A Framework for Eliminating Paradoxical Orders in European Day-Ahead Electricity Markets through Mixed-Integer Linear Programming Strong Duality

A F ramew ork for Eliminating P arado xical Orders in Europ ean Da y-Ahead Electricit y Mark ets through Mixed-In teger Linear Programming Strong Dualit y Zhen W ang a, ∗ , Mohammad Reza Hesamzadeh a , Sh udian Zhao b , Jan Kronqvist b a KTH R oyal Institute of T e chnolo gy, Department of Ele ctric al Engine ering and Computer Scienc e, SE-100 44, Sto ckholm, Swe den b KTH R oyal Institute of T e chnolo gy, Department of Mathematics, SE-100 44, Sto ckholm, Swe den Abstract The presence of in teger v ariables in the Europ ean da y-ahead electricity mark et renders the social w elfare maximization problem non-conv ex and non-diﬀeren tiable, making classical marginal pricing theoretically inconsistent. Existing pricing mechanisms often struggle to balance rev enue adequacy with incen tiv e compatibilit y , typically relying on discriminatory uplift pa ymen ts or suﬀering from w eak dualit y . Lev eraging the Augmented Lagrangian Dualit y (ALD) framew ork, whic h establishes strong duality for Mixed-In teger Linear Programming (MILP), this paper prop oses a nov el ALD pric- ing mec hanism. W e analytically pro ve that this mec hanism is inheren tly incen tiv e-compatible, aligning cen tralized dispatc h with individual incentiv es without requiring side pa ymen ts. Notably , we demon- strate that the ALD price signals in trinsically eliminate Parado xically Rejected Orders (PROs) and P aradoxically Accepted Orders (P AOs) for supply orders. F or the demand side, a suﬃcient condition is introduced to further guaran tee rev enue adequacy , resulting in a transparent and ﬁnancially consis- ten t settlement system. T o ensure computational tractabilit y , we mo dify the Surrogate Absolute-V alue Lagrangian Relaxation (SA VLR) metho d to eﬃcien tly compute the exact p enalt y co eﬃcien ts and op- timal Lagrangian m ultipliers. Numerical exp erimen ts on illustrative examples and the Nordic 12-area electricit y market mo del conﬁrm the sup erior economic prop erties of the ALD pricing mec hanism and the tractabilit y of the mo diﬁed SA VLR algorithm. Keywor ds: OR in energy; Augmen ted Lagrangian and duality; Non-conv ex pricing; Decomp osition algorithms; Incen tive compatibility 1. In tro duction The Nominated Electricity Mark et Op erator (NEMO) provides price signals to coordinate dis- patc h. While conv ex market mo dels possess W alrasian equilibrium and are solv able in p olynomial time (Ah un bay et al., 2024, Section 3.2), the introduction of in teger v ariables, whic h represen t indi- visible physical realities lik e start-up costs and complex "all-or-nothing" bid structures, renders the ∗ Corresp ond ing author Email addr esses: zhenwa@kth.se (Zhen W ang), mrhesa@kth.se (Mohammad Reza Hesamzadeh), shudian@kth.se (Sh udian Zhao), jankr@kth.se (Jan Kronqvist) so cial-w elfare optimization problem non-con v ex. This non-conv exit y creates a duality gap where clas- sical marginal prices cannot guarantee that all market orders remain reven ue-adequate. Consequently , curren t pricing mechanisms rely on ex-p ost uplift pa yments, which are often discriminatory and lac k transparency (Madani et al., 2018). 1.1. Liter atur e R eview The foundational structure of the Europ ean day-ahead energy mark et, encompassing diverse bid t yp es, complex op erational constrain ts, and welfare form ulations for orders, is comprehensiv ely detailed b y Martin et al. (2014); Chatzigiannis et al. (2016). Research works hav e consistently sough t to mak e the mo del b oth comprehensive and computationally tractable. F or instance, Vlachos et al. (2016) in- tro duced ﬂexible hourly proﬁle orders and compared a computationally eﬃcien t Linear Programming (LP) mo del with a ﬂexible yet computationally demanding Mixed Complementarit y Problem (MCP) form ulation. T o enhance compatibilit y with state-of-the-art solv ers, Sleisz et al. (2019) prop osed new MILP formulations for minim um income, scheduled stop, and load gradient conditions. F urthermore, Ilea et al. (2018) utilized a Mixed-In teger Quadratically Constrained Programming (MIQCP) mo del to analyze the correlation b etw een Parado xically A ccepted Orders (P AOs) and Parado xically Rejected Orders (PROs), while Kuang et al. (2019) incorp orated conv ex quadratic deliv erabilit y costs for re- new able integration, though their pricing approach can inadverten tly generate negativ e prices. 