Complex Renewable Energy Networks: Power flows and markets

Complex Renewable Energy Networks: Power flows and markets Mirko Schäfer 14th June 2016 Frankfurt Institute of Advanced Studies (FIAS), Goethe-Universität Frankfurt FIAS Renewable Energy System and Network Analysis (FRESNA) schaefer@fias.uni-frankfurt.de

Table of Contents 1. Consumers and generators in electricity markets 2. Transmission and distribution networks 3. Representing network constraints 4. Efficient dispatch in a two-node system with constraints 5. Representing network constraints in meshed networks 6. Example: Efficient market operation in a 3-node system with constrained transmission 2

Consumers and generators in electricity markets

Consumer behaviour: Theory Suppose for some given period a consumer consumes electricity at a rate of Q MW. Their utility or value function U(Q) in e/h is a measure of their benefit for a given consumption rate Q. For a firm this could be the profit related to this electricity consumption from manufacturing goods. Typical the consumer has a higher utility for higher Q, i.e. the first derivative is positive U (Q) > 0. By assumption, the rate of value increase with consumption decreases the higher the rate of consumption, i.e. U (Q) < 0. 4

Utility: Example A widget manufacturer has a utility function which depends on the rate of electricity consumption Q [e/h] as U(Q) = 0.0667 Q 3 8 Q 2 + 300 Q 3500 3000 2500 Utility U(Q) [ /h] 2000 1500 1000 500 0 0 5 10 15 20 Electricity consumption rate Q [MW] Note that the slope is always positive, but becomes less positive for increasing Q. 5

Optimal consumer behaviour We assume to begin with that the consumer is a price-taker, i.e. they cannot influence the price by changing the amount they consume. Suppose the market price is λ e/mwh. The consumer should adjust their consumption rate Q to maximise their net surplus max [U(Q) λq] Q This optimisation problem is optimised for Q = Q where [Check units: du dq U (Q ) du dq (Q ) = λ has units e/h MW = e/mwh.] I.e. the consumer increases their consumption until they make a net loss for any increase of consumption. U (Q) is known as the inverse demand curve or marginal utility curve, which shows, for each rate of consumption Q what price λ the consumer should be willing to pay. 6

Inverse demand function: Example For our example the inverse demand function is given by U (Q) = 0.2 Q 2 16 Q + 300 300 Inverse demand function U (Q) [ /MWh] 250 200 150 100 50 electricity price 0 0 5 10 15 20 Electricity consumption rate Q [MW] It s called the inverse demand function, because the demand function is the function you get from reversing the axes. 7

Gross consumer surplus The area under the inverse demand curve is the gross consumer surplus, which as the integral of a derivative, just gives the utility function U(Q) again, up to a constant. 300 Inverse demand function U (Q) [ /MWh] 250 200 150 100 50 electricity price gross consumer surplus 0 0 5 10 15 20 Electricity consumption rate Q [MW] 8

Net consumer surplus The more relevant net consumer surplus, or just consumer surplus is the net gain the consumer makes by having utility above the electricity price. 300 Inverse demand function U (Q) [ /MWh] 250 200 150 100 50 electricity price (net) consumer surplus 0 0 5 10 15 20 Electricity consumption rate Q [MW] 9

Generator Cost Function Optimal generator behaviour follows a similar schema to that for consumers. A generator has a cost or supply function C(Q) in e/h, which gives the costs (of fuel, etc.) for a given rate of electricity generation Q MW. Typical the generator has a higher cost for a higher rate of generation Q, i.e. the first derivative is positive C (Q) > 0. For most generators the rate at which cost increases with rate of production itself increases as the rate of production increases, i.e. C (Q) > 0. 10

Cost Function: Example A gas generator has a cost function which depends on the rate of electricity generation Q [e/h] according to C(Q) = 0.005 Q 2 + 9.3 Q + 120 7000 6000 5000 Cost C(Q) [ /h] 4000 3000 2000 1000 0 0 100 200 300 400 500 Electricity generation rate Q [MW] Note that the slope is always positive and becomes more positive for increasing Q. The curve does not start at the origin because of startup 11

