a three-level milp model for generation and transmission ......three-level model formulation we...

A three-level MILP model for generation and transmission expansion planning

David Pozo Cámara (UCLM)

Enzo E. Sauma Santís (PUC)

Javier Contreras Sanz (UCLM)

http://www.1becas.com/pontificia-universidad-catolica-de-chile/logo-universidad-catolica-de-chile-uc/

Contents

1. Introduction

2. Aims and contributions

3. The three-stage transmission planning model:

1. Third stage: Market clearing

2. Second stage: Generation investment equilibria

3. First stage: Transmission investment plan

4. Case studies

5. Conclusions

2

Introduction

Several techniques have been applied to investigate power systemstransmission planning: linear programming, MILP, Bendersdecomposition, dynamic programming. Other authors propose the useof heuristics: genetic algorithms, simulated annealing, agent-basedsystems and game theory.

Other methods integrate transmission expansion planning within apool-based market.

One of this works by Sauma and Oren (2006) introduces amethodology to assess the economic impact of transmissioninvestment anticipating the strategic response of oligopolisticgeneration companies.

3

Introduction

Sauma and Oren (2006) formulate a three-period model for studyinghow the exercise local market power by generation firms affects theequilibrium between generation and transmission investments and thevaluation of different transmission expansion projects.

The methodology is based on an iterative process to find theequilibrium but does not solve the optimal transmission planning, onlyevaluates the social welfare impact of some predeterminedtransmission expansion projects.

Other authors, Motamedi et al. (2010), use an agent-based systemwhere a generation company is a learning agent and uses a heuristicmethod to solve the same problem in four stages: bidding strategies,market clearing, generation investment and transmission expansion.

4

Introduction

Additionally, other works propose multi-period models to characterizeinvestments, like Murphy and Smeers (2005). They use a two-stagemodel of investment in generation capacity. Generation investmentdecisions are made at the first stage (subject to equilibriumconstraints) while spot market operations occurs at the second stage.

Garcés et al. (2009) propose a bilevel model where the transmissionplanner minimizes transmission investment costs in the upper leveland the lower level represents the market clearing. The bilevel modelis reformulated as a mixed-integer linear problem using duality theory.

5

Aims and contributions

We present an approach that extends and transforms the three-level model by Sauma and Oren (2006) into a one-level MILP optimization problem.

Our model integrates transmission planning, generation investment and market operation decisions anticipating both the equilibria of generation investments in a decentralized market and the market clearing equilibria.

We characterize the equilibria of generation investments made by decentralized firms (an EPEC: Equilibrium Problem subject to Equilibrium Constraints) as a set of linear inequalities.

6

Aims and contributions

We consider the generation investment EPEC as a set of linear constraints that the network planner can impose in its transmission planning convex optimization problem.

This makes possible to obtain an optimal transmission plan that anticipates both generation investments and market operation equilibria.

We calculate all possible pure Nash equilibria of generation investment problem (EPEC).

We linearize the entire model obtaining a MILP formulation.

7

Three-level model formulation

We assume that the transmission planning model consists of three stages that are described in reversed order, since the first stage represents the final decision by the transmission planner.

Our model is of the Stackelberg type, where the transmission planner (first level) anticipates generation expansions (second level), and the clearing of the market (third level).

8


In the third level we model the energy market operation equilibrium where the independent system operator (ISO) clears the perfectly competitive market and the generator companies (GENCOs) optimize profits from bidding at marginal costs.

In the second level each GENCO anticipates to the result of the third level in order to plan his own capacity expansion. The problem is modeled as an MPEC (Mathematical Problem with Equilibrium Constraints) per GENCO. We obtain the Nash equilibrium when all firms optimize their investment strategies, each one running an MPEC model. The extension to consider all firms is an EPEC.

In the first level the transmission planner invests on transmission lines anticipating the Nash equilibrium at the second level à la Stackelberg.

9


Transmission Investment

(Maximize social welfare minus

investment cost on lines)

Level 1Optimal decisions:

transmission expansion

plan: fl

Generation Investment

(Maximize GENCOs profits minus

investment cost on capacity)

Level 2

Optimal decisions:

generation capacity

expansion gie

Level 3Optimal decisions: market

operation qie , βi

ePool-Based Market Operation

(Equilibrium of ISO and GENCOs)

10


We assume there is a spot market in which the GENCOs are able to submit their energy bids. Our model produces locational marginal prices (LMPs) as a result of linear network constraints.

Demand is inelastic and there are different demand profiles by selecting a set of equivalent scenarios for each demand profile.

The model considers transmission network constraints through a lossless DC approximation assuming price-taker generators.

LMPs are given by the Lagrange multipliers of the energy constraint at each node.

All nodes can have both demand and generation and all generation capacity at a node is owned by a single GENCO.

11

Third level: Market clearing

In the third level we obtain the equilibrium that occurs when the ISO clears the market for given generation and transmission capacities.

Marginal generation costs are constant and inversely proportional to the new installed capacity:

𝑐𝑖 𝑔𝑖 ,𝑔𝑖0 = 𝑎𝑖 − 𝑏𝑖(𝑔𝑖 − 𝑔𝑖

0).