1.1.1. Pricing Me chanisms and the Uplift Pr oblem Due to inherent non-conv exit y , classical pricing mec hanisms often result in negativ e proﬁts for mark et orders, necessitating frameworks that ensure the market orders remain “in-the-money .” • IP and mo diﬁed IP Pricing: The Integer Programming (IP) pricing mec hanism (O’Neill et al., 2005) solves the centralized problem and ﬁxes binary v ariables to their optimal v alues, utilizing dual v ariables for marginal pricing and uplift paymen ts. The mo diﬁed IP (Bjørndal and Jörnsten, 2008) impro ves up on this by ﬁxing sp eciﬁc v ariables and generating v alid supp ort inequalities to pro vide a non-volatile, non-discriminatory price function for pure integer programming problems. • Alternativ e Rules: Other metho ds include uniform pricing for buy ers and sellers separately to minimize comp ensation costs (T o czyłowski and Zoltowsk a, 2009), and the recen tly prop osed “markup” mechanism by Ahun bay et al. (2024), where sellers receiv e a price signal p α and buyers pa y by another price (1 + α ) p α . Although this mechanism reduces uplift paymen t, it requires sp eciﬁc rounding techniques. More recen tly , the IP and Min-MWP (make-whole paymen t) rule (Ah unba y et al., 2025) w as introduced to minimize side pa yments while main taining congestion signals. A critical comparison of mechanisms up to 2016 is pro vided b y Lib erop oulos and An- drianesis (2016) and extended b y Azizan et al. (2020) through an Equilibrium-Constrained (EC) pricing mec hanism. 2 1.1.2. Convex Hul l Pricing (CHP) and L ong-R un Imp acts Con vex Hull Pricing (CHP) is widely utilized in US markets to minimize uplift while main taining uniform prices (Gribik et al., 2007). Analyses by Stevens et al. (2024) and long-run mark et studies b y By ers and Hug (2023) suggest that CHP p erforms optimally regarding so cial welfare and pro ducer surplus transfer. T echnicall y , Hua and Baldick (2016) prov ed that formulating the conv ex hull is equiv- alen t to solving a Lagrangian relaxation problem. While subgradien t and cutting-plane methods (W ang et al., 2013) can compute an approximation price, the exact conv ex hull formulation remains computa- tionally diﬃcult. S p ecialized solutions exist for sp eciﬁc generators, such as those in the Midcon tinent Indep enden t System Op erator (MISO) system (Y u et al., 2020), or for thermal units (Knuev en et al., 2022). Although Dantzig-W olfe decomp osition (Andrianesis et al., 2021) and extreme p oint meth- o ds (Y ang et al., 2019) can ﬁnd exact conv ex-h ull prices, they face hea vy computational burdens in large-scale m ulti-p erio d mo dels. 1.1.3. Par adoxic al Or ders and Investment Inc entives The long-term health of the electricit y market dep ends heavily on pricing stabilit y and reliable in vestmen t incen tiv es. Researc h comparing capacity expansion and unit commitment mo dels by Ma ys et al. (2021) suggests that pricing mechanisms signiﬁcantly aﬀect the equilibrium capacity mix. In the Europ ean con text, av oiding P A Os is essential; ho wev er, the required reven ue adequacy constrain ts are t ypically nonlinear. Madani and V an V yv e (2015, 2017) developed eﬃcient MIP form ulations and linearized minim um income conditions without introducing auxiliary contin uous or binary v ariables. Nev ertheless, these metho ds struggle computationally when blo ck orders dominate the market clearing. Later work b y Madani and V an V yve (2018) generalized these developmen ts as “Minimum Proﬁt” (MP) conditions and prop osed a decomp osition pro cedure utilizing Benders cuts to handle complex ramping constrain ts. Despite these extensiv e adv ances, existing pricing mechanisms for non-conv ex markets fundamen- tally rely on weak duality , or ex-p ost uplift pa yments. The prop erties of these rules are linked to the optimalit y gap; for