Optimal generator behaviour We assume to begin with that the generator is a price-taker, i.e. they cannot influence the price by changing the amount they generate. Suppose the market price is λ e/mwh. For a generation rate Q, the revenue is λq and the generator should adjust their generation rate Q to maximise their net generation surplus, i.e. their profit: max [λq C(Q)] Q This optimisation problem is optimised for Q = Q where [Check units: dc dq C (Q ) dc dq (Q ) = λ has units e/h MW = e/mwh.] I.e. the generator increases their output until they make a net loss for any increase of generation. C (Q) is known as the marginal cost curve, which shows, for each rate of generation Q what price λ the generator should be willing to supply at. 12

Marginal cost function: Example For our example the marginal function is given by C (Q) = 0.001 Q + 9.3 20 Marginal cost C (Q) [ /MWh] 15 10 5 electricity price 0 0 100 200 300 400 500 Electricity generation rate Q [MW] 13

Generator surplus The area under the curve is generator costs, which as the integral of a derivative, just gives the cost function C(Q) again, up to a constant. The generator surplus is the profit the generator makes by having costs below the electricity price. 20 Marginal cost function C (Q) [ /MWh] 15 10 5 electricity price generator surplus costs 0 0 100 200 300 400 500 Electricity generator rate Q [MW] 14

Consumers and generators Consumers: Their utility or value function U(Q) in e/h is a measure of their benefit for a given consumption rate Q. For a given price λ they adjust their consumption rate Q such that their net surplus is maximised: max [U(Q) λq] Q Generators: A generator has a cost or supply function C(Q) in e/h, which gives the costs (of fuel, etc.) for a given rate of electricity generation Q MW. If the market price is λ e/mwh, the revenue is λq and the generator should adjust their generation rate Q to maximise their net generation surplus, i.e. their profit: max [λq C(Q)] Q 15

The result of optimisation This is the result of maximising the total economic welfare, the sum of the consumer and the producer surplus for consumers with consumption Qi B and generators generating with rate Qi S : [ max U i (Q {Qi B },{Qi S i B ) ] C i (Qi S ) } i i subject to the supply equalling the demand in the balance constraint: Qi B Qi S = 0 λ i i and any other constraints (e.g. limits on generator capacity, etc.). Market price λ is the shadow price of the balance constraint, i.e. the cost of supply an extra increment 1 MW, or reduce generation by an increment of 1 MW. 16

Setting the quantity and price Total welfare (consumer and generator surplus) is maximised if the total quantity is set where the marginal cost and marginal utility curves meet. If the price is also set from this point, then the individual optimal actions of each actor will achieve this result in a perfect decentralised market. Cost [ /MWh] 10 8 6 4 2 Consumer surplus Market price = 5 /MWh Producer surplus Marginal generation cost Marginal consumer utility 0 0 20 40 60 80 100 Electricity amount [MW] 17

Effect of varying renewables: fixed demand, no wind 90 80 70 Wind Nuclear Brown Coal Hard Coal Gas Oil Demand 60 clearing price /MWh 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 GW 18

Effect of varying renewables: fixed demand, 35 GW wind 90 80 70 Wind Nuclear Brown Coal Hard Coal Gas Oil Demand 60 /MWh 50 40 30 20 10 clearing price 0 0 10 20 30 40 50 60 70 80 GW 19

Transmission and distribution networks

Transmission and distribution networks Electricity usually is not consumed where it is produced, so it has to be transported via transmission and distribution networks. Transmission networks: Transport large volumes of electric power over relatively long distances. Distribution networks: Take power from the transmission network and deliver it to a large number of end points in a certain geographic area. 21

Transmission and distribution networks Source: VKU 22

European Transmission Grid Source: ENTSO-E 23

Transmission and distribution networks in Germany Das deutsche Strom-Verteilernetz ist rund 1,7 Millionen Kilometer lang Verteilernetz Übertragungsnetz Niederspannungsnetz: ca. 1.100.000 Kilometer Mittelspannungsnetz: ca. 510.000 Kilometer Hochspannungsnetz: ca. 95.000 Kilometer Höchstspannungsnetz: ca. 35.000 Kilometer Source: BMWi 24

