Javier Contreras Sanz- Universidad de Castilla-La Mancha Jesús María López Lezama- Universidad de Antioquia Antonio Padilha-Feltrin- Universidade Estadual Paulista Jose Ignacio Muñoz-Universidad de Castilla-La Mancha
Introduction Distributed generation General considerations Power flow approximations Inner optimization problem Outer optimization problem Bilevel model Test and results Final 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: P nm.( ) V V V n n m Z nm loss nm = nm + mn P P P Active power flow between nodes n,m P loss nm = ( V V ) 2 n Z nm 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. se gk Min Δ tρ () t P () t + ΔtCp P () t dg gj P, P, V n se dg k gk j gj k K t T j J t T Energy purchaded on the wholesale energy market through the substations. Energy purchased from the DG units. 12
Subject to: Power balance ( ) ( ) Vn(). t Vn() t Vm() t Vm(). t Vm() t Vn() t + Z Z m Ωn nm m Ωn m> n m< n ( V () ()) 2 n t Vm t + Pgn () t Pdn () t = 0 n N, t T : π ( n,) t Z m Ωn nm Power flow limits ( ) Vn(). t Vn() t Vm() t Pnm Pnm; l nm L, t T : φ(l nm, t); φ(l nm, t) Z nm mn 13
Voltage limits V V() t V; n N, t T : ω(,); nt ω(,) nt n n n DG active power limits dg dg dg gj gj gj P P () t P ; j J, t T : β( j,); t β( j,) t Substation active power limits se se se gk gk gk P P () t P ; k K, t T : δ ( k,); t δ( k,) t 14
The outer optimization problem corresponds to the owner of the DG who must maximize profits. Contract price Energy cost ( ) Max Δt Cp c P () t Cp j t T Subject to j J : dg j j gj dg min dg dg max gj gj gj P P () t P ; j J, t T i I { } B = ndg; B 0,1 i i 15
Both problems can be expressed as a bilevel programming problem: P, P, V ( ) Max Δt Cp c P () t Cp j Subject to : B = ndg; B 0,1 { } Min Δ tρ () t P () t + ΔtCp P () t se gk i I t T dg gj i j J n Subject to : dg j j gj i se dg k gk j gj k K t T j J t T 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
Min f ( x) x Subject to : hx ( ) = 0 gx ( ) 0 KKT optimality conditions Stationary condition of the Lagrangean 64444444744444448 P f( x) + λ h ( x) + μ g ( x) = 0 p p= 1 q= 1 h ( x) = 0 p = 1,..., P g ( x) 0 q = 1,..., Q μ g ( x) = 0 q = 1,..., Q q p q q p Primal feasibility condition Complementarity condition Q 644474448 64444 744448 Dual feasibility condition 644474448 0 Q μ q = 1,..., q q q 18
Substituting the inner optimization problem by its KKT optimality conditions, the following single-level optimization problem is obtained: P, P, V, Cp ( ) Max Δt Cp c P () t se dg gk gj n j Subject to : i I t T j J { } B = ndg; B 0,1 i i dg j j gj Primal feasibility conditions 19
Stationary condition of the Lagrangean: ( Vn() t Vm() t ) V () t ( Vn() t Vm() t ) m + + Z m n nm Z m n mn Z Ω Ω m Ωn nm m> n m< n + ω( nt, ) ω( nt, ) + φ( l, t) Vm () t φ( l, t) φ( l, t) Z ( Vn() t Vm() t ) Vm () t + φ( lnm, t) n> m = 0; n N, t T Z nm n< m nm n> m nm n< m nm nm Z nm ( Vn() t Vm() t ) Z nm 20
Complementarity and dual feasibility conditions: Cp π( n, t) + β( j, t) β( j, t) = 0; j J, t T j ρ () t π( n,) t + δ ( k,) t δ( k,) t = 0; k K, t T k ( ) n n ( ) ω( nt, ) V( t) V = 0; ω( nt, ) 0; n N, t T ω( nt, ) V( t) + V = 0; ω( nt, ) 0; n N, t T n ( dg dg ) gj gj ( dg dg ) gj gj ( se se ) gk gk ( se se ) gk gk n β( j, t) P ( t) P = 0; β( j, t) 0; j J, t T β ( jt, ) P ( t) + P = 0; β ( j, t) 0; j J, t T δ ( kt, ) P ( t) P = 0; δ ( kt, ) 0; k K, t T δ( kt, ) P ( t) + P = 0; δ( kt, ) 0; k K, t 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 Load (MW) 55 50 45 40 35 30 25 20 40 60 80 100 Time (%) Energy market price ( /MWh) 65 60 55 50 45 40 35 20 40 60 80 100 Time (%) 23
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. 24
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 Locational marginal price ( /MWh) 75 70 65 60 1 2 3 4 5 6 7 8 9 10 Bus 25
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. 1.06 1.04 Voltage (p.u) 1.02 1 0.98 0.96 0.94 0.92 1 2 3 4 5 6 7 8 9 10 Bus 26
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 10 Contract price: 63.38/MWh Set of possible nodes where to allocate the DG unit. 27
1.06 Voltage profile of the distribution system for the peak hour with and without DG. Voltage (p.u) 1.04 1.02 1 0.98 0.96 0.94 Without DG With DG 0.92 1 2 3 4 5 6 7 8 9 10 Bus Locational marginal prices of the distribution system for the peak hour with and witout DG. Locational marginal price ( /MWh) 75 70 65 Without DG With DG 60 1 2 3 4 5 6 7 8 9 10 Bus 28
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. 2 x 105 1.8 Profit ( ) 1.6 1.4 1.2 1 59 60 61 62 63 64 65 66 67 Price offer ( /MWh) 29
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 9 Contract price DG1: 54.83/MWh DG2: Bus 10 Contract price DG2: 55.82/MWh Set of possible nodes where to allocate the DG units 30
75 With an increasing penetration of DG the locational marginal prices tend to decrease, as shown in the figure. Locational marginal price ( /MWh) 70 65 Without DG Only DG1 DG1 and DG2 60 0 2 4 6 8 10 Bus 1.06 Voltage profile also improves with an increasing penetration of DG, as shown in the figure. Voltage (p.u) 1.04 1.02 1 0.98 0.96 0.94 Without DG Only DG1 DG1 and DG2 0.92 1 2 3 4 5 6 7 8 9 10 Bus 31
Table 1. Optimal location and contract prices Case Bus location Contract price Profits ( ) One DG unit 10 63.38 187,521 Two DG units 9 10 54.83 55.82 101,545 122,359 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. 32
Total payments and energy losses of the DisCo 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 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. 33
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-dispatchable technologies, as well as the inclusion of stochasticity in the model. 34