example, By ers and Hug (2022) observed that as the duality gap narrows, con- sumer surplus t ypically increases. F urthermore, from an in vestmen t p ersp ectiv e, linear pricing rules ha ve generally been sho wn to be more eﬃcient than nonlinear pricing mechanisms (Herrero et al., 2015). T o address these deﬁciencies, this pap er prop oses a transparen t piecewise-linear pricing mechanism for the European da y-ahead mark et. Exploiting MILP strong duality under the ALD framew ork (F eizollahi et al., 2017), the prop osed framework in trinsically eliminates parado xical orders (a suﬃcient condition is needed for demand-side P AO elimination) without the need for explicit rev enue adequacy constrain ts within the optimization mo del. The con tribution of this pap er: This pap er prop oses a no v el pricing mec hanism based on the Augmen ted Lagrangian Dualit y (ALD) approac h for the Europ ean day-ahead market. Our con tribu- 3 tions are: • (C1) Mechanism F ormulation and Economic Prop erties : W e prop ose a piecewise-linear pricing rule (ALD pricing) that eliminates PR Os entirely and preven ts P A Os for supply orders. If demand bids exceed supply oﬀer prices, P AOs among demand orders are also eliminated. This mec hanism is incen tive-compatible and requires zero uplift paymen ts due to transparen t price signals. This mechanism generalizes classical Lo cational Marginal Pricing (LMP) b y extending it from an LP to an MILP mo del, in which the p enalt y co eﬃcient simply v anishes. F urthermore, this pricing mec hanism can also provide congestion prices for T ransmission System Op erators (TSOs). • (C2) Algorithmic Innov ation : W e mo dify the Surrogate Absolute-V alue Lagrangian Relax- ation (SA VLR) metho d (Bragin et al., 2018) to close the MILP duality gap. Using the decom- p osable structure of the welfare maximization problem and motiv ated b y recent results regarding the p olynomial-time computation of exact p enalt y co eﬃcients (Lefeb vre and Sc hmidt, 2024), w e ensure that the prop osed ALD mechanism is computationally tractable. • (C3) Comprehensiv e Comparative Analysis and Numerical V alidation : W e extensiv ely v alidate the prop osed ALD mechanism to bridge theory and practice. First, using an established b enc hmark (Schiro et al., 2016), we compare our mechanism with the existing sc hemes across the four dimensions: individual reven ue adequacy , incentiv e compatibility , economic eﬃciency , and transparency . The analysis demonstrates that the ALD mec hanism outp erforms traditional b enc hmarks b y successfully reconciling these often-conﬂicting economic properties (v alidating C1). F urthermore, through a stylized computation on the Nordic 12-area electricit y mark et, we conﬁrm not only the mechanism’s welfare equiv alence but also the computational tractability of our prop osed algorithm at a practical scale (v alidating C2). A ccording to the discussion ab o v e, we up date (Azizan et al., 2020, p. 481, T able 1) b y adding the ALD pricing mechanism as in T able 1. In addition, the prop osed ALD pricing mechanism formulated in T able 1 has a form ula: λq i + ρq ∗ i − ρ | q i − q ∗ i | , where λ and ρ are the ALD pricing signals to b e introduced in the following sections, and q ∗ i is the optimal solution of the w elfare-maximization problem. It is clear that the prop osed ALD pricing mechanism is piecewise linear. 4 T able 1: Summary of ma jor pricing schemes and their prop erties. Sc heme /prop ert y Price form p i ( q i ) = Prop osed for c i ( q i ) = Mark et clearing Rev enue A dequate Supp orts comp etitiv e equilibrium Economically Eﬃcien t Shado w pricing λ q i Con vex ✓ ✓ ✓ ✓ IP λ q i + u i 1 q i > 0 Start-up plus linear ✓ ✓ ✓ ✓ CHP λ q i + u i 1 q i = q ∗ i Start-up plus linear ✓ ✓ ✓ ✗ SLR λq i Start-up plus linear ✓ ✓ ✗ ✗ PD λq i Start-up plus linear ✓ ✓ ✗ ✗ EC User-sp eciﬁed General ✓ ✓ ✓ ✓ ALD (Prop osed) λ q i + ρ q ∗ i − ρ | q i − q ∗ i | Start-up plus linear ✓ ✓ ✓ ✓ Step 1: Problem F ormulation Non-conv ex Europ ean mark et clearing mo del formulated as a MILP (1) (Prop erties of the dual problem: Lemma 1) Step 2: ALD Pricing Mecha- nism & Economic Prop erties (C1) • Mechanism: Deﬁnition 1 (ALD Price Signals), Deﬁnition 2 (Decentralized Optimization) • Incentiv e Compatibility: Deﬁnition 3, Theorem 2 (IC), Theorem 4 (PRO-F ree) • Reven ue Adequacy: Prop ositions 1 & 2 (TSO of a Radial Netw ork), Theorem 3 (Supply), Corollary 1 (P AO-F ree Condition of Demand) Step 3: Computational Approach (C2) • Decomp osition: Deﬁnition 4 (MILP Decomposition) • Algorithms: Algorithm 1 (Primal Up date), Algorithm 2 (The complete algorithm) • Conv ergence: Theorem 5 (Dual Conv ergence) Step 4: Comparative Analysis (C3) Comparison with b enchmark pricing mechanisms via an illustrative example from (Schiro et al., 2016) Step 5: Numerical V alidation (C3) Nordic 12-area Electricity Market Computational tractability & W elfare equiv alence Identiﬁes pricing chal lenges (e.g., Uplift Payments) in standar d mar ginal pricing Metho dolo gic al F oundation: Zer o Duality Gap of MILP (The or em 1) Employs surro gate optimality con- dition (Deﬁnition 5), c ombination te chnique (R emark 1), and step- size re-initialization (R emark 2) Evaluates four p ersp e ctives: In- c entive Comp atibility, R evenue A de quacy, Ec onomic Eﬃciency, and T r ansp arency Demonstr ates the pr actic al viabil- ity and pr op erties of the pr op ose d ALD mechanism Figure 1: Conceptual roadmap linking the pap er’s organization to the contributions. 5 1.2. Pap er or ganization and the c onc eptual r o admap The remainder of this pap er is organized as follows. Section 2 presen ts a mathematical form ulation of the Europ ean day-ahead electricit y market mo del with nonconv exities retained. In Section 3, w e form ulate this mo del in to a compact MILP framew ork and apply the ALD approac h. The rest of this section in tro duces the prop osed ALD pricing mechanism and provides theoretical proofs of its core prop erties, including incentiv e compatibility , the elimination of parado xical orders. The mo diﬁed SA VLR algorithm, designed to compute the exact p enalt y co eﬃcien t and optimal Lagrangian multipli- ers with a conv ergence guarantee, is detailed in Section 4. Section 5 provides a comparative analysis, ev aluating the prop osed ALD pricing mec hanism against the main existing pricing rules using a stan- dard case study . Section 6 presents the computational results of a st ylized electricity market mo del to demonstrate the tractability of the algorithm and the economic adv antages of the prop osed ALD pricing mec hanism. Finally , Section 7 concludes the pap er and discusses av en ues for future researc h. T o na vigate the theoretical depth of this pap er, Figure 1 illustrates the conceptual roadmap of our study . T o assist in understanding the ALD pricing mechanism, three illustrativ e examples and the pro of of Lemma 1 are av ailable in the supplementary materials. 2. MILP mo del of the Europ ean da y-ahead electricit y mark et 2.1. Notations Mathematical Op erators and Several Sets • ⟨ x, y ⟩ : Inner pro duct of the vectors x and y . • ∥ y ∥ 1 : The L 1 norm of v ector y , deﬁned as P i | y i | . • x ⊙ y : Elemen t-wise pro duct (Hadamard) of v ectors x and y . • A , T , L , LS : Sets of bidding areas, time slots ( |T | = 24 ), interconnections, and small line sets, resp ectiv ely . Order-Related Index Sets • O E , O B O , O P B , O RB , O F H B : The sets of elemen tary orders, blo c k orders, proﬁle blo c k orders, regular blo c k orders, and ﬂexible hourly orders, resp ectiv ely , where O B O = O P B ∪ O RB . • Subscripts s and d denote the supply and demand subsets of an y set O (e.g., O E ,s , O E ,d , where O E = O E ,s ∪ O E ,d ). • O slg : The set of supply orders sub ject to load gradient conditions, where O slg ⊂ O E ,s . • Subscripts a and sup erscript t denote subsets of an y set O sub ject to the area a , or the time slot t , (e.g., O t E ,s,a , O t E ,d,a , where O E ,s = S a ∈A S t ∈T O t E ,s,a , O E ,d = S a ∈A S t ∈T O t E ,d,a ). 