

Loca

tiona

l mar

gina

l pric

e (€

/MW

h)

Without DGWith DG

Voltage profile of the distribution system for the peak hour with and without DG.

Locational marginal prices of the distribution system for the peak hour with and witout DG.

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In this case the DG improves the voltage profile and reduces the locational marginal prices of the system. That is because the net load of the system is reduced due to the presence of the DG.The relationship between price offer and profits for the DG unit located in bus 10 is shown in the figure below.

59 60 61 62 63 64 65 66 671

1.2

1.4

1.6

1.8

2x 105

Price offer (€/MWh)

Pro

fit (€

)

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Case 2: We consider two DG units named as DG1 and DG2. Each unit has a capacity of 4 MW and a production cost of €50/MWh. The DG units can be allocated in any node form 6 to 10.

Solution:DG1: Bus 9Contract price DG1: €

54.83/MWh

DG2: Bus 10Contract price DG2: €

55.82/MWh

Set of possible nodes where to allocate the DG units

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With an increasing penetration of DG the locational marginal prices tend to decrease, as shown in the figure.

0 2 4 6 8 1060

65

70

75

Bus

Loca

tiona

l mar

gina

l pric

e (€

/MW

h)

Without DGOnly DG1DG1 and DG2

1 2 3 4 5 6 7 8 9 100.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Bus

Vol

tage

(p.u

)

Without DGOnly DG1DG1 and DG2Voltage profile also

improves with an increasing penetration of DG, as shown in the figure.

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Table 1. Optimal location and contract prices

Case Bus location Contract price Profits (€)One DG unit 10 63.38 187,521Two DG units 9

1054.8355.82

101,545122,359

It was observed that when two DG units are allocated, the optimal contract price per DG unit reduces, as well as the profit obtained per DG unit. However, total profits are higher with two DG units.

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Case Payments (M€) Energy losses (MWh)

Without DG 19,077 21,297

One DG unit 19,002 18,563

Two DG units 18,718 14,896

Total payments and energy losses of the DisCo

The Disco benefits from the DG units since annual

total payments and energy losses decrease. In case 2 (two DG units) a reduction of 30.05 % of energy losses was obtained.

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It was found that the most suitable location for the DG units are those nodes with the highest locational marginal prices.The DG improves the voltage profile and reduces the locational marginal prices of the network.The benefits of the DG depend on its location and size. A high penetration of DG might lead to reverse power flows with a subsequent increase in power losses.Further work will include non-dispatchabletechnologies, as well as the inclusion of stochasticity in the model.

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