12

ci

ai

gi0

gi

bi

Third level: ISO problem

The ISO problem is modeled as a cost minimization one (dual variables are presented to the right of each constraint):

min𝑞𝑖 ,𝑟𝑖

𝑐𝑖 𝑔𝑖 ,𝑔𝑖0 𝑞𝑖

𝑖

= min𝑞𝑖 ,𝑟𝑖

𝑎𝑖𝑞𝑖 − 𝑏𝑖 𝑔𝑖 − 𝑔𝑖0 𝑞𝑖

𝑖

(1)

subject to:

𝑞𝑖 ≤ 𝑔𝑖 : 𝜉𝑖 ∀𝑖 ∈ 𝑁 (2)

𝑟𝑖𝑖∈𝑁

= 0 : 𝛼

(3)

−𝑓𝑙 ≤ 𝜑𝑙 ,𝑖

𝑖∈𝑁

𝑟𝑖 ≤ 𝑓𝑙 : 𝜆𝑙

−, 𝜆𝑙+

∀𝑙 ∈ 𝐿 (4)

−𝑞𝑖 − 𝑟𝑖 = −𝑑𝑖 : 𝛽𝑖 ∀𝑖 ∈ 𝑁 (5)

𝑞𝑖 ≥ 0 : 𝛾𝑖 ∀𝑖 ∈ 𝑁 (6)

13


The objective function (1) minimizes the total cost of generation.

Constraint (2) establishes the maximum power that the GENCOs can produce.

Constraint (3) indicates that network losses are negligible.

Constraint (4) expresses the maximum flow through lines as a function of the power transfer distribution factors (PTDFs).

Constraint (5) meets demand at every node as a function of the node injection and the flow coming from the lines connected to this node.

Constraint (6) forces power generation to be non negative at every node.

14


The Karush-Kuhn-Tucker (KKT) conditions equivalent to (1)-(6) are

given by:

𝑎𝑖 − 𝑏𝑖 𝑔𝑖 − 𝑔𝑖0 − 𝛾𝑖 − 𝛽𝑖 + 𝜉𝑖 = 0 : 𝑞𝑖 ∀𝑖 ∈ 𝑁 (7)

𝛼 + 𝜆𝑙+ − 𝜆𝑙

−

𝑙∈𝐿

𝜑𝑙 ,𝑖 − 𝛽𝑖 = 0 : 𝑟𝑖 ∀𝑖 ∈ 𝑁 (8)

0 ≤ 𝛾𝑖 ⊥ 𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝑁 (9)

0 ≤ 𝜉𝑖 ⊥ 𝑔𝑖 − 𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝑁 (10)

0 ≤ 𝜆𝑙− ⊥ 𝑓𝑙 + 𝜑𝑙 ,𝑖

𝑖∈𝑁

𝑟𝑖 ≥ 0

∀𝑙 ∈ 𝐿 (11)

0 ≤ 𝜆𝑙+ ⊥ 𝑓𝑙 − 𝜑𝑙 ,𝑖

𝑖∈𝑁

𝑟𝑖 ≥ 0

∀𝑙 ∈ 𝐿 (12)

𝑟𝑖𝑖∈𝑁

= 0 : 𝛼 (13)

−𝑞𝑖 − 𝑟𝑖 = −𝑑𝑖 : 𝛽𝑖 ∀𝑖 ∈ 𝑁 (14)

15

Third level: GENCO problem

Each individual GENCO maximizes its profit considering the income from sales at nodal market prices provided by the ISO cost minimization:

max𝑞𝑖

𝛽𝑖𝑞𝑖 − 𝑎𝑖𝑞𝑖 − 𝑏𝑖 𝑔𝑖 − 𝑔𝑖0 𝑞𝑖

𝑖∈𝑁𝐺

(15)

s.t.

𝑞𝑖 ≤ 𝑔𝑖 : 𝜉𝑖 ∀𝑖 ∈ 𝑁𝐺 (16)

𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝑁𝐺 (17)

16

Third level: GENCO problem

Let’s call primal to the problem in (15)-(17). Thus, from the duality theorem (Luenberger and Ye, 2008), we know that if either the primal or the associated dual problem has an optimal solution, then the other one has the same optimal solution.

Since both primal and dual problems are linear in this case, the problem is convex and we can also apply the strong duality theorem (Luenberger and Ye, 2008). Thus, we get (18) from applying the strong duality theorem:

17

𝛽𝑖𝑞𝑖 − 𝑎𝑖𝑞𝑖 − 𝑏𝑖 𝑔𝑖 − 𝑔𝑖0 𝑞𝑖

𝑖∈𝑁𝐺

= 𝑔𝑖𝜉𝑖𝑖∈𝑁𝐺

∀𝐺 (18)


Using the Fortuny-Amat linearization formula, we have that the set of constraints (19)-(30) fully represents level 3 of our model:

18

𝑎𝑖 − 𝑏𝑖 𝑔𝑖 − 𝑔𝑖0 − 𝛾𝑖 − 𝛽𝑖 + 𝜉𝑖 = 0 : 𝑞𝑖 ∀𝑖 ∈ 𝑁 (19)