6 P arameters • p t a ( i ) , q t a ( i ) : Price and quantit y of order i at area a during p eriod t . • G dn s , G up s : Decrease and increase gradient for supply order s ( M W /min ). • R H , R D : Hourly ( H ) and daily ( D ) ramp limits for net injections, in terconnections, or line sets. • P ini , F ini ; F min lt , F max lt : Initial net injection and initial p ow er ﬂow at the start of the trading day; Minim um and maximum av ailable transmission capacit y for line l at p eriod t . • H lt aa ′ ; A LB , A E G : Po w er T ransfer Distribution F actor (PTDF) matrix for AC interconnections; Incidence matrices for link ed blo c k orders and exclusive groups. Decision V ariables • x t E ,a ( i ) ∈ [0 , 1] : A cceptance ratio for a contin uous elemen tary order i at area a and time t . • x RB ,a ( i ) , x t F H B ,a ( i ) ∈ { 0 , 1 } : Binary acceptance statuses for a regular and ﬂexible hourly blo c k order i at area a and time t . • u P B ,a ( i ) ∈ { 0 , 1 } : Binary execution status for a proﬁle blo ck order i . • p at , f lt : Net injection of area a and pow er ﬂow on interconnection l during time p erio d ( t − 1 , t ) . • e t aa ′ : Bilateral energy exchange b etw een area a and a ′ . • x t a ( i ) : A general order i at area a and time t . (Introduced for ease of notation, either an element of x RB ,a , x P B ,a , x t F H B ,a or x t E ,a ). 2.2. The ele ctricity market mo del and the discr ete optimization pr oblem Note that while we adopt the standard acronym MILP (Mixed-Integer Linear Programming) for con ven tion, our formulation is strictly a Mixed-Binary Linear Programming (MBLP) problem. Uniﬁe d F ormulations: F or eac h area a ∈ A , w e aggregate blo c k order acceptance ratios in to a single v ector x B O,a = [ x P B ,d,a , x RB ,d,a , x P B ,s,a , x RB ,s,a ] ⊤ , where x i ∈ { 0 } ∪ [ R min i , 1] for proﬁle blo c ks ( i ∈ O P B ,a ) and x i ∈ { 0 , 1 } for regular blocks ( i ∈ O RB ,a ). Corresp ondingly , the economic v aluation v ector for order type j ∈ { E , B O , F H B } is deﬁned as c t j,a = p t j,a ⊙ q t j,a (in EUR). Here, c t i ≤ 0 for demand orders and c t i ≥ 0 for supply orders. Note that for ﬂexible hourly orders ( j = F H B ), c F H B ,a is time-in v arian t. So cial w elfare maximization is formulated as a cost-minimization problem, with the mo del b enc h- mark from (Chatzigiannis et al., 2016). T o ensure the mo del reﬂects practical market op erations, w e incorp orate the follo wing conﬁgurations: (i) Storage Assets: Utility-scale storage is decoupled in to separate demand (charging) and supply (disc harging) orders across diﬀeren t p erio ds ( t 1 , t 2 ) . (ii) Cost Structure: Start-up costs are in ternalized within the supply oﬀer prices of supply blo ck orders. (iii) 7 Op erational Assumptions: All interconnections are treated as A C lines op erated by non-proﬁt entities. In addition, the minimum income condition and the sc heduled stop condition are not included in the mo del. The cost minimization problem is given by min X a ∈A X t ∈T  ⟨ c t E ,a , x t E ,a ⟩ + ⟨ c t B O,a , x B O,a ⟩ + ⟨ c F H B ,a , x t F H B ,a ⟩  , (1a) s.t. p t a = ⟨ q t E ,a , x t E ,a ⟩ + ⟨ q t B O,a , x B O,a ⟩ + ⟨ q F H B ,a , x t F H B ,a ⟩ , ∀ t ∈ T , a ∈ T , (1b) X a ∈A p t a = 0 , p t a = X a ′ ∈A ( e t aa ′ − e t a ′ a ) , ∀ t ∈ T , a ∈ A , (1c) f lt = X a,a ′ H lt aa ′ e t aa ′ , F min lt ≤ f lt ≤ F max lt , ∀ l ∈ L , t ∈ T , (1d) R min P B ,a ( i ) u P B ,a ( i ) ≤ x P B ,a ( i ) ≤ u P B ,a ( i ) , ∀ i ∈ O P B ,a , (1e) x B O,a ≤ A LB x B O,a , A E G x B O,a ≤ 1 , ∀ a ∈ A , (1f ) X t ∈T x t F H B ,a ( i ) ≤ 1 , ∀ i ∈ O F H B ,a , (1g) | q t E ,s,a ( i ) x t E ,s,a ( i ) − q t − 1 E ,s,a ( i ) x t − 1 E ,s,a ( i ) | ≤ 60 G up/dn s , ∀ i ∈ O slg ,a , t ∈ T \ { 1 } (1h) | p at − p a,t − 1 − p H,ini at | ≤ R H,up/dn at , ∀ a ∈ A , t ∈ T , (1i) | X t p at − p D,ini a | ≤ R D,up/dn a , ∀ a ∈ A , (1j) | f lt − f l,t − 1 − F H,ini lt | ≤ R H,up/dn lt , ∀ l ∈ L , t ∈ T , (1k) | X l ∈ L ls f lt − X l ∈ L ls f l,t − 1 − F H,ini lt | ≤ R H,up/dn lt , ∀ ls ∈ LS, t ∈ T . (1l) Interpr etations of the optimization pr oblem (1) . • Ob jective F unction (1a): Minimizes the total system costs (or maximizes so cial welfare), aggre- gating bids from Elemen tary orders x t E ,a , Blo ck Orders (BO) x B O,a , and Flexible Hourly Orders (FHB) x t F H B ,a . • P ow er Balance and Netw ork Physics (1b)–(1d): Equation (1b) calculates the net no dal injection p t a b y summing all cleared volumes. Mark et-wide balance and no dal ﬂo w conserv ation are giv en b y (1c). T ransmission constrain ts (1d) enforce physical capacit y limits ( F min / max ) based on the PTDF matrix H . • Constrain ts of Orders (1e)–(1g): Capturing the complex logic of European orders. – Proﬁle and Regular Blocks: (1e) restricts proﬁle block acceptance ratios to x ∈ { 0 } ∪ [ R min , 1] . Regular blo cks are treated as a sp ecial case where R min = 1 , resulting in a binary outcome. – Dep endencies and Exclusivity : (1f) enforces paren t-child requiremen ts for link ed orders, while (1g) ensures m utual exclusivity within predeﬁned groups. 