Transmission and distribution networks in Germany Source: Agora Energiewende / RAP 25

TSOs in Germany Source: Wikipedia (Francis McLloyd) 26

Transmission grid near Frankfurt Source: ENTSO-E 27

Source: NRM Netzdienste Rhein-Main 28 Schwanheim Flughafen Rödelheim Griesheim Niederursel Main Praunheim Hausen Kalbach-Riedberg Gallus Heddernheim Bockenheim Niederrad Ginnheim Gutleutviertel Nieder-Eschbach Eschersheim Westend-Süd Bonames Dornbusch Westend-Nord Frankfurter Berg Eckenheim Nordend-West Innenstadt Altstadt Bahnhofsviertel Sachsenhausen-N. Harheim Sachsenhausen-S. Preungesheim Nieder-Erlenbach Berkersheim Nordend-Ost Bornheim Ostend Seckbach Oberrad Riederwald Bergen-Enkheim Fechenheim Frankfurts DSO Stromverteilnetz der NRM Netzdienste Rhein-Main GmbH Frankfurt am Main Stand: 01/2013 NRM Netzdienste Rhein-Main (subsidiary company of Mainova)

Power grids and electricity markets The (physical) balancing of supply and demand has to respect the network constraints of the system. These constraints have to be implemented by the system operator, but to some extent can also be included into the market design. Transmission and distribution networks are (almost?) natural monopolies, which leads to substantial market power. These networks are typically state owned, cooperatives or heavily regulated (many interesting problems with respect to incentives, tariffs, etc.). Network expansion is part of the long-term efficient operation of the system. Note the interdependency between network and generation investment. 29

Representing network constraints

Physical limits on networks Thermal limits: Relate to the maximum amount of power which can be transmitted via a transmission line. Voltage stability limits: Relate to the supply of reactive power to keep the system voltage close to a specific level. Dynamic and transient stability limits: Relate to the stability of the system frequency, and the stability of the synchronized operation of the generators. 31

Representing transmission networks Terminology: We represent the transmission grid as an network, consisting of nodes and links. The nodes may represent individual generators, groups of generators and consumers, whole geographic regions or just a point where different transmission lines meet. The links represent transmission lines, or more generally the possibility to transfer electric power between the respective nodes connected to the respective link. Source: PyPSA 32

Basic implementation of thermal limits A link l connecting two nodes allows to transport electrical power as a power flow F l from one node to the other. We implement the thermal limits on a line l as the capacity K l, which gives the upper limit of power flow F l on l: F l K l F l K l This looks just like another constraint for KKT. Unfortunately, the power flows F l are usually not free parameters, but are connected via physical laws from the generation and consumption pattern Qi S, Qi B at the nodes. 33

Spoiler: Nodal benefit function Suppose at node k there are some consumers and generators i N k, with generation Q S i and consumption Q B i. We define the benefit function B k (Z k ) of node k as follows: [ ] B k (Z k ) = max U i (Q {Qi B,Qi S i B ) C i (Qi S ) } i N k i N k where we have introduced a new variable Z k for the total nodal power imbalance (supply - demand) at the node Z k = Qi B λ k i N k i N k Q S i The optimisation of the benefit function B k (Z k ) yields the optimal dispatch for the consumers and generators at node k under the constraint that this dispatch leads to a net injection Z k at this node. The parameter λ k gives the change in the objective function when we relax the respective constraint - i.e. the marginal price at this node. 34

Spoiler: Full optimisation problem Note: the values of the Z k are not yet fixed by the scheme. Now we fix the values by maximising total economic welfare given constraints for the nodal injections (determined by the transmission constraints): [ ] max B k (Z k ) {Z k } subject to k Z k = 0 λ k h l ({Z k }) d l µ l with [ B k (Z k ) = max U i (Q {Qi B,Qi S i B ) } i N k subject to Z k = i N k Q S i i N k C i (Q S i ) ] i N k Q B i λ k 35

