Electricity prices and tariffs to keep everyone happy: a framework for fixed and nodal prices coexistence in distribution grids with optimal tariffs for investment cost recovery
aa r X i v : . [ ec on . GN ] J a n Electricity prices and tariffs to keep everyone happy: a framework for compatible fixed andnodal structures to increase efficiency
Iacopo Savelli a, ∗ , Thomas Morstyn b a Smith School of Enterprise and the Environment, University of Oxford, South Parks Road, Oxford, UK b Department of Engineering Science, University of Oxford, Parks Road, Oxford, UK
Abstract
Some consumers, such as householders, are unwilling to face volatile electricity prices, and perceive as unfair price differentiationsbased on location. For these reasons, nodal prices in distribution networks are rarely employed. However, the increasing availabilityof renewable resources in distribution grids, and emerging price-elastic behaviour, pave the way for the effective introduction ofmarginal nodal pricing schemes in distribution networks. The aim of the proposed framework is to show how traditional non-flexibleconsumers can coexist with flexible users in a local distribution area, where the latter pay nodal prices whereas the former are chargeda fixed price, which is derived by the underlying nodal prices. In addition, it determines how the distribution system operator shouldmanage the local grid by optimally determining the lines to be expanded, and the collected network tariff levied on network users,while accounting for both congestion rent and investment costs. The proposed framework is formulated as a non-linear integer bilevelmodel, which is then recast as an equivalent single optimization problem, by using integer algebra and complementarity relations.The power flows in the distribution area are modelled by resorting to a second-order cone relaxation, whose solution is exact forradial networks under mild assumptions. The final model results in a mixed-integer quadratically constrained program, which can besolved with off-the-shelf solvers. Numerical test cases based on a 5-bus and a 33-bus networks are reported to show the effectivenessof the proposed method.
Keywords: distribution network; nodal prices; network tariffs and charges; fixed cost recovery; flexible consumers; ∗ Corresponding author
Email addresses: [email protected] (Iacopo Savelli), [email protected] (Thomas Morstyn)
Preprint submitted to Elsevier January 14, 2020 omenclature
Sets and Indices L set of distribution lines M set of possible lumpy capacity expansions for thedistribution lines N set of distribution nodes, with N = { , . . . , N } ,where N is the total number of nodes N + set of all nodes except the slack bus (node zero), i.e. N + = N \{ }T set of time periodsΩ Dt,n set of flexible consumers at time t in node n Ω Gt,n set of generators at time t in node n Parameters a i,j,m conductance of the line ( i, j ) ∈ L c pt, price of active power traded with the transmissiongrid at time tc qt, price of reactive power traded with the transmissiongrid at time tc upt, price of upward reserve at time tc downt, price of downward reserve at time tc dt,n,k demand bid price for flexible consumers k in node n at time tc gt,n,k supply bid price for generators k in node n at time td p,maxt,n,k maximum quantity of active power demanded byflexible consumer k in node n at time td p,mint,n,k minimum quantity of active power demanded byflexible consumers k in node n at time td q,maxt,n,k maximum quantity of reactive power demanded byflexible consumers k in node n at time td q,mint,n,k minimum quantity of reactive power demanded byflexible consumers k in node n at time tD t,n fixed demand required by non-flexible consumers innode n at time te i,j,m negative of the susceptance for the line ( i, j ) ∈ L F i,j,m lumpy capacity expansion m ∈ M for line ( i, j ) ∈ L F maxi,j maximum power flow leaving node i over the line( i, j ) F mini,j maximum power flow leaving node j over the line( i, j ) g p,maxt,n,k maximum quantity of active power supplied by gen-erators k in node n at time tg p,mint,n,k minimum quantity of active power supplied by gen-erators k in node n at time tg q,maxt,n,k maximum quantity of reactive power supplied bygenerators k in node n at time tg q,mint,n,k minimum quantity of reactive power supplied bygenerators k in node n at time tK vari,j,m variable cost of expansion m ∈ M for the line ( i, j ) ∈L K fixi,j,m fixed cost of expansion m ∈ M for the line ( i, j ) ∈ L K op non-investment costs paid by the distribution oper-ator during the considered time periods K tot total budget for investments v maxt,i maximum voltage magnitude in node n at time t , inp.u. v mint,i minimum voltage magnitude in node n at time t , inp.u. ψ n sensitivity parameter accounting for the location ofnode n with respect to node zero Variables d pt,n,k allocated active power demand for flexible consumer k in node n at time td qt,n,k allocated reactive power demand for flexible con-sumer k in node n at time tg pt,n,k allocated active power for generator k in node n attime tg qt,n,k allocated reactive power for generator k in node n at time tp t,n active power injection in node n at time tq t,n reactive power injection in node n at time tr upt,n upward reserve provided in node n at time tr downt,n downward reserve provided in node n at time tu i,j,m binary variable equal to one if the expansion m ∈ M is applied to line ( i, j ) ∈ L W iit,i voltage magnitude squared at node i at time tW ij,pt,i,j real component of the product V i ¯ V j , where V i is thevoltage at node i and ¯ V j the complex voltage conju-gate at node jW ij,qt,i,j imaginary component of the product V i ¯ V j , where V i is the voltage at node i and ¯ V j the complex voltageconjugate at node jτ network tariff π D fixed price charged to non-flexible consumers π pt,n nodal price for active power2 . Introduction By 2050, half of all European Union citizen could produce energy from renewable energy resources, with a total electricitygeneration capable to satisfy the 45% of the European energy demand [1]. This shift from fossil fuel to low-carbon emission sourcesis also desirable, as it could contribute to achieve greenhouse gas-reduction targets [2]. Householders are expected to become moreaware and engaged into the energy system [3]. For example, by providing services to the electricity grid (such as reserve), and bydynamically adapting their consumption through demand response programs and more price-sensitive behaviours [4]. However, thedeployment of such large quantity of energy at the local level will require the development of adequate market mechanisms with achange from centralised to decentralized schemes [5], and improvements in pricing approaches to effectively reflect these dynamicsat the distribution level [6]. In addition, the current electricity network should be reconsidered, as the traditional design assumespower flows are from large generators connected upstream at transmission level, to final consumers downstream at distribution level[7]. By contrast, the future energy system will experience both active consumers and small local generators willing to trade energyamong themselves, and to provide services and sell electricity up to main grid [8]. Therefore, both novel market mechanisms, andnew network planning strategies must be developed in order to allow local consumers and distributed generations to be activelyengaged and empowered.In the literature, a seminal contribute in network planning is due to Paul Joskow and Jean Tirole [9]. They present an extensiveanalysis of merchant transmission investments, with a focus on real-world issues, as for example the effect of lumpiness in transmissionexpansions. That is, an electricity line can be expanded only by discrete steps. Recently, the same problem in connection with theeffect network charges has been investigated in [10]. In particular, a new scheme termed ex-ante dynamic network tariff is proposed,where the effect of network charges is explicitly accounted during a market clearing process. Greenhouse gas emission targets canalso play a leading role in shaping the future energy system. In this sense, reference [11] presents a two-stage stochastic optimizationmodel to determine the optimal mix of generation and transmission capacity, while incorporating environmental constraints. Inparticular, two different carbon policies are considered to decarbonize electricity production. The first is based on an explicit capon carbon emissions, whereas the latter is modelled as a Pigouvian-tax levied upon fossil fuel generators. Along this line reference[12] compares 36 different scenarios to evaluate policies and pathways to decarbonize the energy sector, by resorting to a least costoptimization model for a simplified European power sector with price-inelastic demands. Uncertainty is considered in [13], where astochastic two-stage optimisation for transmission planning is presented. Several economic, technology, and regulatory scenarios areconsidered to depict uncertain future outcomes. Some metrics such as the estimate the value of information, and the cost of ignoringuncertainty are also reported.A crucial problem in transmission planning is due to the full recovery of the investment costs, also termed revenue adequacycondition . In particular, the congestion rent collected by the network operator is not sufficient to guarantee the recovery of thefixed costs [14]. To overcome this issue, network tariffs must be introduced. Reviews of the most common network charge schemesare reported in [15] and [16], where the postage stamp, MW-mile, incremental cost, and marginal cost methods are reviewed andcompared. Along this line, reference [17] analyses the impact of four different tariff schemes on network planning outcomes, byresorting to a long-term transmission expansion problem, which is applied to a realistic case study based on the actual power systemin Chile.The future energy system, where flexible consumers and small generators can dynamically adapt their consumption and generationas prices change, requires the adoption of an efficient and responsive pricing scheme, in particular at the distribution level [7].However, despite the clear evidence of the marginal nodal pricing as a means to guarantee allocative efficiency and welfare increase[18], several countries still model even their wholesale electricity markets as a single-zone market (e.g. Great Britain, France,Germany, Poland, Hungary [19]). Furthermore, to the best of the authors knowledge, nodal prices at the distribution level are notyet implemented anywhere. This leads to a series of inefficiencies at the local level as the prices do not react to congestions by risingtheir level in importing zones and lowering it in exporting zone. As a consequence, a distorted incentive to higher consumption andlower production can be observed, which can translate in greater balancing and reinforcement costs to guarantee grid stability andreliability [7]. Two of main reasons against the deployment of nodal prices are the granularity and the volatility of nodal prices[20]. That is, nodal price can vary from one location to another even within a relatively small local area, and they can fluctuatesignificantly from one period to another, as they reflect the actual grid congestions. For this reason, fixed prices can be preferredby non-flexible traditional consumers [21], as they are unprepared for forecasting volatile prices or even predicting their own futureconsumptions [22]. By contrast, flexible consumers and generators can benefit from time periods with favourable energy prices andincrease their revenues by exploiting their flexibility [23]. This motivates the need for new mechanisms such that nodal prices forflexible users and fixed prices for traditional consumers could coexist effectively at the distribution level. Furthermore, to achievethe full benefit of flexibility, the distribution network operator needs to explicitly account for the additional power flows from flexibleusers when making long-term network investment decisions.The aim of the proposed framework is to address emerging economic problems in a local distribution area related to the coexistenceof traditional consumers, flexible prosumers, and a distribution system operator (DSO) who manages the local electricity network. Inparticular, the main objectives can be summarized as follows. First, we show how flexible prosumers can provide reserve to the maingrid under a nodal pricing scheme, while coexisting with traditional consumers paying fixed prices (that do not vary by location)for their energy demand. The importance of nodal pricing relies on the efficient allocation of scarce resources including energy, linecapacity, and technical constraints [14]. In particular, marginal nodal pricing allows both the cost of congestions and the effect ofpower losses in distribution lines to be internalised into the electricity prices [7]. Then, a fixed price for the traditional consumersbased on the underlying nodal prices is determined, which therefore embodies their efficiency and cost reflectiveness. In this settings,3he DSO operates the local distribution grid, and it also has the capability to expand the network by building new lines, subject tosome fixed and variable costs. A line expansion affects the power flows by increasing the maximum power transmitted through theline and by changing the line admittance, which in turn affects the power losses. The full recovery of the investment costs for theDSO is guaranteed by determining an optimal network tariff levied on grid users, while accounting for the collected congestion rentand expansion costs. The network tariff is charged to all users proportionally to their grid access capacity. The proposed frameworkis structured as a non-linear integer bilevel model. The lower level problem represents a market clearing problem, where the electricalgrid is modelled by using a second-order cone relaxation, whose solution is exact for radial networks under mild conditions [24]. Theupper level problem represents an optimal long-term distribution grid planning problem, which determines the lines to be built orexpanded, and the appropriate network tariff to be collected. The bilevel model is then recast as a single, equivalent mixed-integerquadratically constrained program (MIQCP) by resorting to complementarity relations and integer algebra, which can be solvedwith off-the-shelf solvers. To summarize, the main novelties of the proposed framework are:1. computation of the fixed price charged to inelastic consumers according to the underlying distribution nodal marginal prices,while ensuring monetary payments are in equilibrium;2. provision of flexibility (in terms of both active and reactive power) as reserve to the main grid, furnished by flexible consumersand producers with price-elastic behaviour.3. market clearing formulation for the distribution network within a long-term investment planning problem by resorting to abilevel model, which is then recast as an equivalent MIQCP problem;4. determination of the optimal network tariff collected by the network operator and levied on distribution grid users, whileaccounting for congestion rent and grid expansion costs, and subject to the exact modelling of the distribution power flows byresorting to a second-order cone relaxation;The remaining sections are structured as follows. Section 2 describes in detail the proposed bilevel framework. Section 3 describeshow the bilevel model can be recast as an equivalent MIQCP model by resorting to complementarity relations and integer algebra.Section 4 reports four test cases base on a 5-bus network to highlight the main properties of the presented model, and a numericaltest based on a widely used 33-bus distribution network. Finally, Section 5 outlines the main conclusions.