8 – Flexible Or ders : (1g) conﬁrms that ﬂexible hourly orders are sc heduled in at most one time p eriod across the trading day . • Op erational Gradients: Constraints (1h) impose load gradien t limits on supply elementary orders. • System Ramping: The set of inequalities (1i)–(1l) c haracterizes multi-temporal ramping limits, including hourly and daily constrain ts on net injections ((1i), (1j)), individual interconnection ﬂo w (1k), and ﬂo ws for sp eciﬁed line sets (1l). Due to the blo ck orders in (1e)–(1g), the standard marginal pricing often requires discriminatory uplift pa yments. 3. The ALD-based pricing mo del T o simplify the application of the ALD approach, w e ﬁrst formulate the constrain ts of (1) into compact feasible sets. De c omp ose d V ariable and the F e asible Set. W e decomp ose the decision v ariables into blo ck vectors x E , x B O , x F H B , p represen ting elemen tary , blo c k, ﬂexible hourly orders, and the net injections across all areas and p eriods. The feasible sets of each decomp osed v ariable are formulated as follows: • X E : Deﬁned by x t E ,a ( i ) ∈ [0 , 1] and load gradien t constraints (1h). • X B O : x P B ,a ( i ) with disjunctive set { 0 } ∪ [ R min B O , 1] (e.g, (1e) and x RB ,a ( i ) ∈ { 0 , 1 } , sub ject to logic constrain ts (1f). • X F H B : x t F H B ,a ( i ) ∈ { 0 , 1 } with temp oral constraint (1g). • P : p t a deﬁned b y the p o w er balance (1c), ﬂo w limits (1d), and all system ramp limits (1i)–(1l), whic h is detailed in Section 4.2. The feasible set of the problem (1) is deﬁned by X : = X E × X B O × X F H B , where X E and P are p olyhedra; X B O and X F H B con tain integer constraints, characterizing the problem as an MILP . Comp act formulation. Using aggregated notation, the optimization problem (1) is formulated as min x ∈ X,p ∈ P X a ∈A X t ∈T  ⟨ c t E ,a , x t E ,a ⟩ + ⟨ c t B O,a , x B O,a ⟩ + ⟨ c F H B ,a , x t F H B ,a ⟩  (2a) s.t. p t a = ⟨ q t E ,a , x t E ,a ⟩ + ⟨ q t B O,a , x B O,a ⟩ + ⟨ q F H B ,a , x t F H B ,a ⟩ , ∀ a, t. (2b) where (2b) represen ts the global coupling constrain ts that link the acceptance decisions of all orders with the net injection set P . 9 ALD F ormulation. W e no w apply the ALD approach b y dualizing the coupling constraints (2b) with m ultipliers λ t a and augmenting the ob jectiv e with an L 1 -norm p enalt y term with co eﬃcient ρ . The ALD problem is deﬁned as max λ ∈ R |A|×|T | z ALD ρ ( λ ) : = max λ min x ∈ X,p ∈ P L ρ ( x, p, λ ) , (3) where the Augmen ted Lagrangian function L ρ is L ρ ( x, p, λ ) := X a,t   X j ∈ E ,B O ,F H B ⟨ c t j,a , x t j,a ⟩   + X a,t λ t a   p t a − X j ⟨ q t j,a , x t j,a ⟩   + ρ X a,t       p t a − X j ⟨ q t j,a , x t j,a ⟩       1 . (4) 3.1. Str ong duality of the ALD appr o ach Strong duality is an imp ortan t prop ert y in pricing mechanisms, as argued in Section 1. This section b egins with a lemma that introduces the prop erties of the dual optimization problem for (3). Lemma 1 (Characterization of the MILP Dual) . The dual optimization pr oblem max λ ∈ R |A|×|T | z ALD ρ ( λ ) of (3) is c onc ave, wher e the obje ctive function is pie c ewise line ar and sub-diﬀer entiable. Note that Lemma 1 is a classical result, and the pro of has b een omitted. The prop erties of the dual problem of a Lagrange relaxation of the MILP are stated in (Bragin et al., 2015, p.117), and the additional p enalt y term do es not inﬂuence these prop erties, see the detailed pro of in the Supplement. In the follo wing, w e will in tro duce a strong duality theory for the ALD approac h describ ed abov e for the MILP problem. In addition, readers interested in closing the dualit y gap can refer to (Chen and Chen, 2010; Lefebvre and Schmidt, 2024). Note that, unlik e the augmented Lagrangian relaxation pro cedure, the approach of (Chen and Chen, 2010) requires only a p enalt y . Theorem 1 (Exact Penalt y Co eﬃcien t) . L et ( x ∗ , p ∗ ) b e an optimal solution to the MILP pr oblem (2) with optimal obje ctive value z M I P . Deﬁne z ALD ρ ( λ ) as the optimal value of the dual pr oblem (3) . 