Efficient dispatch in a two-node system with constraints

Example [The following example is taken from the book by Strbac and Kirschen.] Consider two nodes representing regions, each with different total demand, using different types of generators: First node: Fixed demand Q1 B = 500 MW. The (inverse) supply function for the generators is given by π 1 = MC 1 = 10 + 0.01Q 1 [e/mwh] Second node: Fixed demand Q2 B = 1500 MW. The (inverse) supply function for the generators is given by π 2 = MC 2 = 13 + 0.02Q 2 [e/mwh] For simplicity we assume that at both nodes the total generation limit is 5 GW. Transmission line from node 1 to node 2 with capacity K. 37

Example: Separate markets Transmission capacity K = 0: First node: Fixed demand Q B 1 = 500 MW. The competitive price is λ 1 = MC 1 (Q B 1 ) = 10 + 0.01 500 = 15 [e/mwh] Second node: Fixed demand Q B 2 = 1500 MW. The competitive price is λ 2 = MC 2 (Q B 2 ) = 13 + 0.02 1500 = 43 [e/mwh] 38

Example: Separate markets 70 60 Supply curves Local demand 50 Cost [ /MWh] 40 30 20 10 0 costs 1 costs 2 Q i = Q B i 39

Example: Single market Transmission capacity K = : There is now a total demand Q B = Q1 B + QB 2, leading to a single market clearing price λ. The generators at node 1 and node 2 adjust their output such that under the constraint that MC 1 (Q 1 ) = MC 2 (Q 2 ) = λ, Q 1 + Q 2 = Q B This leads to Q 1 = 1433 MW, Q 2 = 567 MW, and λ = 24.33 e/mwh. 40

Example: Single market 70 60 Supply curves Local demand 50 Cost [ /MWh] 40 30 20 10 0 costs 1 costs 2 Q B i Q i Q B i 41

Example: Single market The power flow F from node 1 to node 2 is given by F = ( Q 1 Q1 B ) ( = Q2 Q2 B ) = (1433 500) MW = (567 1500) MW = Z 1 = Z 2 = 933 MW with {Z 1, Z 2 } = {933 MW, 933 MW} denoted as the injection pattern. From the balancing condition it follows Z 1 + Z 2 = 0. We call a node with Z > 0 a source, and a node with Z < 0 a sink. If Z 1 > Z 2, we have a flow from node 1 to node 2, if Z 2 > Z 1, we have a flow from node 2 to node 1 (source to sink). 42

Example: Single market Separate markets Single market Q B 1 [MW] 500 500 Q 1 [MW] 500 1433 Z 1 [MW] 0 +933 λ 1 [e/mwh] 15 24.33 Q B 2 [MW] 1500 1500 Q 2 [MW] 1500 567 Z 2 [MW] 0-933 λ 2 [e/mwh] 43 24.33 F 1 2 [MW] 0 933 i λ i Q i [e] 72000 48660 i λ i Q B i [e] 72000 48660 43

Example: constraint transmission Two nodes representing two regions, each with different total demand, using different types of generators: First node: Fixed demand Q1 B = 500 MW. The (inverse) supply function for the generators is given by π 1 = MC 1 = 10 + 0.01Q 1 [e/mwh] Second node: Fixed demand Q2 B = 1500 MW. The (inverse) supply function for the generators is given by π 2 = MC 2 = 13 + 0.02Q 2 [e/mwh] For simplicity we assume that at both nodes the total generation limit is 5 GW. Transmission line from node 1 to node 2 with capacity K = 400 MW. 44

Example: Constrained single market 70 60 Supply curves Local demand 70 60 Supply curves Local demand 50 50 Cost [ /MWh] 40 30 Cost [ /MWh] 40 30 20 20 10 0 costs 1 costs 2 Qi = Qi B 10 0 costs 1 costs 2 Q B i Qi Q B i 70 60 Supply curves Local demand 50 Cost [ /MWh] 40 30 20 10 0 costs 1 costs 2 Q B i Qi Q B i 45

Example: Constrained market Transmission capacity K = 400 MW: The transmission capacity is less than the power flow occuring for a single market with unconstrained transmission. The (cheaper) generators at node 1 export power until the line is congested. The (more expensive) generators then cover the remaining load at node 2. MC 1 (Q B 1 + K) = λ 1 MC 2 (Q B 2 K) = λ 2 This leads to Q 1 = 900 MW, λ 1 = 19 e/mwh, Q 2 = 1100 MW, and λ 2 = 35 e/mwh. 46