2. The proposed framework
Figure 1: Distribution network with 33 buses (or nodes), where node zero represents the local power substation (which is connected upstream with thetransmission grid), whereas the remaining nodes downstream depict the access points for local users connected to the distribution grid (e.g. householders,flexible consumers, and generators).
Figure 1 shows an example of distribution grid. The local substation is located at node zero, and is connected upstream with thetransmission grid, whereas the remaining nodes downstream depict the access points for local users connected to the distributiongrid (e.g. householders, flexible consumers, and generators). In the proposed framework, the users of the network are classifiedas flexible and non-flexible. Non-flexible users are traditional householders, who demand fixed quantities regardless of the price,i.e. inelastic demand [25]. By contrast, flexible consumers/generators have elastic demand/supply. They bid different quantitiesat different prices, and are capable to exploit their flexibility by selling upward and downward reserve capacity to the main grid.Upward reserve is the amount of additional power that can be injected in a node, whereas downward reserve is the further amount ofpower that can be withdrawn from a node. In the proposed model, upward reserve can be provided by generators using their sparepower capacity, and by flexible consumers decreasing their consumption. Similarly, downward reserve can be provided by generatorsdecreasing their power output, and by flexible consumers increasing their consumption.The approach required to model power flows in distribution grids differs significantly from the one usually employed in transmissionnetworks. High-voltage transmission networks can be approximately described by using the so-called DC load flow method [26].The key advantage of this approach is that power flows are represented by linear relations. It is based on three assumptions:(i) the line resistances are negligible; (ii) the per-unit voltage magnitude in each node is one; (iii) and the voltage phase angledifferences across line ends are relatively small. These assumptions have important consequences. The first assumption impliesthat power losses are completely neglected. The second one implies that reactive power flowing across lines is always zero. Theseassumptions approximately hold for high-voltage transmission lines, but do not hold in distribution grids. That is, both power4osses and reactive power must be considered in describing low and medium voltage distribution networks. This requires non-linearmodelling approaches. In the following, we consider distribution networks with a radial topology, as in Figure 1. In this case, thepower flows at distribution level can be represented by resorting to a second-order cone relaxation, which ensures the exactness ofthe obtained solutions for these networks under mild technical assumptions [27, 24], which hold in all the examples and test casesanalysed in the following sections.
The proposed framework is structured as a bilevel problem, which involves the optimization of two nested mathematical programs,termed upper and lower level problems [28]. Formally, a bilevel model is defined as follows:max u ∈ U F ( u, x ∗ ) (1a)s.t. x ∗ ∈ arg max x ∈X f ( x ; u ) , (1b)where (1a) represents the upper level problem, whereas (1b) characterizes the lower level problem. The functions F and f are theirrespective objective functions, and U , X are constraint sets. A key property of bilevel programming is that all the upper level decisionvariables, labelled as u in (1a)-(1b), enter the lower level as fixed parameters. The variables x ∗ represent the optimal solution of thelower level problem, which depends on the upper level variables u , i.e., x ∗ = x ∗ ( u ). However, for ease of reading this dependenceis usually not formally expressed. Historically, bilevel programming has been extensively used in the field of game theory to modelStackelberg games [29]. However, in power system economics bilevel programs (as well as complementary problems [30]), are notusually used to actually build a game, but as a means of accessing dual variables [31], which are related to the electricity nodalprices under the marginal pricing framework [32, 33]. Figure 2: Bilevel framework. The upper level model represents a long-term investment planning problem accounting for participants’ surplus, networktariff, investment costs, and revenues from reserve provision. It optimally determines the tariff τ levied on network users, the grid expansion level u i,j,m ,and the fixed price π D paid by traditional inelastic consumers. Then, the lower level problem clears the market given the selected grid expansion u i,j,m ,while accounting for the exact power flows at distribution level. The optimal solution of the lower level problem, x ∗ in (1b), encompasses the allocateddemand and generation quantities d t,n,k , g t,n,k , the power injections p t,n , and the market prices π t,n , which are used in turn by the upper level to compute π D , and to verify the revenue adequacy condition. Figure 2 depicts the structure of the proposed bilevel framework. The upper level problem is a long-term investment planningproblem, which accounts for participants’ surplus, network tariff, reserve provision, and grid investment costs. It determines theunique price π D paid by non-flexible consumers (i.e. the traditional householders), the optimal network tariff τ collected by thenetwork operator, and the optimal grid expansion u i,j,m . Given the selected network expansion u i,j,m , which affects both themaximum power flow over the distribution lines and their admittance, the lower level solves a market clearing problem to obtainthe optimal solution, labelled as x ∗ in (1b). That is, it determines the optimal allocated quantities d t,n,k , g t,n,k , the nodal powerinjections p t,n , and the nodal prices π t,n , for both active and reactive power.5 .2.1. Upper Level This section describes in detail the upper level problem introduced in Section 2.2, which is defined as follows.max τ,π D ,u i,j,m X t ∈T X n ∈N + (cid:16) X k ∈ Ω Dt,n c dt,n,k d pt,n,k ∗ − X k ∈ Ω Gt,n c gt,n,k g pt,n,k ∗ (cid:17) − X t ∈T (cid:16) c pt, p ∗ t, + c qt, q ∗ t, (cid:17) + X t ∈T (cid:16) c upt, X n ∈N + ψ n r upt,n ∗ + c downt, X n ∈N + ψ n r downt,n ∗ (cid:17) − τ X t ∈T X n ∈N + (cid:16) D t,n + X k ∈ Ω Dt,n d p,maxt,n,k + X k ∈ Ω Gt,n g p,maxt,n,k (cid:17) − X ( i,j ) ∈L X m ∈M u i,j,m ( K fixi,j,m + K vari,j,m F i,j,m ) (2a) s.t.π D X t ∈T X n ∈N + D t,n = X t ∈T X n ∈N + π pt,n ∗ D t,n (2b) − X t ∈T (cid:16) X n ∈N π pt,n ∗ p ∗ t,n + c qt, q ∗ t, (cid:17) + τ X t ∈T X n ∈N + (cid:16) D t,n + X k ∈ Ω Dt,n d p,maxt,n,k + X k ∈ Ω Gt,n g p,maxt,n,k (cid:17) ≥ K op + X ( i,j ) ∈L X m ∈M u i,j,m ( K fixi,j,m + K vari,j,m F i,j,m ) (2c) X ( i,j ) ∈L X m ∈M u i,j,m ( K fixi,j,m + K vari,j,m F i,j,m ) ≤ K tot (2d) X m ∈M u i,j,m = 1 ∀ ( i, j ) ∈ L (2e) u i,j,m ∈ H (2f) u i,j,m ∈ { , } , π D ∈ R , τ ≥ x ∗ in (1). The upper level objectivefunction (2a) is composed by five groups of terms enclosed by parentheses, which represent in order: (i) the surplus of flexibleconsumers and producers, bidding different quantities at different prices c dt,n,k and c gt,n,k ; (ii) the revenues (costs) from selling(buying) any excess (deficit) of active and reactive power to (from) the upstream transmission grid at substation level, i.e. at nodezero; (iii) the income from upward r upt,n and downward r downt,n reserve provision to the main grid, paid respectively c upt, and c downt, ,whereas ψ n is a scaling factor to account for the different location between the substation and the node n where the reserve is actuallyinjected into the distribution grid; (iv) the total network tariff levied on all grid users (non-flexible consumers, flexible consumers,and flexible generators) connected to the local distribution grid, paid proportionally to their access capacity; (v) the fixed K fixi,j,m andvariable K vari,j,m investment costs to undertake the expansion m on the line ( i, j ) if a specific expansion is selected, i.e. if u i,j,m = 1.Discount factors are not included for ease of reading. As described in Section 1, power lines can be augmented