1. If λ ∗ maximizes the dual function, ther e exists a ﬁnite thr eshold 0 < ρ ∗ < ∞ such that z ALD ρ ( λ ∗ ) = z M I P ∀ ρ ≥ ρ ∗ , eﬀe ctively closing the duality gap (F eizol lahi et al., 2017, The or em 4). 2. F or any arbitr ary λ ∈ R |A|×|T | , ther e exists a ﬁnite ρ ( λ ) > 0 such that z ALD ρ ( λ ) ( λ ) = z M I P (F eizol- lahi et al., 2017, Pr op osition 8). Theorem 1 establishes that for a giv en λ ∗ , a threshold ρ ∗ exists that eliminates the dualit y gap. F urthermore, for an y ρ > ρ ∗ , the optimal solution set of the ALD-based problem (3) coincides with that of the original MILP (2). In this context, λ ∗ is said to supp ort an exact p enalt y represen tation for the MILP problem (F eizollahi et al., 2017, Proposition 1). Here, a fundamental question arises, for λ ∗ ∈ R |A|×|T | , one can set a very large penalty co eﬃcien t to close the duality gap. How ev er, suc h a p enalt y coeﬃcient will cause computational issues, and 10 more imp ortan tly , it is meaningless when in terpreting in practice, see the discussion in (Lefebvre and Sc hmidt, 2024). Hence, our aim is to determine a v alid p enalt y co eﬃcient ρ ∈ [ ρ ∗ , ρ ] that is larger than the threshold ρ ∗ , but remains as close to this lo wer b ound as computationally feasible. 3.2. The pr op ose d ALD pricing me chanism Assume that the exact p enalt y co eﬃcient ρ for the ALD problem (3) has b een determined such that ρ ∈ [ ρ ∗ , ρ ] , ensuring strong duality . Let Λ denote the set of optimal dual v ariables λ t, ∗ ∈ R |A|×|T | for problem (1). Since problem (2) is an MILP , its optimal solution ( x ∗ , p ∗ ) , which do es not need to b e unique, can b e eﬃciently obtained using commercial solvers such as Gurobi or CPLEX. Based on these parameters, w e deﬁne the ALD pricing mechanism as follows: Deﬁnition 1 (ALD Price Signals) . A t e ach time slot t ∈ T , the lo c ational mar ginal pric e (LMP) of the bidding ar e a a ∈ A , denote d by LM P t a , is deﬁne d as LM P t a : = η t a + min λ t, ∗ a ∈ Λ λ t, ∗ i , s.t. λ t, ∗ a + η t a ∈ Λ , η t a ≤ ρ, a ∈ A , t ∈ T , (5) wher e η t a is a p ar ameter use d to derive c ongestion pric e signals, while exc essive tr ansmission inc entives ar e mitigate d by the upp er b ound ρ . F or a supply(demand) or der i , its r evenue(c ost) c onsists of the c ommo dity r evenue(c ost): LM P t a q t ( i ) x t ( i ) , a non-c onvex r ewar d(char ge): ρq t ( i ) x t, ∗ ( i ) , and a deviation p enalty: ρ | q t ( i )( x t ( i ) − x t, ∗ ( i )) | . The T r ansmission System Op er ator (TSO) c ol le cts c ongestion r evenue b ase d on the Financial T r ans- mission R ight (FTR), deﬁne d as F T R t aa ′ : = LM P t a ′ − LM P t a , wher e p ower ﬂows thr ough line l fr om no de a to no de a ′ . A ccording to Def. 1, the aggregate daily surplus for an order i is S i = X t ∈T  LM P t a q t ( i ) x t ( i ) − ρ | q t ( i )  x t ( i ) − x t, ∗ ( i )  | + ρq t ( i ) x t, ∗ ( i ) − p t ( i ) q t ( i ) x t ( i )  . (6) The ov erall daily w elfare for the TSO, assuming no ﬂow deviation, is P l ∈L P t ∈T f ∗ lt F T R t l , where f ∗ lt is the ﬂo w of line l during the time p eriod [ t − 1 , t ] . Note that in Eq. (5), the minimal optimal dual v ariables are selected to ensure the economic eﬃciency of the pricing mechanism. F urthermore, the introduction of η t a facilitates congestion price disco very when netw ork constraints are activ e, and its selection is problem-dep enden t. 3.2.1. De c entr alize d optimization, inc entive c omp atibility and c ongestion r ents The problem in (2) represents the cen tralized mark et-clearing form ulation, from whic h w e deriv e the individual optimization problems. F or notational consistency , λ t, ∗ a denotes the lo cational marginal price (LMP) as determined b y the expression (5). 