−

𝑙∈𝐿

𝜑𝑙 ,𝑖 − 𝛽𝑖 = 0 : 𝑟𝑖 ∀𝑖 ∈ 𝑁 (20)

𝑟𝑖𝑖∈𝑁

= 0 : 𝛼 (21)

−𝑞𝑖 − 𝑟𝑖 = −𝑑𝑖 : 𝛽𝑖 ∀𝑖 ∈ 𝑁 (22)

0 ≤ 𝛾𝑖 ≤ 𝐵𝑀𝛾𝑖(𝜂𝑖𝛾

) ∀𝑖 ∈ 𝑁 (23)

0 ≤ 𝑞𝑖 ≤ 𝐵𝑀𝛾𝑖(1 − 𝜂𝑖𝛾

) ∀𝑖 ∈ 𝑁 (24)

0 ≤ 𝜆𝑙− ≤ 𝐵𝑀𝜆𝑙

−(𝜂𝑙

𝜆−) ∀𝑙 ∈ 𝐿 (25)


19

0 ≤ 𝑓𝑙 + 𝜑𝑙 ,𝑖

𝑖∈𝑁

𝑟𝑖 ≤ 𝐵𝑀𝜆𝑙−

(1 − 𝜂𝑙𝜆−)

∀𝑙 ∈ 𝐿 (26)

0 ≤ 𝜆𝑙+ ≤ 𝐵𝑀𝜆𝑙

+(𝜂𝑙

𝜆+) ∀𝑙 ∈ 𝐿 (27)

0 ≤ 𝑓𝑙 − 𝜑𝑙 ,𝑖

𝑖∈𝑁

𝑟𝑖 ≤ 𝐵𝑀𝜆𝑙+

(1 − 𝜂𝑙𝜆+

) ∀𝑙 ∈ 𝐿 (28)

0 ≤ 𝜉𝑖 ≤ 𝐵𝑀𝜉𝑖(𝜂𝑖𝜉

) ∀𝑖 ∈ 𝑁 (29)

0 ≤ 𝑔𝑖 − 𝑞𝑖 ≤ 𝐵𝑀𝜉𝑖(1 − 𝜂𝑖𝜉

) ∀𝑖 ∈ 𝑁 (30)

Second level: Generation investment

In the second level, each GENCO determines the investments in generation capacity to increase its profits due to the linear decrease in the generation marginal costs:

20

max𝑔𝑖

𝑈𝐺 = 𝛽𝑖𝑞𝑖 − 𝑐𝑖 𝑔𝑖 ,𝑔𝑖0 𝑞𝑖 − 𝐶𝐼𝐺 𝑔𝑖 ,𝑔𝑖

0

𝑖∈𝑁𝐺

= 𝛽𝑖𝑞𝑖 − 𝑎𝑖𝑞𝑖 − 𝑏𝑖 𝑔𝑖 − 𝑔𝑖0 𝑞𝑖 –𝐾𝑖 𝑔𝑖 − 𝑔𝑖

0

𝑖∈𝑁𝐺

(31)

s.t.

(19) - (30)

Using (18), we can rewrite the problem as:

max𝑔𝑖

𝑈𝐺 = 𝑔𝑖𝜉𝑖 –𝐾𝑖 𝑔𝑖 − 𝑔𝑖0

𝑖∈𝑁𝐺

(32)

s.t.

(19) - (30)


21

The only non-linear term in (32) is 𝑔𝑖𝜉𝑖 . Since the 𝑔𝑖 variables are controlled by the generation

firms and it is possible for the generation expansion to be done in discrete steps, then we use a

binary expansion (Pereira et al., 2005) to discretize 𝑔𝑖 :

𝑔𝑖 = 𝑔𝑖0 + Δ𝑔𝑖

2𝑘𝑦𝑘𝑖

Λ 𝑖

𝑘=0

𝑖 ∈ 𝑁𝐺 (33)

Accordingly, the non-linear product 𝑔𝑖𝜉𝑖 can be replaced by the expression:

𝑔𝑖𝜉𝑖 = 𝑔𝑖0𝜉𝑖 + Δ𝑔𝑖

2𝑘𝑦 𝑘𝑖

Λ 𝑖

𝑘=0

𝑖 ∈ 𝑁𝐺 (34)

where we define 𝑦 𝑘𝑖 by constraints (35) and (36), using the Fortuny-Amat linearization formula:

0 ≤ 𝜉𝑖 − 𝑦 𝑘𝑖 ≤ 𝐵𝑀(1 − 𝑦𝑘𝑖 ) ∀𝑖 ∈ 𝑁𝐺 , 𝑘 = 0,1, ,…Λi (35)

0 ≤ 𝑦 𝑘𝑖 ≤ 𝐵𝑀(𝑦𝑘𝑖 ) ∀𝑖 ∈ 𝑁𝐺 , 𝑘 = 0,1, ,…Λi (36)


22

The generation expansion problem of each GENCO can be set as a linear Mathematical Program

subject to Equilibrium Constraints (MPEC), as shown in (37) – (51):

max𝑔𝑖

𝑈𝐺 = 𝑔𝑖0𝜉𝑖 + Δ𝑔𝑖

2𝑘𝑦 𝑘𝑖

Λ 𝑖

𝑘=0

–𝐾𝑖 Δ𝑔𝑖 2𝑘𝑦𝑘𝑖

Λ 𝑖

𝑘=0

𝑖∈𝑁𝐺

(37)

s.t.