Example: Constrained market 70 60 Supply curves Local demand 50 Cost [ /MWh] 40 30 20 10 0 costs 1 costs 2 Q B i Q i Q B i 47

Example: Constrained market Q B 1 Separate markets Single market Constrained market [MW] 500 500 500 Q 1 [MW] 500 1433 900 Z 1 [MW] 0 +933 +400 λ 1 [e/mwh] 15 24.33 19 Q B 2 [MW] 1500 1500 1500 Q 2 [MW] 1500 567 1100 Z 2 [MW] 0-933 -400 λ 2 [e/mwh] 43 24.33 35 F 1 2 [MW] 0 933 400 i λ i Q i [e] 72000 48660 55600 i λ i Q B i [e] 72000 48660 62000 48

Locational marginal pricing Due to the congestion of the transmission line, the marginal cost of producing electricity is different at node 1 and node 2. The competitive price at node 2 is higher than at node 1 this corresponds to locational marginal pricing, or nodal pricing. Since consumers pay and generators get paid the price in their local market, in case of congestion there is a difference between the total payment of consumers and the total revenue of producers this is the merchandising surplus or congestion rent, collected by the market operator. For each line it is given by the price difference in both regions times the amount of power flow between them: Congestion rent = λ F 49

Spoiler: LMP in a meshed network Source: PyPSA (Python for Power System Analysis) 50

Redispatch Another way to handle congestion is to correct the single market outcome retrospectively using redispatch. Consider the previous example with line capacity K = 400 MW. Single market result: Q 1 = 1433 MW, Q 2 = 567 MW, market price λ = 24.33 e/mwh, power flow 933 MW. System operator has to adjust the dispatch: Q 1 = 533 MW Q 2 = +533 MW 51

Redispatch 70 60 Supply curves Local demand 50 Cost [ /MWh] 40 30 20 10 0 costs 1 costs 2 Q B i Cost of redispatch: 0.5 (35 19) 533 = 4264 [e/mwh] 52

Redispatch vs. Nodal pricing 70 60 50 Supply curves Local demand 70 60 50 Supply curves Local demand Cost [ /MWh] 40 30 Cost [ /MWh] 40 30 20 20 10 0 costs 1 costs 2 Q B i 10 0 costs 1 costs 2 Q B i Qi Q B i Note that the cost of dispatch for the generators is identical for redispatch (left) and nodal pricing (right). 53

Redispatch in Germany Source: Bundesnetzagentur/ Bundeskartellamt 54

Redispatch in Germany Source: Bundesnetzagentur/ Bundeskartellamt 55

Representing network constraints in meshed networks

The DC Load Flow model Modelling the power flows in the transmission grid accurately requires to solve the physical AC equations, which is complicated. For stable network operation, the DC Load Flow model is a reasonable approximation. In this model, the power flow F l over a link l is a linear function of the nodal imbalances Z k : F l = k H lk Z k, or in matrix notation (with F the vector of power flows, and Z the injection pattern) F = HZ. The matrix H is denoted as the power transfer distribution factors (PTDF) matrix. It is calculated from the topology of the network and the physical properties of the connections (impedance) via Kirchhoff s Current and Voltage law (KCL, KVL). 57

Some degrees of freedom for the PTDF matrix For a balanced injection pattern the nodal imbalances sum up to zero: Z k = 0. k For balanced injection patterns we have some degrees of freedom for the PTDF matrix entries: F l = k = k = k (H lk + c l ) Z k H lk Z k + c l ( k H lk Z k. Z k ) Adding the constant c l to all entries of row l does not change the result, that is instead of the PTDF matrix H lk we can also use the PTDF matrix H lk + c l with arbitrary values of c l. 58

The reference node From any PTDF matrix H lk we can construct a PTDF matrix H (n) lk has zero entries H (n) ln = 0 in row n: H (n) lk = H lk H ln. In this case node n is denoted as the reference node. which 59