only by discreteamount, i.e. lumpy expansions, and the parameter F i,j,m represents the additional discrete line capacity increase. As a consequence,the set M of the possible capacity expansions is a finite set. Relation (2b) defines the fixed price π D charged to all non-flexibleconsumers as the average of the distribution nodal prices π pt,n , weighted by their fixed demand quantities D t,n , over all nodes and timeperiods considered. Equivalently, the fixed price π D is the price which ensures that the same total monetary amount is collected as ifthe non-flexible consumers paid nodal prices. As a consequence, constraint (2b) generates implicitly a subsidy between non-flexibleconsumers, where those in lower-priced zones pay more to compensate those in higher-priced zones. Constraint (2c) enforces therevenue adequacy condition. The first terms inside the parenthesis in the left-hand side represent the merchandising surplus [14],where the minus is due to the sign convention used to define the nodal injections as generation minus demand. The merchandisingsurplus represents the difference between the payments made by consumers and the revenues collected by generators, and it is asource of revenues for the network operator. In lossless power flow models (e.g. DC load flow), the merchandising surplus coincideswith the congestion rent [7]. Therefore, the condition (2c) ensures that the income from merchandising surplus and collected tariffsis greater than or equal to the total investment costs paid by the network operator, where the term K op accounts for any residualnon-investment cost. Constraint (2d) enforces an upper bound on the total monetary expenses for grid investments, where K tot is thetotal amount available for the investments. Relation (2e) ensures that for each line ( i, j ) only one expansion m is undertaken, with m ∈ M = { , . . . , } , where m = 0 represents the current line state (i.e. no expansion), whereas m = 1 represents the expansion of100% of the current capacity (i.e. the line is doubled). Finally, the relation (2f) accounts for any additional constraints on investmentdecisions, dictated for example by environmental concerns, policies on investments, strategies on line expansion, or limits due toright-of-ways, where H is the abstract constraint set representing these conditions.6 .2.2. Lower level This section describes in detail the lower level problem sketched in Section 2.2. The purpose of this optimization program isto solve a market clearing problem, given the grid expansion u i,j,m selected by the upper level problem, while providing reserve tothe main grid and exactly modelling the power flows at distribution level. The optimal power flows are modelled by resorting to asecond-order cone relaxation, which is exact for radial networks under mild assumptions on both line impedances and voltage limits,that holds for all the reported test cases (see [24, 27] for a technical discussion on resistance, reactance, and voltage magnitude upperbound requirements). The total power injected in a single node, termed apparent power s t,n ∈ C is a complex number, where thereal part is termed active power p t,n ∈ R , whereas the imaginary part is termed reactive power q t,n ∈ R . That is, s t,n := p t,n + iq t,n where i is the imaginary unit. Similarly, the voltage V t,n in each node n is a complex number. To cast the optimization problem inthe real domain, the following additional definitions are introduced: W iit,i := V t,i ¯ V t,i ∀ t ∈ T , ∀ i ∈ N (3) W ijt,i,j := V t,i ¯ V t,j = W ij,pt,i,j + iW ij,qt,i,j ∀ t ∈ T , ∀ ( i, j ) ∈ L (4) W jit,j,i := V t,j ¯ V t,i = W ji,pt,j,i + iW ji,qt,j,i ∀ t ∈ T , ∀ ( i, j ) ∈ L (5) y i,j,m := a i,j,m − ie i,j,m ∀ ( i, j ) ∈ L , ∀ m ∈ M (6)where y i,j,m is the admittance of the line ( i, j ) as a result of the expansion m , V t,i is the complex voltage in node i at time t , thenotation ¯ V t,i refers to the complex voltage conjugate, W iit,i ∈ R is the voltage magnitude in node n , and W ij,pt,i,j ∈ R , W ij,qt,i,j ∈ R , W ji,pt,j,i ∈ R , W ji,qt,j,i ∈ R , a i,j,m ∈ R , and e i,j,m ∈ R , are auxiliary variables introduced to remove the complex terms. Note that themodelling approach for distribution grids, based on the second-order cone relaxation introduced in [24, 27], requires that the matrix: " W iit,i W ijt,i,j W jit,j,i W iit,j (7)be positive semi-definite [34]. By Sylvester criterion [35], this condition is equivalent to requiring both W iit,i ≥ W ijt,i,j W jit,j,i ≤ W iit,i W iit,j ⇐⇒ ( W ij,pt,i,j (cid:1) + (cid:0) W ij,qt,i,j (cid:1) + (cid:16) W iit,i − W iit,j (cid:17) ≤ (cid:16) W iit,i + W iit,j (cid:17) ⇐⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W ij,pt,i,j W ij,qt,i,j ( W iit,i − W iit,j ) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ W iit,i + W iit,j W iit,i + W iit,j ≥ W iit, ( · ) ≥
0. Note that the matrix (7) is Hermitian by construction.Considering the above definitions, and the grid expansion u i,j,m selected by the upper level (which enters the lower level as aparameter, see Figure 2), the lower level problem introduced in Section 2.2 is defined as follows.( d pt,n,k ∗ , g pt,n,k ∗ , p ∗ t, , q ∗ t, , r upt,n ∗ , r downt,n ∗ , p ∗ t,n , [ π pt,n ∗ ]) =arg max X t ∈T X n ∈N + (cid:16) X k ∈ Ω Dt,n c dt,n,k d pt,n,k − X k ∈ Ω Gt,n c gt,n,k g pt,n,k (cid:17) − X t ∈T c pt, p t, − X t ∈T c qt, q t, + X t ∈T (cid:16) c upt, X n ∈N + ψ n r upt,n + c downt, X n ∈N + ψ n r downt,n (cid:17) (9a)7 .t.p t,n = X k ∈ Ω Gt,n g pt,n,k − X k ∈ Ω Dt,n d pt,n,k − D t,n ∀ t ∈ T , ∀ n ∈ N + [ π pt,n ∈ R ] (9b) q t,n = X k ∈ Ω Gt,n g qt,n,k − X k ∈ Ω Dt,n d qt,n,k ∀ t ∈ T , ∀ n ∈ N + [ π qt,n ∈ R ] (9c) p t,n = X ( i,j ) ∈L : i = n X m ∈M u i,j,m (cid:0) W iit,i a i,j,m − W ij,pt,i,j a i,j,m + W ij,qt,i,j e i,j,m (cid:1) + X ( i,j ) ∈L : j = n X m ∈M u i,j,m (cid:0) W iit,j a i,j,m − W ji,pt,j,i a i,j,m + W ji,qt,j,i e i,j,m (cid:1) ∀ t ∈ T , ∀ n ∈ N [ λ pt,n ∈ R ] (9d) q t,n = X ( i,j ) ∈L : i = n X m ∈M u i,j,m (cid:0) W iit,i e i,j,m − W ij,pt,i,j e i,j,m − W ij,qt,i,j a i,j,m (cid:1) + X ( i,j ) ∈L : j = n X m ∈M u i,j,m (cid:0) W iit,j e i,j,m − W ji,pt,j,i e i,j,m − W ji,qt,j,i a i,j,m (cid:1) ∀ t ∈ T , ∀ n ∈ N [ λ qt,n ∈ R ] (9e) X m ∈M u i,j,m (cid:0) W iit,i a i,j,m − W ij,pt,i,j a i,j,m + W ij,qt,i,j e i,j,m (cid:1) ≤ F maxi,j + X m ∈M u i,j,m F i,j,m ∀ t ∈ T , ∀ ( i, j ) ∈ L [ µ maxt,i,j ≥
0] (9f) X m ∈M u i,j,m (cid:0) W iit,j a i,j,m − W ji,pt,j,i a i,j,m + W ji,qt,j,i e i,j,m (cid:1) ≤ F mini,j + X m ∈M u i,j,m F i,j,m ∀ t ∈ T , ∀ ( i, j ) ∈ L [ µ maxt,i,j ≥
0] (9g) W iit, + 2 X ( i,j ) ∈P n X k : i ∈P k X m ∈M u i,j,m (cid:0) a i,j,m a i,j,m + e i,j,m p t,k + e i,j,m a i,j,m + e i,j,m q t,k (cid:1) ≤ ( v maxt,n ) ∀ t ∈ T , ∀ n ∈ N + [ β t,n ≥
0] (9h) W iit,i ≤ ( v maxt,i ) ∀ t ∈ T , ∀ i ∈ N + [ χ maxt,n ≥
0] (9i) − W iit,i ≤ − ( v mint,i ) ∀ t ∈ T , ∀ i ∈ N + [ χ mint,n ≥
0] (9j) d pt,n,k ≤ d p,maxt,n,k ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Dt,n [ ϕ d,p,maxt,n,k ≥
0] (9k) − d pt,n,k ≤ − d p,mint,n,k ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Dt,n [ ϕ d,p,mint,n,k ≥
0] (9l) g pt,n,k ≤ g p,maxt,n,k ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Gt,n [ ϕ g,p,maxt,n,k ≥
0] (9m) − g pt,n,k ≤ − g p,mint,n,k ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Gt,n [ ϕ g,p,mint,n,k ≥
0] (9n) d qt,n,k ≤ d q,maxt,n,k ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Dt,n [ ϕ d,q,maxt,n,k ≥
0] (9o) − d qt,n,k ≤ − d q,mint,n,k ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Dt,n [ ϕ d,q,mint,n,k ≥
0] (9p) g qt,n,k ≤ g q,maxt,n,k ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Gt,n [ ϕ g,q,maxt,n,k ≥
0] (9q) − g qt,n,k ≤ − g q,mint,n,k ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Gt,n [ ϕ g,q,mint,n,k ≥