𝑎𝑖 − 𝑏𝑖 Δ𝑔𝑖 2𝑘𝑦𝑘𝑖

Λ 𝑖

𝑘=0

− 𝛾𝑖 − 𝛽𝑖 + 𝜉𝑖 = 0 : 𝑞𝑖 ∀𝑖 ∈ 𝑁 (38)


−

𝑙∈𝐿

𝜑𝑙 ,𝑖 − 𝛽𝑖 = 0 : 𝑟𝑖 ∀𝑖 ∈ 𝑁 (39)

𝑟𝑖𝑖∈𝑁

= 0 : 𝛼 (40)

−𝑞𝑖 − 𝑟𝑖 = −𝑑𝑖 : 𝛽𝑖 ∀𝑖 ∈ 𝑁 (41)


23

0 ≤ 𝛾𝑖 ≤ 𝐵𝑀𝛾𝑖(𝜂𝑖𝛾

) ∀𝑖 ∈ 𝑁 (42)

0 ≤ 𝑞𝑖 ≤ 𝐵𝑀𝛾𝑖(1 − 𝜂𝑖𝛾

) ∀𝑖 ∈ 𝑁 (43)

0 ≤ 𝜆𝑙− ≤ 𝐵𝑀𝜆𝑙

−(𝜂𝑙

𝜆−) ∀𝑙 ∈ 𝐿 (44)

0 ≤ 𝑓𝑙 + 𝜑𝑙 ,𝑖

𝑖∈𝑁

𝑟𝑖 ≤ 𝐵𝑀𝜆𝑙−

(1 − 𝜂𝑙𝜆−)

∀𝑙 ∈ 𝐿 (45)

0 ≤ 𝜆𝑙+ ≤ 𝐵𝑀𝜆𝑙

+(𝜂𝑙

𝜆+) ∀𝑙 ∈ 𝐿 (46)

0 ≤ 𝑓𝑙 − 𝜑𝑙 ,𝑖

𝑖∈𝑁

𝑟𝑖 ≤ 𝐵𝑀𝜆𝑙+

(1 − 𝜂𝑙𝜆+

) ∀𝑙 ∈ 𝐿 (47)

0 ≤ 𝜉𝑖 ≤ 𝐵𝑀𝜉𝑖(𝜂𝑖𝜉

) ∀𝑖 ∈ 𝑁 (48)

0 ≤ 𝑔𝑖0 + Δ𝑔𝑖

2𝑘𝑦𝑘𝑖

Λ 𝑖

𝑘=0

− 𝑞𝑖 ≤ 𝐵𝑀𝜉𝑖(1 − 𝜂𝑖𝜉

)

∀𝑖 ∈ 𝑁 (49)

0 ≤ 𝜉𝑖 − 𝑦 𝑘𝑖 ≤ 𝐵𝑀(1 − 𝑦𝑘𝑖 ) ∀𝑖 ∈ 𝑁𝐺 ,

𝑘 = 0,1, ,…Λi (50)

0 ≤ 𝑦 𝑘𝑖 ≤ 𝐵𝑀(𝑦𝑘𝑖 ) ∀𝑖 ∈ 𝑁𝐺 ,

𝑘 = 0,1, ,…Λi (51)


Level 2 problem can be formulated as an Equilibrium Problem with Equilibrium Constraints (EPEC) in which each firm a mixed integer linear programming (MILP) MPEC problem faces given the other firms’ commitments and the system operator’s import/export decisions).

This EPEC represents the equilibrium when all the GENCOs expand their capacity simultaneously subject to the market equilibrium of level 3.

We enumerate the GENCOs’ investment strategies and express the Nash equilibria conditions as a finite set of inequalities.

24


25

To characterize the EPEC equilibria as a set of linear inequalities, we discretize all generation

investment strategic variables of the problem. The general expression for the Nash equilibrium is

given by:

𝑈𝐺𝑒(𝑔𝑖

𝑒 ,∀𝑖 ∈ 𝑁) ≥ max𝑔𝑖

𝑈𝐺 𝑔𝑖 ,𝑔−𝑖𝑒 ,∀𝑖 ∈ 𝑁𝐺 ,∀ − 𝑖 ∉ 𝑁𝐺 ∀𝐺 ∈ Ψ (52)

where, for all GENCOs, 𝑈𝐺𝑒(𝑔𝑖

𝑒) is the utility function of each GENCO 𝐺 given its strategic decision

variable 𝑔𝑖𝑒 in the Nash equilibrium, which is always better than any other utility resulting from a

different strategy, assuming that the other GENCOs use their Nash equilibrium strategies, 𝑔−𝑖𝑒 .