Example: Two nodes One line with label l = 1 from node 1 to node 2, which have nodal imbalances Z 1 and Z 2. Choose node 2 as the reference node. The PTDF matrix is a 1 2 matrix with entries ( ) H = 1 2 1 0 The power flow on line 1 is calculated as F 1 = H 11 Z 1 + H 12 Z 2 = Z 1. 60

Example: Three nodes I Reference node 3. ( ) H = 1 3 1 0 0 2 3 0 1 0 Power flows: F 1 3 = Z 1 F 2 3 = Z 2 61

Example: Three nodes II Reference node 3. 1 2 1/3 1/3 0 H = 1 3 2/3 1/3 0 2 3 1/3 2/3 0 Power flows: F 1 2 = 1 3 (Z 1 Z 2 ) F 1 3 = 1 3 (2Z 1 + Z 2 ) F 2 3 = 1 3 (Z 1 + 2Z 2 ) 62

Example: Three nodes III Remark: Different line impedances (reactances) change the PTDF matrix and thus the power flow! 1 2 2/5 1/5 0 H = 1 3 3/5 1/5 0 2 3 2/5 4/5 0 Power flows: F 1 2 = 1 5 (2Z 1 Z 2 ) F 1 3 = 1 5 (3Z 1 + Z 2 ) F 2 3 = 2 5 (Z 1 + 2Z 2 ) 63

Transmission constraints Thermal limits as transmission constraints for the optimisation problem can be represented as general inequality constraints: h l ({Z k }) d l. In our treatment, we model these constraints as capacity limits, that is as upper boundaries for the power flow on a link: F l (Z k ) K l F l (Z k ) K l Using the PTDF matrix, this reads [ ] H lk Z k K l k [ ] H lk Z k K l k 64

Feasible injections The set of feasible injections represents the collection of all balanced injection patterns {Z k } which satisfy the transmission constraint equations [ ] H lk Z k K l k [ ] H lk Z k K l k 65

Example: Three nodes I Reference node 3. ( ) H = 1 3 1 0 0 2 3 0 1 0 Z2 30 20 10 0 K2 3 Power flows: 10 K1 3 F 1 3 = Z 1 F 2 3 = Z 2 20 30 15 10 5 0 5 10 15 Z1 Capacity limits: K 1 3 = 10 MW K 2 3 = 20 MW 66

Example: Three nodes II Reference node 3. 1 2 1/3 1/3 0 H = 1 3 2/3 1/3 0 2 3 1/3 2/3 0 Power flows: F 1 2 = 1 3 (Z 1 Z 2 ) F 1 3 = 1 3 (2Z 1 + Z 2 ) F 2 3 = 1 3 (Z 1 + 2Z 2 ) Capacity limits: (K 1 2, K 2 3, K 1 3 ) = (10, 20, 10) MW Z2 30 20 K2 3 10 0 K1 3 10 20 30 20 15 10 5 0 5 10 15 20 Z1 67

Example: Efficient market operation in a 3-node system with constrained transmission

An exemplary 3-node system Source: Kirschen and Strbac 69

An exemplary 3-node system Example taken from Kirschen and Strbac, Chapter 6. Generator Capacity Marginal cost (MW) (e/mwh) A 140 7.5 B 285 6 C 90 14 D 85 10 Line Reactance Capacity (p.u.) (MW) 1 2 0.2 126 1 3 0.2 250 2 3 0.1 130 70

Power flows and feasible injections 1000 500 Z2 0 1 2 2/5 1/5 0 H = 1 3 3/5 1/5 0 2 3 2/5 4/5 0 Power flows: F 1 2 = 1 5 (2Z 1 Z 2 ) Z2 500 1000 1000 500 0 500 1000 Z1 300 200 100 0 100 F 1 3 = 1 5 (3Z 1 + Z 2 ) F 2 3 = 2 5 (Z 1 + 2Z 2 ) 200 300 400 200 0 200 400 Z1 71