0] (9r) W ij,pt,i,j = W ji,pt,j,i ∀ t ∈ T , ∀ ( i, j ) ∈ L [ ǫ pt,i,j ∈ R ] (9s) W ij,qt,i,j = − W ji,qt,j,i ∀ t ∈ T , ∀ ( i, j ) ∈ L [ ǫ qt,i,j ∈ R ] (9t) r upt,n = X k ∈ Ω Gt,n ( g p,maxt,n,k − g pt,n,k ) + X k ∈ Ω Dt,n d pt,n,k ∀ t ∈ T , ∀ n ∈ N + [ ρ upt,n ∈ R ] (9u) r downt,n = X k ∈ Ω Gt,n g pt,n,k + X k ∈ Ω Dt,n ( d p,maxt,n,k − d pt,n,k ) ∀ t ∈ T , ∀ n ∈ N + [ ρ downt,n ∈ R ] (9v) (cid:0) W ij,pt,i,j (cid:1) + (cid:0) W ij,qt,i,j (cid:1) + (cid:16) W iit,i − W iit,j (cid:17) ≤ (cid:16) W iit,i + W iit,j (cid:17) ∀ t ∈ T , ∀ ( i, j ) ∈ L (9w)The terms enclosed in squared brackets are dual variables. The lower level decision variables are d pt,n,k ∈ R , g pt,n,k ∈ R , d qt,n,k ∈ R , g qt,n,k ∈ R , p t,n ∈ R , q t,n ∈ R , r upt,n ∈ R , r downt,n ∈ R , W iit,i ∈ R , W ij,pt,i,j ∈ R , W ij,qt,i,j ∈ R , W ji,pt,j,i ∈ R , and W ji,qt,j,i ∈ R . The objectivefunction (9a) maximizes the consumers and producers surplus, while accounting for the revenues from upward r upt,n and downward r downt,n reserve provision, and considering the trades with the transmission grid at substation level. The positive (negative) active p t, and reactive q t, power represent the flow withdrawn from (injected into) the upstream transmission grid at time t . Constraints(9b)-(9c) define the active and reactive power injections for each node n ∈ N + . Constraints (9d)-(9e) enforce the Kirchhoff’s firstlaw, requiring the power injection to be equal to the sum of power inflows and outflows. Constraints (9f)-(9g) set the limits on activepower flows over each line ( i, j ) ∈ L , where the amount F i,j,m represents the additional capacity introduced by the expansion u i,j,m .Note that the expansion affects also the line admittance through the terms a i,j,m and e i,j,m . Constraint (9h)-(9j) enforce voltage8agnitude limits, where (9h) is a technical condition to ensure the obtained optimal power flows are exact (see Lemma 2 in [24]).Constraints (9k)-(9r) set the demand and supply lower and upper bounds for each flexible consumers and generators, for both activeand reactive power. Constraints (9s)-(9t) enforce the Hermitian property for the matrix (7). Constraints (9u)-(9v) define both theupward and downward reserve, as described in Section 2.1. Finally, as shown in (8), the second-order cone constraint (9w) enforcesthe positive semi-definite condition for matrix (7). The (auxiliary) dual variables associated with the second-order cone components W ij,pt,i,j , W ij,qt,i,j , ( W iit,i − W iit,j ) /
2, and − ( W iit,i + W iit,j ) /
2, are η at,i,j ∈ R , η bt,i,j ∈ R , η ct,i,j ∈ R , and γ t,i,j ≥
0, respectively. Note that, given u i,j,m , the lower level problem is a second-order cone program, and therefore it is a convex optimization problem [34].
3. Resolution method
When a lower level problem (1b) is a convex optimization program, and satisfies at least one constraint qualification condition(such as the Slater’s condition [34]), then it can be equivalently represented, within the bilevel model, by resorting to its first ordernecessary and sufficient Karush-Kuhn-Tucker (KKT) conditions. However, the KKT complementary slackness conditions are non-linear non-convex relations, which introduce non-linearities into the problem. To overcome this issue, a key optimization propertycan be exploited. That is, the complementary slackness conditions hold if the strong duality property holds [34]. The strong dualityrequires the equivalence between the objective function values of both the primal and dual problems. Therefore, as long as thestrong duality holds, a convex lower level problem (1b) can be equivalently represented within the bilevel model (1) by resorting toits primal constraints, dual constraints, and the strong duality property, leading to a single and equivalent optimization problem, asoutlined in Figure 3.
Figure 3
The lower level problem introduced in Section 2.2.2 is a second-order cone program, and therefore a convex optimization problem.In the following, we assume the Slater’s condition holds, which is a reasonable assumption for any well formed conic problems in realworld applications (see [36, p. 89] for a discussion on this point). Therefore, for conic programs, the (generalized) Slater’s conditionimplies the strong duality property [34], which in turn implies the (generalized) complementary slackness conditions [37]. Therefore,the bilevel framework introduced in Section 2.2 can be recast as a single and equivalent optimization program, as follows:max X t ∈T X n ∈N + (cid:16) X k ∈ Ω Dt,n c dt,n,k d pt,n,k − X k ∈ Ω Gt,n c gt,n,k g pt,n,k (cid:17) − X t ∈T c pt, p t, − X t ∈T c qt, q t, + X t ∈T (cid:16) c upt, X n ∈N + ψ n r upt,n + c downt, X n ∈N + ψ n r downt,n (cid:17) − τ X t ∈T X n ∈N + (cid:16) D t,n + X k ∈ Ω Dt,n d p,maxt,n,k + X k ∈ Ω Gt,n g p,maxt,n,k (cid:17) − X ( i,j ) ∈L X m ∈M u i,j,m ( K fixi,j,m + K vari,j,m F i,j,m ) (10a) s.t. (2b) − (2g) (10b)(9b) − (9w) (10c)(A.1) − (A.16) (10d)(B.1) (10e)where the objective function (10a) is the same as in the upper level problem (2). Constraint (10b) represents the upper levelconstraints, (10c) the lower level constraints, and (10d) its dual constraints (see Appendix A). Finally, (10e) refers to the strongduality property, reported in Appendix B. Note that in the single level problem there is no distinction between upper and lower9evel variables. Therefore, the decision variables of the single level program (10) are the decision variables of the upper level, as wellas those of the primal and dual lower level problems. The single level problem (10) is a non-linear integer optimization problem. The main sources of non-linearities are due to thepresence of bilinear terms. In particular, there are two types of bilinear terms:1. the product of the binary variable u i,j,m and a continuous variable, such as u i,j,m W iit,i in (9f);2. the product π pt,n p t,n in (2c).Note that the second-order cone constraints (9w) and (A.16) are convex relations, so they are not an issue. The first type ofnon-linearity can be removed by using standard integer algebra. Indeed, the product ux involving the binary u and the continuousbounded variable − M ≤ x ≤ M can be equivalently replaced by introducing the auxiliary variable y ux defined as follows: − M u ≤ y ux ≤ M u (11a) − M (1 − u ) ≤ x − y ux ≤ M (1 − u ) (11b)By using (11a)-(11b) all the bilinear terms in the single level program (10) involving binary variables can be removed. A detaileddescription of the introduced auxiliary variables is reported in Appendix C. In order to remove the non-linearity of the second type,the following lemma is stated, where the proof is given in Appendix D. Lemma 1.
The following relation holds at the optimum of the lower level problem: π pt,n p t,n = X k ∈ Ω Gt,n ( ϕ g,p,maxt,n,k g p,maxt,n,k − ϕ g,p,mint,n,k g p,mint,n,k + c upt, ψ n g pt,n,k − c downt, ψ n g pt,n,k + c gt,n,k g pt,n,k ) − X k ∈ Ω Dt,n ( − ϕ d,p,maxt,n,k d p,maxt,n,k + ϕ d,p,mint,n,k d p,mint,n,k + c upt, ψ n d pt,n,k − c downt, ψ n d pt,n,k + c dt,n,k d pt,n,k ) − π pt,n D t,n ∀ n ∈ N + (12)and π t, p t, = c pt, p t, by definition.By using Lemma 1 and the auxiliary variables (C.1)-(C.24), the single level problem (10) can be equivalently recast as a mixed-integer quadratically constrained program, fully reported in Appendix E, which can be solved with off-the-shelf solvers.
4. Numerical results and discussion
This section reports five test cases (based on a 5-bus and a 33-bus network) obtained by using the MIQCP described in Section 3,and fully reported in Appendix E. Unless stated otherwise, the following settings are used. The set of the line expansion is definedas M = { , . , } , i.e., three different line expansions are possible, respectively of 0%, 50%, and 100% of the current capacity. Indetail, the expansion u i,j,m = 1 enforces the line expansion m ∈ M on the line ( i, j ) ∈ L , through the following parameters: a i,j,m = a i,j, (1 + m ) (13) e i,j,m = e i,j, (1 + m ) (14) F i,j,m = m F maxi,j (15) K fixi,j,m = m K fixi,j, (16) K vari,j,m F i,j,m = m K vari,j, F maxi,j (17)where a i,j, is the line conductance and e i,j, the negative of the line susceptance, i.e., y i,j, = a i,j, − ie i,j, , where y i,j, is theadmittance before any expansion. The term F maxi,j is maximum power flow over the line ( i, j ) without expansion. The parameter K fixi,j, and K vari,j, represent the fixed and variable costs of building a full new line. Note that the investment costs are appliedproportionally to the expansion m , however the effect of economies of scale can be implemented by modifying (16)-(17) accordingly.Voltages are expressed in per-unit, with substation voltage magnitude (slack bus) equal to one, and maximum and minimum voltagemagnitude in the remaining nodes set as v maxt,i = 1 .
20 and v mint,i = 0 .