Hence, the Nash equilibrium in (52) is solved by approximating its solution using discrete

strategies. In doing that, we replace expression (52) by a set of inequalities, where the strategic

variables are discretized for each GENCO.


26

The Nash equilibria of the GENCOs’ capacity investment decisions are given

by the following set of inequalities:

𝑈𝐺𝑒 𝑔𝑖

𝑒 ,∀𝑖 ∈ 𝑁

≥ 𝑈𝐺𝑠𝐺 𝑔𝑖

𝑠𝐺 ,𝑔−𝑖𝑒 ,∀𝑖

∈ 𝑁𝐺 ,∀ − 𝑖 ∉ 𝑁𝐺

∀𝐺 ∈ 𝛹,∀𝑠𝐺 ∈ 𝒮𝐺

(53)

where we have to distinguish between the left hand side (LHS) and the right

hand side (RHS) of (53).


27

The LHS in (53) is the utility function of each GENCO given its strategic decision variable in the

Nash equilibrium. That is, the definition of the utility function for GENCO 𝐺 in the equilibrium is

given by:

𝑈𝐺𝑒 = 𝑔𝑖

0𝜉𝑖𝑒 + Δ𝑔𝑖

2𝑘𝑦 𝑘𝑖𝑒

Λ 𝑖

𝑘=0

–𝐾𝑖 Δ𝑔𝑖 2𝑘𝑦𝑘𝑖

𝑒

Λ 𝑖

𝑘=0

𝑖∈𝑁𝐺

∀𝐺 ∈ Ψ (54)

subject to the linearized constraints of stage 3 in the equilibrium, which correspond to

constraints (38)-(51), but replacing 𝑦𝑘𝑖 , 𝑦 𝑘𝑖 , 𝑞𝑖 , 𝑟𝑖 , 𝛾𝑖 ,𝛽𝑖 , 𝜉𝑖 ,𝛼, 𝜆𝑙+, 𝜆𝑙

−, 𝜂𝑖𝛾

, 𝜂𝑙𝜆+

, 𝜂𝑙𝜆− , and 𝜂𝑖

𝜉 by

𝑦𝑘𝑖𝑒 , 𝑦 𝑘𝑖

𝑒 , 𝑞𝑖𝑒 , 𝑟𝑖

𝑒 , 𝛾𝑖𝑒 ,𝛽𝑖

𝑒 , 𝜉𝑖𝑒 ,𝛼𝑒 , 𝜆𝑙

+𝑒 , 𝜆𝑙−𝑒 , 𝜂𝑖

𝛾 𝑒, 𝜂𝑙

𝜆+ 𝑒, 𝜂𝑙

𝜆− 𝑒, and 𝜂𝑖

𝜉 𝑒, respectively, and considering

(50) and (51) for all 𝑖 ∈ 𝑁.


28

The RHS in (53) is the utility function of each GENCO given a particular value of the strategic

decision variable. That is, considering firm 𝐺 chooses strategy 𝑠𝐺 (which involves investing in

generation capacity at node i up to the capacity 𝑔𝑖𝑠𝐺 , with 𝑖 ∈ 𝑁𝐺 ,∀𝐺 ∈ Ψ), the definition of

the utility function for GENCO 𝐺 is given by:

subject to the corresponding constraints of stage 3, which correspond to constraints (38)-

(51), but considering them ∀𝐺 ∈ Ψ,∀𝑠𝐺 ∈ 𝒮𝐺 , replacing

𝑦𝑘𝑖 ,𝑦 𝑘𝑖 , 𝑞𝑖 , 𝑟𝑖 , 𝛾𝑖 ,𝛽𝑖 , 𝜉𝑖 ,𝛼, 𝜆𝑙+, 𝜆𝑙

−, 𝜂𝑖𝛾

, 𝜂𝑙𝜆+

, 𝜂𝑙𝜆− , and 𝜂𝑖

𝜉 by

𝑦𝑘𝑖𝑠𝐺 ,𝑦 𝑘𝑖

𝑠𝐺 , 𝑞𝑖𝑠𝐺 , 𝑟𝑖

𝑠𝐺 , 𝛾𝑖𝑠𝐺 ,𝛽𝑖

𝑠𝐺 , 𝜉𝑖𝑠𝐺 ,𝛼𝑠𝐺 , 𝜆𝑙

+𝑠𝐺 , 𝜆𝑙−𝑠𝐺 , 𝜂𝑖

𝛾 𝑠𝐺 , 𝜂𝑙𝜆+ 𝑠𝐺 , 𝜂𝑙

𝜆− 𝑠𝐺 , and 𝜂𝑖𝜉 𝑠𝐺 , respectively,

and replacing (38) by (56) and (57), (49) by (58) and (59), (50) by (60), and (51) by (61):

𝑈𝐺𝑠𝐺 = 𝑔𝑖

𝑠𝐺𝜉𝑖𝑠𝐺 –𝐾𝑖 𝑔𝑖

𝑠𝐺 − 𝑔𝑖0

𝑖∈𝑁𝐺

∀𝐺 ∈ Ψ,∀sG ∈ 𝒮𝐺 (55)