Economic dispatch Market price: λ = 7.5 e/mwh QA S = 125 MW QB S = 285 MW QC S = 0 MW QD S = 0 MW K 1 2 = 126 MW K 1 3 = 250 MW K 2 3 = 130 MW 300 200 100 F 1 2 = 1 5 (2Z 1 Z 2 ) = 156 MW Z2 0 100 F 1 3 = 1 5 (3Z 1 + Z 2 ) = 204 MW 200 300 400 200 0 200 400 Z1 F 2 3 = 2 5 (Z 1 + 2Z 2 ) = 96 MW Z 1 = 360 MW, Z 2 = 60 MW. 72

Economic redispatch QA S = 125 MW QA S = 50 MW QB S = 285 MW QB S = 285 MW QC S = 0 MW QC S = 0 MW QD S = 0 MW QD S = 75 MW Redispatch cost: 187.5 e/h 300 200 100 K 1 2 = 126 MW K 1 3 = 250 MW K 2 3 = 130 MW F 1 2 = 1 5 (2Z 1 Z 2 ) = 126 MW Z2 0 100 200 300 400 200 0 200 400 Z1 F 1 3 = 1 5 (3Z 1 + Z 2 ) = 159 MW F 2 3 = 2 5 (Z 1 + 2Z 2 ) = 66 MW Z 1 = 275 MW, Z 2 = 60 MW. 73

Nodal prices The nodal marginal price is equal to the minimal system cost of supplying an additional megawatt of load at this node. QA S = 50 MW QB S = 285 MW QC S = 0 MW QD S = 75 MW Nodal prices: λ 1 = c A = 7.5 e/mwh λ 3 = c D = 10 e/mwh λ 2 = 1.5 c D 0.5 c A = 11.25 e/mwh K 1 2 = 126 MW K 1 3 = 250 MW K 2 3 = 130 MW F 1 2 = 1 5 (2Z 1 Z 2 ) = 126 MW F 1 3 = 1 5 (3Z 1 + Z 2 ) = 159 MW F 2 3 = 2 5 (Z 1 + 2Z 2 ) = 66 MW 74

Nodal prices Economic operation of the three-node system using nodal pricing. Node 1 Node 2 Node 3 System Consumption (MW) 50 60 300 410 Production (MW) 335 0 75 410 Nodal marginal price (e/mwh) 7.5 11.25 10 - Consumer payments (e/h) 375 675 3000 4050 Generator revenue (e/h) 2512.5 0 750 3262.5 Congestion rent (e/h) 787.5 75

Nodal prices Congestion rent: Connection Flow From price To price Surplus (MW) (e/mwh) (e/mwh) (e/h) 1 2 126 7.5 11.25 427.5 1 3 159 7.5 10 397.5 2 3 66 11.25 10-82.5 Total 787.5 Note the counter-intuitive flow from node 2 (higher price) to node 3 (lower price)! 76

Example slightly changed The nodal marginal price is equal to the minimal system cost of supplying an additional megawatt of load at this node. QA S = 47.5 MW QB S = 285 MW QC S = 0 MW QD S = 77.5 MW Nodal prices: K 1 2 = 126 MW K 1 3 = 250 MW K 2 3 = 65 MW F 1 2 = 1 5 (2Z 1 Z 2 ) = 125 MW λ 1 = c A = 7.5 e/mwh λ 3 = c D = 10 e/mwh λ 2 = 2 c A 1 c D = 5 e/mwh F 1 3 = 1 5 (3Z 1 + Z 2 ) = 157.5 MW F 2 3 = 2 5 (Z 1 + 2Z 2 ) = 65 MW The nodal marginal price at node 2 is lower than the marginal cost of any generator! 77

Negative nodal prices Generator A has marginal costs 60e/MWh, generator B has marginal costs 30e/MWh. The line between E and D is constrained to 25 MW. The additional load of 10 MW at node E reduces the system cost by 300 e/h, so λ E = 30 e/mwh! Source: Harris 78

Copyright Unless otherwise stated the graphics and text is Copyright c Mirko Schäfer and Tom Brown, 2016. We hope the graphics borrowed from others have been attributed correctly; if not, drop a line to the authors and we will correct this. The source L A TEX, self-made graphics and Python code used to generate the self-made graphics are available on the course website: http://fias.uni-frankfurt.de/~brown/courses/electricity_ markets/ The graphics and text for which no other attribution are given are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. cba 79