80, respectively. The parameter ψ n is set to one, i.e., the reserveprovided is paid the same price regardless of the actual node n of injection into the distribution grid. For ease of exposition, weassume a single time period, and one flexible consumer and one generator per node, i.e., T = { } , Ω Dt,n = { } , and Ω Gt,n = { } .Condition (2f) is checked ex-post. The MIQCP model has been implemented with Pyomo 5.6 [38], and solved with Cplex 12.9 [39]on a 64-core CPU with 256 GB of RAM. 10
12 3 4
Figure 4: 5-bus distribution grid.
This paragraph reports four test cases based on the prototypical 5-bus grid depicted in Figure 4. As described in Section 2.1, thesubstation is located at node zero (slack bus) and is connected upstream with the main transmission network. In these examples, foreach node n ∈ N + with N + = 1 , , ,
4, the amount of fixed power required by non-flexible consumers (e.g. group of householders)is D t,n = 100 kW. Furthermore, flexible consumers demand up to d p,maxt,n,k = 100 kW in each node. Similarly, generators offer up to g p,maxt,n,k = 100 kW in each node. Flexible demand bid price c dt,n,k and generation bid prices c gt,n,k are reported in Table 1. Table 1: 5-bus network: demand and generation bid prices
Nodes c dt,n,k c gt,n,k The first test case on the 5-bus grid shows the effect of reserve provision on flexible consumers and generators. To focus on thiseffect, no line expansion is allowed, i.e. M = 0. Active power can be traded with the transmission network at price c pt, = 5 £ /kWh,whereas the price for reactive power is set to zero, i.e., c qt, = 0 £ /kWh, and F maxi,j = 1000 kW for each line. Table 2: 5-bus network: effect of reserve provision on flexible users n c dt,n,k c gt,n,k c upt, = 0 c upt, = 5 π pt,n d pt,n,k g pt,n,k π pt,n d pt,n,k g pt,n,k c upt, from zero to 5 £ /kWh. The second and thirdcolumns report the bid price for demand c dt,n,k and generation c gt,n,k in each node n for ease of comparison, whereas the remainingcolumns show the obtained nodal prices π pt,n , and the cleared quantities d pt,n,k and g pt,n,k for flexible consumers and generators underthe two different reserve prices. In both cases, no line is congested, and therefore the differences in nodal prices are only due to theeffect of power losses (for a detailed description of the effect of power losses see Section 4.2.3). The important point to highlightin this example is the quantity allocated in node 3. When the upward reserve price is equal to zero, the nodal price in node 3 is30 . £ /kWh. The demand bid price in that node (which represents how much flexible consumers are willing to pay) is 30 £ /kWh.As a consequence, the demand bid order is out-of-the-money [40], and fully rejected. The generation bid price in node 3 (whichrepresents the generator’s marginal cost assuming perfect competition [41, 14]) is also 30 £ /kWh. In this case, the supply bid orderis in-the-money and fully executed, as it yields a profit of 0 . £ /h for each kW traded. For ease of reading, the definition of upwardreserve in (9u) is reported again here: r upt,n = X k ∈ Ω Gt,n ( g p,maxt,n,k − g pt,n,k ) + X k ∈ Ω Dt,n d pt,n,k (18)That is, upward reserve can be provided by using the spare generator capacity g p,maxt,n,k − g pt,n,k rising the power output, and by flexibleconsumers reducing their allocated consumption d pt,n,k . Table 2 shows that when the price of upward reserve c upt, rises from zero to5 £ /kWh, the optimal allocation is to fully reject the supply order, and fully accept the demand order. In the previous case, the11enerator in node 3 makes a profit of 0 . £ /h for each kW traded. Now, by having its capacity unallocated, it can provide itsfull capacity as upward reserve, which is paid 5 £ /kWh, and therefore leads to a greater profit. Similarly, in the previous case thedemand bid was fully rejected, whereas now the demand order is fully executed, even though its bid of 30 £ /kWh is smaller than the(new) nodal price of 31 . £ /kWh. The reason is that flexible consumers also collect revenues from the upward reserve provision,which is paid 5 £ /kWh, leading to a surplus of 30 − .
08 + 5 = 3 . £ /h for each kW. Note that as a consequence of the shiftin the allocated demand and generation, there is a net reduction of power supplied, which leads to an increase of the nodal prices.However, this is not an issue for flexible consumers or generators, as they now also collect the reserve price. Table 3: 5-bus network: effect of reserve provision on surplus and revenues c upt, = 0 c upt, = 5Surplus cons. 1410 1251Reserve cons. 0 1500Surplus gen. 1642 1641Reserve gen. 0 1000Total 3052 5392In detail, Table 3 shows the surplus from power trades and the revenues from reserve provision, for both flexible consumers andgenerators. As can be observed, despite the loss in surplus due to the increase of nodal prices, the revenues from reserve allow flexibleconsumers and generators to achieve a significant welfare increase. However, the increase of nodal prices leads to an increase of theprice π D charged to non-flexible consumers, which is linked to nodal prices through (2b). In this case, it rises from 30 . £ /kWhto 30 . £ /kWh, without being offset by reserve revenues (non-flexible consumers cannot provide reserve as their demand is fixed).Note however that the provision of downward reserve instead of upward reserve, would lead to an opposite pattern, lowering theprice π D . This example shows how the possibility of providing reserve represents an opportunity to make profits for flexible usersin a local distribution network, and should represent a further incentive for non-flexible consumers to adopt more price-sensitivebehaviours and technology allowing them to be flexible. The second test case on the 5-bus grid shows the effect of how a different proportion of flexible and non-flexible consumers canaffect nodal prices and allocated quantities. As in the previous case, no line expansion is possible, i.e., M = { } . Both active andreactive power can be traded with the main transmission grid through the substation at 30 £ /kWh. The maximum power flow ofthe lines is set to F maxi,j = 300 kW. No reserve is provided. Three scenarios are compared, where:i) D t,n = 150 and d p,maxt,n,k = 50;ii) D t,n = 100 and d p,maxt,n,k = 100;iii) D t,n = 50 and d p,maxt,n,k = 150. Table 4: 5-bus network: effect of different proportion flexible/non-flexible consumers on prices
Case (i) D t,n = 150, d p,maxt,n,k = 50 Case (ii) D t,n = 100, d p,maxt,n,k = 100 Case (iii) D t,n = 50, d p,maxt,n,k = 150 π pt, π pt, π pt, π pt, π D able 5: 5-bus network: effect of different proportion flexible/non-flexible consumers on allocated quantities Node n Case (i) D t,n = 150, d p,maxt,n,k = 50 Case (ii) D t,n = 100, d p,maxt,n,k = 100 Case (iii) D t,n = 50, d p,maxt,n,k = 150 D t,n d pt,n,k g pt,n,k D t,n d pt,n,k g pt,n,k D t,n d pt,n,k g pt,n,k π pt,n for each node, whereas the last rowshows the fixed price π D levied on non-flexible consumers. Table 5 shows the demanded quantities for non-flexible consumers D t,n ,the allocated demand d pt,n,k of flexible consumers, and the cleared generation g pt,n,k for each node (in the first four rows), and thetotal sum for the whole grid in the last row. As can be observed from Table 4, the increase of flexible consumption has a significanteffect on prices. In particular, note how the increase of flexible demand leads to a reduction of the price π D paid by the non-flexibleconsumers. This effect can be explained observing (see Table 5) that the flexible demand d pt,n,k is never executed in node 3 and4, where the nodal prices are significantly greater than the submitted bid c dt,n,k which is 30 £ /kWh and 25 £ /kWh, respectively.By contrast, non-flexible consumption must be always satisfied regardless of the price, which lead to a significant amount of powerdemanded in high-priced nodes. The line between node 1 and node 0 is congested in both case (i) and case (ii), which contributesfurther to explain the greater nodal prices. This test case shows that an increase in the proportion between flexible and non-flexibleconsumers in a distribution network can have a positive impact in terms of lower prices. In particular, it demonstrates that it canbe beneficial also for the non-flexible consumers, i.e., the traditional householders. The third test case on the 5-bus grid shows the effect of congestions and power losses on allocated quantities and prices. As in theprevious case, no line expansion is possible, i.e., M = { } . Active power can be traded with the transmission grid at 5 £ /kWh, andreserve provision is not considered. Here, two different scenarios are compared. In the first case, the maximum power flow capacityof each line is F maxi,j = 820 kW, whereas in the second case F maxi,j = 800 kW. That is, in the first case the maximum allowed powerflow is 2 .
5% greater.
Table 6: Flows with F maxi,j = 820 and F maxi,j = 800 (i,j) F maxi,j = 820 F maxi,j = 800(0,1) 819.60 800.00(1,2) 201.18 201.18(1,3) 405.14 405.14(3,4) 201.11 201.10Table 6 shows the flow of active power from node i to node j over the line ( i, j ). In the first case, the power flow over the line(0 ,
1) is 819 .
60 kW. Therefore, in the second case, when F maxi,j is reduced to 800 kW the line results congested. Table 7: Prices and allocated quantities n F maxi,j = 820 F maxi,j = 800 π pt,n D t,n d pt,n,k g pt,n,k π pt,n D t,n d pt,n,k g pt,n,k π pt,n , the requested non-flexible demand D t,n , the allocated flexible demand d pt,n,k , and the suppliedgeneration g pt,n,k for both the considered cases. As can be observed, in the first case no generator is dispatched. The whole localdemand is satisfied by importing power from the upstream transmission grid at the cheaper price of c pt, = 5 £ /kWh. The maingrid acts therefore as marginal unit [14], setting the price at the importing node zero as π t, = 5 £ /kWh. Note that, the nodalprices π pt,n in the remaining nodes are greater than this price, despite no congestion is present in the whole grid. This is due to theeffect of power losses, which introduce a loss price component into the nodal prices [42], similarly to the shadow price of congestion13hen a line is congested [7]. In the second case, the total demand satisfied is again 800 kW. We recall that in this case the line(0 ,
1) is congested, and only 800 kW can be imported from the main transmission network. In principle, the imported power isexactly equal to the demanded quantity, therefore, no generators should be dispatched. However, the optimal solution shows thatthe generator in node one has to produce 18 .