29

𝑎𝑖 − 𝑏𝑖 𝑔𝑖𝑠𝐺 − 𝑔𝑖

0 − 𝛾𝑖𝑠𝐺 − 𝛽𝑖

𝑠𝐺 + 𝜉𝑖𝑠𝐺 = 0 ∀𝑖 ∈ 𝑁𝐺 ,∀𝐺 ∈ Ψ,∀𝑠𝐺 ∈ 𝒮𝐺 (56)

𝑎𝑖 − 𝑏𝑖 Δ𝑔𝑖 2𝑘𝑦𝑘𝑖

𝑒

Λ 𝑖

𝑘=0

− 𝛾𝑖𝑠𝐺 − 𝛽𝑖

𝑠𝐺 + 𝜉𝑖𝑠𝐺 = 0

∀𝑖 ∉ 𝑁𝐺 ,∀𝐺 ∈ Ψ,∀𝑠𝐺 ∈ 𝒮𝐺 (57)

0 ≤ 𝑔𝑖𝑠𝐺 − 𝑞𝑖

𝑠𝐺 ≤ 𝐵𝑀𝜉𝑖(1 − 𝜂𝑖𝜉 𝑠𝐺 ) ∀𝑖 ∈ 𝑁𝐺 ,∀𝐺 ∈ Ψ,∀𝑠𝐺 ∈ 𝒮𝐺 (58)

0 ≤ 𝑔𝑖0 + Δ𝑔𝑖

2𝑘𝑦𝑘𝑖𝑒

Λ 𝑖

𝑘=0

− 𝑞𝑖𝑠𝐺 ≤ 𝐵𝑀𝜉𝑖(1 − 𝜂𝑖

𝜉 𝑠𝐺 )

∀𝑖 ∉ 𝑁𝐺 ,∀𝐺 ∈ Ψ,∀𝑠𝐺 ∈ 𝒮𝐺 (59)

0 ≤ 𝜉𝑖𝑠𝐺 − 𝑦 𝑘𝑖

𝑠𝐺 ≤ 𝐵𝑀(1 − 𝑦𝑘𝑖𝑒 )

𝑘 = 0,1, ,…Λi ,∀𝑖 ∉ 𝑁𝐺 ,∀𝐺 ∈ Ψ,

∀𝑠𝐺 ∈ 𝒮𝐺 (60)

0 ≤ 𝑦 𝑘𝑖𝑠𝐺 ≤ 𝐵𝑀(𝑦𝑘𝑖

𝑒 ) 𝑘 = 0,1, ,…Λi ,∀𝑖 ∉ 𝑁𝐺 ,∀𝐺 ∈ Ψ,

∀𝑠𝐺 ∈ 𝒮𝐺 (61)

First level: Transmission investment

In level 1, the network planner –which we model as a Stackelbergleader in our 3-level game– maximizes the social welfare subject to the transmission constraints while anticipating the solutions from levels 2 and 3.

Since we have considered inelastic demands, this problem is equivalent to minimize the total cost: sum of generation dispatch costs and transmission investment costs.

Thus, the objective function of the transmission planner in level 1 is:

30

min𝑓𝑙 ,𝑙 ∈𝐿𝑖𝑛𝑣

𝑐𝑖 𝑔𝑖 ,𝑔𝑖0 𝑞𝑖

𝑖

+ 𝐶𝐼𝐿 𝑓𝑙 , 𝑓𝑙0

𝑙∈𝐿𝑖𝑛𝑣

= min𝑓𝑙 ,𝑙 ∈𝐿𝑖𝑛𝑣

𝑎𝑖𝑞𝑖 − 𝑏𝑖 𝑔𝑖 − 𝑔𝑖0 𝑞𝑖

𝑖

+ 𝐾𝑙 𝑓𝑙 − 𝑓𝑙0

𝑙∈𝐿𝑖𝑛𝑣

(62)


31

The problem of minimizing (62) subject to the transmission constraints and the constraints

representing the solutions from levels 2 and 3 is non linear. Moreover, the variables that

represent the solution to the EPEC are equilibrium results, thus, having 𝑞𝑖𝑒 instead of 𝑞𝑖 , and

𝑔𝑖𝑒 instead of 𝑔𝑖 . The non-linear term in the objective function, 𝑔𝑖

𝑒𝑞𝑖𝑒 , can be decomposed

using the binary expansion applied to 𝑔𝑖𝑒 and linearized using the Fortuny-Amat formulation.

This yields:

𝑔𝑖𝑒𝑞𝑖

𝑒 = 𝑔𝑖0𝑞𝑖

𝑒 + Δ𝑔𝑖 2𝑘𝑦 𝑘𝑖

𝑒

Λ 𝑖

𝑘=0

∀𝑖 ∈ 𝑁𝐺 (63)

0 ≤ 𝑞𝑖𝑒 − 𝑦 𝑘𝑖

𝑒 ≤ 𝐵𝑀𝑞(1 − 𝑦𝑘𝑖𝑒 ) ∀𝑖 ∈ 𝑁𝐺 , 𝑘 = 0,1, ,…Λi (64)

0 ≤ 𝑦 𝑘𝑖𝑒 ≤ 𝐵𝑀𝑞(𝑦𝑘𝑖

𝑒 ) ∀𝑖 ∈ 𝑁𝐺 , 𝑘 = 0,1, ,…Λi (65)

where 𝑦 𝑘𝑖𝑒 is a continuous variable taking the values of either 0 or 𝑞𝑖

𝑒 .