97 kW. The reason is again due to the power losses. In this second case, the (active)power loss, given by P n p t,n ∀ n ∈ N , is equal to 18 .
97 kW. Therefore, the generator in node one is dispatched with only the purposeof covering power losses within the local distribution area, which accounts for nearly 2 .
5% of the total demand. A further importantconsequence is that the generator in node one becomes the marginal generator. Therefore, its marginal generation cost c gt, ,k sets thenodal price in that node (i.e. π pt, = c gt, ,k = 20 £ /kWh), which affects all the remaining nodes. As a result, the fixed price π D paidby non-flexible consumers rises from 5 . £ /kWh to 20 . £ /kWh. This example shows how a small change in the line capacity canhave a significant impact on the distribution nodal prices, with non-negligible effects for both flexible and non-flexible consumers.It also shows how a relative small amount of power such as losses, could trigger expensive generation in constrained network, witha considerable impact on prices. The fourth test case on the 5-bus grid shows the effect of investment decisions on prices and network tariff. In this example,the generation g p,maxt,n,k is set to zero, whereas all the other settings are same as in Section 4.2.3, with maximum flow capacity F maxi,j = 800 kW. The investment cost parameters are K fixi,j, = 100 £ /h, K vari,j, = 0 . £ /kWh, K op = 0 £ /h, and K tot = 10000 £ /h.From the previous test case reported in Section 4.2.3, under this settings the only congestion is on the line (0 , M and M are compared. In the first set, the possible expansions are M = { , . , . , . , } , i.e., all thenetwork lines can be expanded with step increment equal to 25% of their current capacity. In the second set, the possible expansionare M = { , . , } . Table 8: Merchandising surplus, total network tariff, fixed and variable investment costs, and profit M M M. Surplus 88 78Tot. tariff 0 12Fixed costs 25 50Var. costs 20 40Profit 43 0As expected, the optimal solution shows that in both cases the line (0 ,
1) is expanded, with expansion m = 0 .
25 in the first case,and m = 0 .
50 in the second case. Table 8 shows the merchandising surplus and the total tariff collected by the network operator, aswell as both fixed and variable investment costs, and the resulting profit. Fixed and variable costs are determined as in (15)-(17).The total investment cost in the first case is equal to 45 £ /h, whereas the collected merchandising surplus is 88 £ /h, leading to aprofit collected by the network operator equal to 43 £ /h. This extra profit is due to the lumpiness of network expansions, i.e. in realworld instances, a line can only be increased by discrete amount (for an extensive discussion on this point see [9, 10]). In the secondcase, the expansion of 50% leads to a total investment cost of 90 £ /h. The merchandising surplus is only 78 £ /h, and therefore atariff τ equal to 0 . £ /kWh is charged to network users, which allows the network operator to recover the remaining amountof 12 £ /h. Note that the merchandising surplus decreases from 88 to 78. The reason is that a line expansion also has an effect onthe admittance, as modelled in (13)-(14). In particular, the line resistance decreases. As a consequence, network loss drops from16 . £ /h to 15 . £ /h, which in turn affects nodal prices (as shown in Section 4.2.3), and therefore influences the merchandisingsurplus. Table 9: Prices and allocated quantities n M M Difference π pt,n π pt,n τ = 0 . £ /kWh. Similarly, the fixed price π D charged to non-flexible consumers decreases from 5 . £ /kWhto 5 . £ /kWh, with a reduction of 0 . £ /kWh. Therefore, despite the tariff paid, the expansion of 50% is more favourable forboth flexible and non-flexible consumers, because yields a greater reduction of prices paid. By contrast, the profit for the networkoperator drops to zero. This test cases shows that a line expansion has a double effect: (i) it increases the maximum power flowover the line, and (ii) it reduces the network losses. Furthermore, it highlights how different network planning strategies can have14 significant impact on surplus redistribution among grid participants, and demonstrates the potential conflict of interest of thenetwork operator. This test case is based on the IEEE 33-bus distribution network [43], which has been introduced in Figure 1. Bid prices forflexible consumers and generators are sampled from a normal distribution with standard deviation of 10 £ /kWh and mean equalto 50 £ /kWh and 20 £ /kWh, respectively. Negative prices are set to zero. Flexible demand and generation account for half of thenon-flexible demand. Active and reactive power can be traded with the main grid at 30 £ /kWh. Reserve provision is not considered.The maximum power flow capacity of the lines is F maxi,j = 2000 kW. The base voltage is 12 . K fixi,j, = 5 , £ /h, K vari,j, = 1 £ /kWh, K op = 0 £ /h, and K tot = 100 , £ /h. In this testcase, an example of possible utilization of constraint (2f) is given by introducing the following requirements: u i,j, ≤ u h,k, ∀ ( i, j ) ∈ L , ∀ ( h, k ) ∈ P j (19)That is, if a line ( i, j ) is expanded, then all the upstream line ( h, k ) ∈ P j connected to this line up to the substation must beexpanded. In this test case, two scenarios are compared. In the first one, no expansion is allowed. In the second one, the set of thepossible expansions is M = { , } , i.e. a line (and all lines upstream of it due to (19)), can be either doubled or left as they are. Table 10: Power flows over the first four lines in the 33-bus distribution network (i,j) M = { } M = {
0, 1 } (0,1) 2000 3513(1,2) 1715 3047(2,3) 1185 2000(3,4) 1119 1868()Table 10 reports the power flows for the first four lines in the two scenarios. When no expansion is possible, the line (0 ,
1) iscongested (power flow equal to F maxi,j = 2000 kW). In the second case, both the line (0 ,
1) and (1 ,
2) are expanded, whereas the line(2 ,
3) is now congested.
No expansionLines (0,1), (1, 2) expanded
NodeNodalprices(£/kWh)
Figure 5: Nodal prices in the 33-bus distribution grid. The solid red line depicts the nodal prices in case of no expansion. The dashed blue line representsthe nodal prices when the lines (0 ,
1) and (1 ,
2) are expanded.
Figure 5 shows the nodal prices in the two cases, where the solid red line depicts the no expansion case, whereas the dashed blueline represents the case when both the lines (0 ,
1) and (1 ,
2) are expanded. As can be observed, the nodal prices are significantlylower in the second case, in particular for the nodes located in the area labelled by A in Figure 6, i.e., all the nodes upstream of thecongested line (2 , A B
Figure 6: 33-bus distribution grid. In the case with M = { , } , the line connecting node 2 to node 3 is congested. The area labelled by A includes all thenode upstream of the congested line. π D paid by non-flexible consumers drops from 66 . £ /kWh to 37 . £ /kWh. This test case, based on thewidely used 33-bus network scheme, shows how grid expansion plans can have significant impacts on both prices paid by networkusers and their electricity consumptions. Furthermore, it displays how a single congestion in a line can have a cascade effect on allthe downstream nodes in distribution networks, due to their radial topology [7].
5. Conclusion
The increasing share of flexible consumers, who are willing to pay a different price for each quantity demanded, as well asthe significant availability of renewable energy resources and distributed generation, foster the introduction of nodal prices at thedistribution level. Marginal nodal pricing is a desirable scheme as it leads to an efficient allocation of scarce resources in termsof both energy and line capacities. The proposed framework, based on a bilevel model, aims to shed light on how a distributionnetwork could be managed in this context, by providing also new profit sources for its users. In particular, it shows how flexibleand non-flexible consumers can coexist, where the former pay nodal prices and the latter are charged a fixed price based on theunderlying nodal prices. Flexibility can be exploited by consumers and producers to obtain new streams of revenues by providingreserve to the main grid. The results obtained show that the increase of flexible consumers can be also beneficial for the non-flexibleones, as it could decrease their price paid. Moreover, an accurate planning of the distribution network is required to account forthe increased power flows. In this sense, three main aspects should be highlighted. First, small congestion within the distributionnetwork can have a substantial impact on prices, and this effect is amplified by the radial topology of these networks. Second, powerlosses cannot be neglected. They have a non-negligible impact on prices and could trigger expensive generation in locally constrainedareas. Third, line expansions can lead to a surplus redistribution between consumers, producers and network operator, where thelumpy nature of line capacity expansions exacerbates the network operator’s conflict of interest.Future work will aim to provide extensive policy analyses and recommendations based on the proposed framework. Additionallines of research include the extension of the presented model to consider upstream wholesale and ancillary service markets.
Acknowledgement
The present work has been supported by the EPSRC grant EP/S000887/1 and by the EPSRC grant EP/S031901/1, titledEnergyREV - Market Design for Scaling up Local Clean Energy Systems. Icons used in Figure 1 are from Freepik by Flaticon.
Appendix A. Dual constraints
This section reports the dual constraints of the lower level problem described in Section 2.2.2. π pt,n + ϕ d,p,maxt,n,k − ϕ d,p,mint,n,k − ρ upt,n + ρ downt,n = c dt,n,k ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Dt,n [ d pt,n,k ∈ R ] (A.1) π qt,n + ϕ d,q,maxt,n,k − ϕ d,q,mint,n,k = 0 ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Dt,n [ d qt,n,k ∈ R ] (A.2) − π pt,n + ϕ g,p,maxt,n,k − ϕ g,p,mint,n,k + ρ upt,n − ρ downt,n = − c gt,n,k ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Gt,n [ g pt,n,k ∈ R ] (A.3) − π qt,n + ϕ g,q,maxt,n,k − ϕ g,q,mint,n,k = 0 ∀ t ∈ T , ∀ n ∈ N + , ∀ k ∈ Ω Gt,n [ g qt,n,k ∈ R ] (A.4) λ pt, = − c pt, ∀ t ∈ T [ p t, ∈ R ] (A.5) λ qt, = − c qt, ∀ t ∈ T [ q t, ∈ R ] (A.6) π pt,n + λ pt,n + 2 X ( i,j ) ∈P n X k : i ∈P k X m ∈M u i,j,m a i,j,m a i,j,m + e i,j,m β t,k = 0 ∀ t ∈ T , ∀ n ∈ N + [ p t,n ∈ R ] (A.7) π qt,n + λ qt,n + 2 X ( i,j ) ∈P n X k : i ∈P k X m ∈M u i,j,m e i,j,m a i,j,m + e i,j,m β t,k = 0 ∀ t ∈ T , ∀ n ∈ N + [ q t,n ∈ R ] (A.8) ρ upt,n = c upt, ψ n ∀ t ∈ T , ∀ n ∈ N + [ r upt,n ∈ R ] (A.9) ρ downt,n = c downt, ψ n ∀ t ∈ T , ∀ n ∈ N + [ r downt,n ∈ R ] (A.10) X m ∈M u i,j,m (cid:16) a i,j,m λ pt,i + e i,j,m λ qt,i − a i,j,m µ maxt,i,j (cid:17) + ǫ pt,i,j + η at,i,j = 0 ∀ t ∈ T , ∀ ( i, j ) ∈ L [ W ij,pt,i,j ∈ R ] (A.11) X m ∈M u i,j,m (cid:16) a i,j,m λ pt,j + e i,j,m λ qt,j − a i,j,m µ mint,i,j (cid:17) − ǫ pt,i,j = 0 ∀ t ∈ T , ∀ ( i, j ) ∈ L [ W ji,pt,j,i ∈ R ] (A.12) X m ∈M u i,j,m (cid:16) − e i,j,m λ pt,i + a i,j,m λ qt,i + e i,j,m µ maxt,i,j (cid:17) + ǫ qt,i,j + η bt,i,j = 0 ∀ t ∈ T , ∀ ( i, j ) ∈ L [ W ij,qt,i,j ∈ R ] (A.13) X m ∈M u i,j,m (cid:16) − e i,j,m λ pt,j + a i,j,m λ qt,j + e i,j,m µ mint,i,j (cid:17) + ǫ qt,i,j = 0 ∀ t ∈ T , ∀ ( i, j ) ∈ L [ W ji,qt,j,i ∈ R ] (A.14)16 ( i,j ) ∈L : i = n (cid:16) X m ∈M u i,j,m (cid:0) − a i,j,m λ pt,i − e i,j,m λ qt,i + µ maxt,i,j a i,j,m (cid:1) + η ct,i,j − γ t,i,j (cid:17) + X ( i,j ) ∈L : j = n (cid:16) X m ∈M u i,j,m (cid:0) − a i,j,m λ pt,j − e i,j,m λ qt,j + µ mint,i,j a i,j,m (cid:1) − η ct,i,j + γ t,i,j (cid:17) + χ maxt,n − χ mint,n = 0 ∀ t ∈ T , ∀ n ∈ N + [ W iit,n ∈ R ] (A.15)( η at,i,j ) + ( η bt,i,j ) + ( η ct,i,j ) ≤ ( γ t,i,j ) ∀ t ∈ T , ∀ ( i, j ) ∈ L (A.16) Appendix B. Strong Duality
The strong duality property requires the equivalence between the primal and dual objective function values, and is defined asfollows. X t ∈T X n ∈N + (cid:16) X k ∈ Ω Dt,n c dt,n,k d pt,n,k − X k ∈ Ω Gt,n c gt,n,k g pt,n,k (cid:17) − X t ∈T c pt, p t, − X t ∈T c qt, q t, + X t ∈T (cid:16) c upt, X n ∈N + ψ n r upt,n + c downt, X n ∈N + ψ n r downt,n (cid:17) = X t ∈T X n ∈N + − π pt,n D t,n + χ maxt,n ( v maxt,i ) − χ mint,n ( v mint,i ) + X t ∈T X n ∈N + (cid:0) X k ∈ Ω Dt,n d p,maxt,n,k ϕ d,p,maxt,n,k + d q,maxt,n,k ϕ d,q,maxt,n,k − d p,mint,n,k ϕ d,p,mint,n,k − d q,mint,n,k ϕ d,q,mint,n,k + ρ downt,n d p,maxt,n,k (cid:1) + X t ∈T X n ∈N + (cid:0) X k ∈ Ω Gt,n g p,maxt,n,k ϕ g,p,maxt,n,k + g q,maxt,n,k ϕ g,q,maxt,n,k − g p,mint,n,k ϕ g,p,mint,n,k − g q,mint,n,k ϕ g,q,mint,n,k + ρ upt,n g p,maxt,n,k (cid:1) + X t ∈T X ( i,j ) ∈L F maxi,j µ maxt,i,j + F mini,j µ mint,i,j + X m ∈M u i,j,m F i,j,m ( µ maxt,i,j + µ mint,i,j )+ X t ∈T X ( i,j ) ∈L : i =0 X m ∈M W iit, u i,j,m (cid:0) a i,j,m λ pt,i + e i,j,m λ qt,i − a i,j,m µ maxt,i,j (cid:1) + X t ∈T X ( i,j ) ∈L : j =0 X m ∈M W iit, u i,j,m (cid:0) a i,j,m λ pt,j + e i,j,m λ qt,j − a i,j,m µ mint,i,j (cid:1) + X t ∈T X n ∈N + (cid:0) ( v maxt,n ) − W iit, (cid:1) β t,n + X t ∈T X ( i,j ) ∈L : i =0 W iit, − η ct,i,j + γ t,i,j X t ∈T X ( i,j ) ∈L : j =0 W iit, η ct,i,j + γ t,i,j Appendix C. Auxiliary variables
This section reports the auxiliary variables used to recast the single level problem (10) as a MIQCP program.17 ( v maxt,i ) u i,j,m ≤ y w ij p,ut,i,j,m ≤ ( v maxt,i ) u i,j,m (C.1) − ( v maxt,i ) (1 − u i,j,m ) ≤ W ij,pt,i,j − y w ij p,ut,i,j,m ≤ ( v maxt,i ) (1 − u i,j,m ) (C.2) − ( v maxt,i ) u i,j,m ≤ y w ij q,ut,i,j,m ≤ ( v maxt,i ) u i,j,m (C.3) − ( v maxt,i ) (1 − u i,j,m ) ≤ W ij,qt,i,j − y w ij q,ut,i,j,m ≤ ( v maxt,i ) (1 − u i,j,m ) (C.4) − ( v maxt,i ) u i,j,m ≤ y w ji p,ut,i,j,m ≤ ( v maxt,i ) u i,j,m (C.5) − ( v maxt,i ) (1 − u i,j,m ) ≤ W ji,pt,j,i − y w ji p,ut,i,j,m ≤ ( v maxt,i ) (1 − u i,j,m ) (C.6) − ( v maxt,i ) u i,j,m ≤ y w ji q,ut,i,j,m ≤ ( v maxt,i ) u i,j,m (C.7) − ( v maxt,i ) (1 − u i,j,m ) ≤ W ji,qt,j,i − y w ji q,ut,i,j,m ≤ ( v maxt,i ) (1 − u i,j,m ) (C.8)0 ≤ y µ maxt,i,j ,ut,i,j,m ≤ M u i,j,m (C.9)0 ≤ µ maxt,i,j − y µ maxt,i,j ,ut,i,j,m ≤ M (1 − u i,j,m ) (C.10)0 ≤ y µ mint,i,j ,ut,i,j,m ≤ M u i,j,m (C.11)0 ≤ µ mint,i,j − y µ mint,i,j ,ut,i,j,m ≤ M (1 − u i,j,m ) (C.12)( v mint,i ) u i,j,m ≤ y w ii ,ut,i,j,m ≤ ( v maxt,i ) u i,j,m (C.13)( v mint,i ) (1 − u i,j,m ) ≤ W iit,i − y w ii ,ut,i,j,m ≤ ( v maxt,i ) (1 − u i,j,m ) (C.14) − M u i,j,m ≤ y λ,p,ut,i,j,m ≤ M u i,j,m (C.15) − M (1 − u i,j,m ) ≤ λ pt,i − y λ,p,ut,i,j,m ≤ M (1 − u i,j,m ) (C.16) − M u i,j,m ≤ y λ,q,ut,i,j,m ≤ M u i,j,m (C.17) − M (1 − u i,j,m ) ≤ λ qt,i − y λ,q,ut,i,j,m ≤ M (1 − u i,j,m ) (C.18) M u i,j,m ≤ y p,ut,i,j,m,k ≤ M u i,j,m (C.19) M (1 − u i,j,m ) ≤ p t,k − y p,ut,i,j,m,k ≤ M (1 − u i,j,m ) (C.20) M u i,j,m ≤ y q,ut,i,j,m,k ≤ M u i,j,m (C.21) M (1 − u i,j,m ) ≤ q t,k − y q,ut,i,j,m,k ≤ M (1 − u i,j,m ) (C.22)0 ≤ y β,ut,i,j,m,k ≤ M i,j,m u i,j,m (C.23)0 ≤ β t,k − y β,ut,i,j,m,k ≤ M i,j,m (1 − u i,j,m ) (C.24)where constraints (C.1)-(C.12) are defined ∀ t ∈ T , ∀ ( i, j ) ∈ L , ∀ m ∈ M , constraints (C.13)-(C.18) are defined ∀ t ∈ T , ∀ ( i, j ) ∈ L∪ ˜ L , ∀ m ∈ M with ˜ L = { ( j, i ) : ( i, j ) ∈ L} , constraints (C.19)-(C.24) are defined ∀ t ∈ T , ∀ ( i, j ) ∈ L , ∀ m ∈ M , ∀ k ∈ N + . Moreover, M