32

In (62), we have considered that there is a set of transmission lines that are

candidates for investment 𝐿𝑖𝑛𝑣 . That means that the previously-constant

maximum active flows 𝑓𝑙 are now variables of the problem in level 1.

Note that, contrary to the assumptions in (Sauma and Oren, 2006), the

network planner now solves level 1 for the optimal transmission expansion

capacities in both new and existing lines within the set of candidate locations.

Therefore, we can formulate level 1 problem as a mixed integer linear

programming optimization program subject to EPEC and other equilibrium

constraints.

Finding all Nash equilibria in level 2

The EPEC for the level 2 problem may have multiple equilibria.

The model described finds only one EPEC equilibrium, but we could be interested in detecting more than one equilibrium, or even all of them.

We modify the previous section model of the level 2 problem in order to find all pure strategy EPEC equilibria.

To do that, we generate holes in the feasible region for each solution found within the set of discrete strategies: yki

e .

Given a solution vector of the EPEC problem of level 2, we include a new constraint to generate a hole in the solution already found.

33

Finding all Nash equilibria in level 2

34

(𝑦𝑘𝑖∗ (𝑛) − 𝑦𝑘𝑖

𝑒 )2

𝑖 ,𝑘

≥ 𝜖 ∀𝑛 (66)

Each one of the quadratic terms in (66) is expanded as:

𝑦𝑘𝑖∗ − 𝑦𝑘𝑖

𝑒 2 = 𝑦𝑘𝑖∗ 2 + 𝑦𝑘𝑖

𝑒 2 − 2 𝑦𝑘𝑖∗ 𝑛 𝑦𝑘𝑖

𝑒 (67)

and, using the fact that 𝑦𝑘𝑖∗ and 𝑦𝑘𝑖

𝑒 are binary numbers, (67) is equivalent to:

𝑦𝑘𝑖∗ + 𝑦𝑘𝑖

𝑒 − 2 𝑦𝑘𝑖∗ 𝑛 𝑦𝑘𝑖

𝑒 (68)

which is a linear expression. Thus, (67) becomes:

𝑦𝑘𝑖∗ + 𝑦𝑘𝑖

𝑒 − 2 𝑦𝑘𝑖∗ 𝑛 𝑦𝑘𝑖

𝑒

𝑖 ,𝑘

≥ 𝜖2 ∀𝑛 (69)

Variation of line impedance in level 1

35

Link impedance as a function of transmission capacity.

𝑓𝑙0

𝑥𝑙

Impedance

Line capacity

𝑥𝑙2

2𝑓𝑙0

PTDFs calculation in level 1

In order to apply this modification to the model proposed for stage 1, and keep the level-1 formulation as a mixed integer linear programming optimization program, continuously changing the power transfer distribution factors (PTDFs) seems unviable due to the nonlinearities involved.

Instead, we consider a discretization of the equivalent impedance in the potentially-expanded lines and calculate the associated PTDFs.

36

Variation of line impedance in level 1

37

If the initial transmission capacity of link l is 𝑓𝑙0 and investments can be done

up to a capacity whose value is 𝑓𝑙𝑚𝑎𝑥 , we can approximate the equivalent

impedance by performing a discrete approximation, between 𝑓𝑙0 and 𝑓𝑙

𝑚𝑎𝑥

𝑓𝑙0

𝑥𝑙0

Equivalent

impedance

Final capacity

𝑥𝑙1

𝑓𝑙3 = 𝑓𝑙

𝑚𝑎𝑥 𝑓𝑙1

𝑥𝑙2

𝑓𝑙2

𝑥𝑙3

Discretization of the equivalent impedance as a function of installed transmission capacity.

Case studies: 3-node example

38

1

2

3

ℓ3 ℓ1

ℓ2

3-node case study data

Node

Demand

Generation units:

Production costs parameters

Unit cost

of investment

𝑖 𝑑𝑖 [MW] 𝑔𝑖0[MW] 𝑎𝑖[$/MWh] 𝑏𝑖[$/(MW·MWh)] 𝐾𝑖[$/MW]

1 30 30 25 0.3 0.02

2 25 30 24 0.3 0.02

3 20 30 24 0.3 0.02


39

PTDFs for the four considered states in the 3-node network, when investing in line 1 only

Case of no link

investment

Case of investment in

interval [7 8.4] MW


interval [8.4 10.5] MW


interval [10.5 14] MW

0 0.667 0.333 0 0.686 0.343 0 0.727 0.363 0 0.774 0.387

0 0.333 0.667 0 0.314 0.657 0 0.273 0.636 0 0.226 0.613

0 -0.333 0.333 0 -0.314 0.343 0 -0.273 0.363 0 -0.226 0.387


The thermal capacity for each line is 7 MW and the unit transmission investment cost (Kl) is $25/MW for each line.