is set equal to 3000 £ /kWh, which represents the maximum bid price currently allowed in the European wholesale markets, whereas: M = X k ∈ Ω Gt,n g p,mint,n,k − X k ∈ Ω Dt,n d p,maxt,n,k − D t,n (C.25) M = X k ∈ Ω Gt,n g p,maxt,n,k − X k ∈ Ω Dt,n d p,mint,n,k − D t,n (C.26) M = X k ∈ Ω Gt,n g q,mint,n,k − X k ∈ Ω Dt,n d q,maxt,n,k (C.27) M = X k ∈ Ω Gt,n g q,maxt,n,k − X k ∈ Ω Dt,n d q,mint,n,k (C.28) M i,j,m = M a i,j,m + e i,j,m min( a i,j,m , e i,j,m ) (C.29)18 ppendix D. Proof of Lemma 1 By using the definition of p t,n in (9b) we have: π pt,n p t,n = π pt,n ( X k ∈ Ω Gt,n g pt,n,k − X k ∈ Ω Dt,n d pt,n,k − D t,n ) = X k ∈ Ω Gt,n π pt,n g pt,n,k − X k ∈ Ω Dt,n π pt,n d pt,n,k − π pt,n D t,n (D.1)The term π pt,n g pt,n,k can be rewritten as follow by using (A.3): π pt,n g pt,n,k =( ϕ g,p,maxt,n,k − ϕ g,p,mint,n,k + ρ upt,n − ρ downt,n + c gt,n,k ) g pt,n,k = ϕ g,p,maxt,n,k g pt,n,k − ϕ g,p,mint,n,k g pt,n,k + ρ upt,n g pt,n,k − ρ downt,n g pt,n,k + c gt,n,k g pt,n,k (D.2)The strong duality condition (B.1) ensures that all the complementary slackness conditions hold. Therefore, by using thecomplementary slackness associated with the constraints (9m) and (9n) we obtain:( g pt,n,k − g p,maxt,n,k ) ϕ g,p,maxt,n,k = 0 ⇐⇒ g pt,n,k ϕ g,p,maxt,n,k = g p,maxt,n,k ϕ g,p,maxt,n,k (D.3)( g p,mint,n,k − g pt,n,k ) ϕ g,p,mint,n,k = 0 ⇐⇒ g pt,n,k ϕ g,p,mint,n,k = g p,mint,n,k ϕ g,p,mint,n,k (D.4)Therefore, by using (D.3)-(D.4), and the dual conditions (A.9)-(A.10), the following linear relation can be obtained: π pt,n g pt,n,k = ϕ g,p,maxt,n,k g p,maxt,n,k − ϕ g,p,mint,n,k g p,mint,n,k + c upt, ψ n g pt,n,k − c downt, ψ n g pt,n,k + c gt,n,k g pt,n,k (D.5)Similarly, by using (A.1), the complementary slackness condition of the constraints (9k)-(9l), and (A.9)-(A.10), the followingcondition can be obtained: π pt,n d pt,n,k = − ϕ d,p,maxt,n,k d p,maxt,n,k + ϕ d,p,mint,n,k d p,mint,n,k + c upt, ψ n d pt,n,k − c downt, ψ n d pt,n,k + c dt,n,k d pt,n,k (D.6)By substituting (D.5) and (D.6) in (D.1), the relation stated in Lemma 1 is obtained. Appendix E. Final MIQCP model
This section reports the final mixed-integer quadratically constrained program.max X t ∈T X n ∈N + (cid:16) X k ∈ Ω Dt,n c dt,n,k d pt,n,k − X k ∈ Ω Gt,n c gt,n,k g pt,n,k (cid:17) − X t ∈T c pt, p t, − X t ∈T c qt, q t, + X t ∈T (cid:16) c upt, X n ∈N + ψ n r upt,n + c downt, X n ∈N + ψ n r downt,n (cid:17) − τ X t ∈T X n ∈N + (cid:16) D t,n + X k ∈ Ω Dt,n d p,maxt,n,k + X k ∈ Ω Gt,n g p,maxt,n,k (cid:17) − X ( i,j ) ∈L X m ∈M u i,j,m ( K fixi,j,m + K vari,j,m F i,j,m ) (E.1) s.t. (2b) (E.2) X t ∈T X n ∈N + (cid:16) X k ∈ Ω Gt,n ( − ϕ g,p,maxt,n,k g p,maxt,n,k + ϕ g,p,mint,n,k g p,mint,n,k − c upt, ψ n g pt,n,k + c downt, ψ n g pt,n,k − c gt,n,k g pt,n,k )+ X k ∈ Ω Dt,n ( − ϕ d,p,maxt,n,k d p,maxt,n,k + ϕ d,p,mint,n,k d p,mint,n,k + c upt, ψ n d pt,n,k − c downt, ψ n d pt,n,k + c dt,n,k d pt,n,k ) + π pt,n D t,n (cid:17) − X t ∈T c pt, p t, + τ X t ∈T X n ∈N + (cid:16) D t,n + X k ∈ Ω Dt,n d p,maxt,n,k + X k ∈ Ω Gt,n g p,maxt,n,k (cid:17) ≥ K op + X j ∈M u i,j,m ( K fixi,j,m + K vari,j,m F i,j,m ) (E.3)(2d) − (2g) (E.4)(9b) − (9b) (E.5)(E.6)19 t,n = X ( i,j ) ∈L : i = n X m ∈M (cid:0) y w ii ,ut,i,j,m a i,j,m − y w ij p,ut,i,j,m a i,j,m + y w ij q,ut,i,j,m e i,j,m (cid:1) + X ( i,j ) ∈L : j = n X m ∈M (cid:0) y w ii ,ut,j,i,m a i,j,m − W ji,pt,j,i u i,j,m a i,j,m + W ji,qt,j,i u i,j,m e i,j,m (cid:1) ∀ t ∈ T , ∀ n ∈ N (E.7) q t,n = X ( i,j ) ∈L : i = n X m ∈M (cid:0) y w ii ,ut,i,j,m e i,j,m − y w ij p,ut,i,j,m e i,j,m − y w ij q,ut,i,j,m a i,j,m (cid:1) + X ( i,j ) ∈L : j = n X m ∈M (cid:0) y w ii ,ut,j,i,m e i,j,m − y w ji p,ut,i,j,m e i,j,m − y w ij q,ut,i,j,m a i,j,m (cid:1) ∀ t ∈ T , ∀ n ∈ N (E.8) X m ∈M (cid:0) y w ii ,ut,i,j,m a i,j,m − y w ij p,ut,i,j,m a i,j,m + y w ij q,ut,i,j,m e i,j,m (cid:1) ≤ F maxi,j + X m ∈M u i,j,m F i,j,m ∀ t ∈ T , ∀ ( i, j ) ∈ L (E.9) X m ∈M (cid:0) y w ii ,ut,j,i,m a i,j,m − y w ji p,ut,i,j,m a i,j,m + y w ji q,ut,i,j,m e i,j,m (cid:1) ≤ F mini,j + X m ∈M u i,j,m F i,j,m ∀ t ∈ T , ∀ ( i, j ) ∈ L (E.10) W iit, + 2 X ( i,j ) ∈P n X k : i ∈P k X m ∈M (cid:0) a i,j,m y p,ut,i,j,m,k a i,j,m + e i,j,m + e i,j,m y q,ut,i,j,m,k a i,j,m + e i,j,m (cid:1) ≤ ( v maxt,n ) ∀ t ∈ T , ∀ n ∈ N + (E.11)(9i) − (9w) (E.12)(A.1) − (A.6) (E.13) π pt,n + λ pt,n + 2 X ( i,j ) ∈P n X k : i ∈P k X m ∈M a i,j,m y β,ut,i,j,m,k a i,j,m + e i,j,m = 0 ∀ t ∈ T , ∀ n ∈ N + (E.14) π qt,n + λ qt,n + 2 X ( i,j ) ∈P n X k : i ∈P k X m ∈M e i,j,m y β,ut,i,j,m,k a i,j,m + e i,j,m = 0 ∀ t ∈ T , ∀ n ∈ N + (E.15)(A.9) − (A.10) (E.16) X m ∈M (cid:16) a i,j,m y λ,p,ut,i,j,m + e i,j,m y λ,q,ut,i,j,m − a i,j,m y µ maxt,i,j ,ut,i,j,m (cid:17) + ǫ pt,i,j + η at,i,j = 0 ∀ t ∈ T , ∀ ( i, j ) ∈ L (E.17) X m ∈M (cid:16) a i,j,m y λ,p,ut,j,i,m + e i,j,m y λ,q,ut,j,i,m − a i,j,m y µ mint,i,j ,ut,i,j,m (cid:17) − ǫ pt,i,j = 0 ∀ t ∈ T , ∀ ( i, j ) ∈ L (E.18) X m ∈M (cid:16) − e i,j,m y λ,p,ut,i,j,m + a i,j,m y λ,q,ut,i,j,m + e i,j,m y µ maxt,i,j ,ut,i,j,m (cid:17) + ǫ qt,i,j + η bt,i,j = 0 ∀ t ∈ T , ∀ ( i, j ) ∈ L (E.19) X m ∈M (cid:16) − e i,j,m y λ,p,ut,j,i,m + a i,j,m y λ,q,ut,j,i,m + e i,j,m y µ mint,i,j ,ut,i,j,m (cid:17) + ǫ qt,i,j = 0 ∀ t ∈ T , ∀ ( i, j ) ∈ L (E.20) X ( i,j ) ∈L : i = n (cid:16) X m ∈M (cid:0) − a i,j,m y λ,p,ut,i,j,m − e i,j,m y λ,q,ut,i,j,m + y µ maxt,i,j ,ut,i,j,m a i,j,m (cid:1) + η ct,i,j − γ t,i,j (cid:17) + X ( i,j ) ∈L : j = n (cid:16) X m ∈M (cid:0) − a i,j,m y λ,p,ut,j,i,m − e i,j,m y λ,q,ut,j,i,m + y µ mint,i,j ,ut,i,j,m a i,j,m (cid:1) − η ct,i,j + γ t,i,j (cid:17) + χ maxt,n − χ mint,n = 0 ∀ t ∈ T , ∀ n ∈ N + (E.21)(A.16) (E.22)(B.1) (E.23)(C.1) − (C.24) (E.24) References [1] B. 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