We consider 4 possibilities for transmission investment: investment in line 1, investment in line 2, investment in line 3, and investment affecting lines 1, 2, and 3 simultaneously.

The flow limit according to our discretization process is 14 MW and there are four states for the expansion line capacity: no investment, line flow bound between 7 and 8.4 MW, line flow bound between 8.4 and 10.5 MW, and line flow bound between 10.5 and 14 MW.

For the second level of our model, we assume the three GENCOs can invest in generation capacity from 30 MW up to 54 MW at intervals of 1.6 MW.

40


41

Link impedance as a function of the capacity in line 1.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

Capacity investment factor

Lin

k im

ped

an

ce(p

.u.)


Solving the problem of level 1, for the case of investing in line 1 only, we obtain that the optimal value is to invest up to 7.4 MW of capacity for line 1.

The GENCO in node 1 invests 14.4 MW in generation capacity, meaning that its total production becomes 44.4 MW in level 3.

The GENCO in node 1 becomes the most economic unit, whose marginal cost is $20.68/MWh, and the production of this GENCO is partially consumed at node 1 (30 MWh) and partially sent through lines 1 and 2.

This yields the same LMPs for all the nodes and the minimum cost of dispatch.

42


43

Optimal market clearing values given the solutions of level 1 and 2 in the 3-node network

Node

Profits [$]

Available capacity for

each GENCO [MW]

LMP [$/MWh]

Production [MWh]

1 147.41 44.4 24 44.4

2 0 30 24 17.817

3 0 30 24 12.783

Optimal values of the problem for level 1 of the 3-node network

Case Cost for the transmission

planner [$] Line capacity [MW]

Available capacity [MW]

𝑔1 𝑔2 𝑔3

ℓ1 1662.592 7.4 (ℓ1) 44.4 30 30

ℓ2 1662.592 7.4 (ℓ2) 44.4 30 30

ℓ3 1678.448 7 (ℓ3) 42.8 30 30

ℓ1, ℓ2 & ℓ3 1662.592 7.4 (ℓ1) 44.4 30 30

7.4 (ℓ2) 44.4 30 30


Note that if we fix the investment in line 1 and we solve the EPEC problem by applying the methodology to find all pure Nash equilibria, we obtain two more equilibria.

In the optimization process of level 1, the transmission planner attempts to anticipate the EPEC equilibrium by choosing the best possible solution for level 2. However, this cannot be guaranteed.

Hence, we solve the level 1 problem using what we call an optimistic solution for the transmission planner, which considers that the transmission planner anticipates the best (from the social welfare viewpoint) EPEC equilibria.

There is also a pessimistic solution for the transmission planner, which considers the worst EPEC equilibria.

44


45

Optimistic and pessimistic level-1 solutions for the case of investing only in line 1.

7 7.4 7.6 8 9 10 11 12 13 141.650

1662.6

1723.4

1700

1.750

1.800

Capacity of line 1

Dis

patc

h p

lus lin

e investm

ent

cost


46

4

ℓ4 1

2

3

ℓ3 ℓ1

ℓ2

4-node example data

Node

Demand

Generation units:

Production cost parameters

Unit cost

of investment

𝑖 𝑑𝑖 [MW] 𝑔𝑖0[MW] 𝑎𝑖[$/MWh] 𝑏𝑖[$/(MW·MWh)] 𝐾𝑖[$/MW]

1 30 30 25 0.3 0.02

2 25 30 24 0.3 0.02

3 20 30 24 0.3 0.02

4 25 30 24 0.3 0.02


47

4-node example line data

Line

Initial thermal

limit capacity

Unit transmission

investment cost

Maximum thermal

limit capacity

ℓ 𝑓𝑙0 [MW] 𝐾𝑙 [$/MW] 𝑓𝑙

𝑚𝑎𝑥 [MW]

1 7 25 14

2 7 25 14

3 7 25 14

4 0 25 14


48

Optimal values of the problem for level 1 of the 4-node network

Case Cost for the transmission

planner [$] Line capacity [MW]

Available capacity [MW]

𝑔1 𝑔2 𝑔3 𝑔4

ℓ1 , ℓ2, ℓ3& ℓ4 2262.592 7.4 (ℓ1) 44.4 30 30 30

7.4 (ℓ2) 44.4 30 30 30

0.4 (ℓ4) 44.4 30 30 30

Conclusions

A MILP three-level model for transmission investment is proposed:Level 1: Transmission plannerLevel 2: Equilibrium in generation expansionLevel 3: Market clearing

The transmission planner anticipates the generation expansiondecisions and market clearing à a la Stackelberg.

The Nash equilibrium of generation expansion uses an EPECframework.

All possible pure Nash equilibrium are obtained.

Line impedance approximation as a function of installed capacity .

49

A three-level MILP model for generation and transmission expansion planning

David Pozo Cámara (UCLM)

Enzo E. Sauma Santís (PUC)

Javier Contreras Sanz (UCLM)

http://www.1becas.com/pontificia-universidad-catolica-de-chile/logo-universidad-catolica-de-chile-uc/

a three-level milp model for generation and transmission ......three-level model formulation we...

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