11 Deﬁnition 2 (Decen tralized Optimization) . The individual optimization pr oblem for e ach or der i in bidding ar e a a ∈ A is formulate d as fol lows: min x a ( i ) ∈ X i X t ∈T  c t a ( i ) x t a ( i ) − λ t, ∗ a q t a ( i ) x t a ( i ) + ρ   q t a ( i )∆ x t a ( i )   − ρq t a ( i ) x t, ∗ a ( i )  , or e quivalently max x a ( i ) ∈ X i X t ∈T  λ t, ∗ a q t a ( i ) x t a ( i ) − c t a ( i ) x t a ( i ) − ρ   q t a ( i )∆ x t a ( i )   + ρq t a ( i ) x t, ∗ a ( i )  , s.t. ∆ x t a ( i ) = x t a ( i ) − x t, ∗ a ( i ) , ∀ t ∈ T , a ∈ A , λ t, ∗ a by Def. 1 , ρ ∈ [ ρ ∗ , ρ ] , (7) wher e X i is the fe asible r e gion of x a ( i ) = [ x 1 a ( i ) , · · · , x |T | a ( i )] , and for blo ck or ders: x 1 a ( i ) = · · · = x |T | a ( i ) . Note that in Eq. 7, the sets X i ma y exhibit interdependencies, as seen in the blo c k order constraints (1f)–(1g). F or link ed blo c k orders, a hierarc hical solving sequence is required: the optimization prob- lems for paren t orders m ust b e solv ed b efore those for child orders. F urthermore, orders sub ject to exclusiv e group constraints are coupled and must b e solved sim ultaneously within a single optimization framew ork to ensure feasibility . Building on the deﬁnition of Incentiv e Compatibilit y in the literature, the theorem b elo w establishes a k ey prop ert y inherent in the prop osed ALD pricing mec hanism. Deﬁnition 3 (Incen tive Compatibilit y) . A pricing me chanism is inc entive-c omp atible if no market p articip ant has a ﬁnancial inc entive to deviate fr om its al lo c ate d quantity (the c entr alize d solution), (Milgr om and W att, 2022, Se ction 4.3). Theorem 2 (Incentiv e Compatibility of ALD) . The ALD pricing me chanism intr o duc e d in Def. 1 is inc entive-c omp atible (IC). Pr o of. Consider the minimization problem in (7), then the ob jective function represents the additiv e in verse of the order i ’s surplus. W e establish incentiv e compatibilit y b y proving that the market-clearing solution { x t, ∗ a ( i ) } t ∈T minimizes the individual optimization (7). Let P a,t,i denote P a,t P i ∈O t a . Step 1: The Str ong Duality of the Centr alize d ALD. Under the ALD pricing mec hanism with λ ∗ and ρ ∈ [ ρ ∗ , ρ ] , the dual problem (3) achiev es its maximum at z M I P , [see Theorem 1]. F or an y optimal decision v ariable ( x ∗ , p ∗ ) , the cen tralized ob jectiv e (4) satisﬁes L ρ ( x, p, λ ∗ ) = X a,t,i c t a ( i ) x t a ( i ) + X a,t  λ t, ∗ a r t a ( x, p ) + ρ | r t a ( x, p ) |  = z M I P . (8) where r t a ( x, p ) = p t a − P i ∈O t a q t a ( i ) x t a ( i ) is the no dal p ow er balance residual. 12 Step 2: A ggr e gation of the Individual Optimization Pr oblem. Reform ulate the ob jectiv e function de- noted as f a,i ( x a ( i )) in (7) b y adding P t λ t, ∗ a q t a ( i ) x t, ∗ a ( i ) − P t λ t, ∗ a q t a ( i ) x t, ∗ a ( i ) (=0): X t c t a ( i ) x t a ( i ) + X t λ t, ∗ a q t a ( i ) x t, ∗ a ( i ) − X t λ t, ∗ a q t a ( i ) x t a ( i ) + ρ X t   q t a ( i ) x t a ( i ) − q t a ( i ) x t, ∗ a ( i )   − ρ X t q t a ( i ) x t, ∗ a ( i ) − X t λ t, ∗ a q t a ( i ) x t, ∗ a ( i ) . (9) By aggregating the ob jective functions in (9) across all orders, w e observe that the summation of the second to last term satisﬁes ρ X a,t,i q t a ( i ) x t, ∗ a ( i ) = X t ρ X a X i q t a ( i ) x t, ∗ a ( i ) = 0 , (10) whic h is due to the p o wer balance equation (1c) ∀ t ∈ T . The summation of the ﬁnal term in (9), sp eciﬁcally − P a,t,i λ t, ∗ a q t a ( i ) x t, ∗ a ( i ) , represen ts the total congestion rev en ue allo cated to the transmission system operator (TSO). This relationship is formally established and demonstrated in Prop osition 1. Therefore, the summation of Eq. (9) ov er all orders minus the rev enue of the transmission system op erator will b e equal to Φ ag gr e ( x ) = X a,t,i c t a ( i ) x t a ( i ) + X a,t,i λ t, ∗ a  q t a ( i ) x t, ∗ a ( i ) − q t a ( i ) x t a ( i )  + ρ X a,t,i   q t a ( i ) x t a ( i ) − q t a ( i ) x t, ∗ a ( i )   . (11) Step 3: The T riangle Ine quality of the Centr alize d ALD. Note that given the optimal decision v ariables, the residuals r t, ∗ a v anish, which implies p t, ∗ a = P i q t a ( i ) x t, ∗ a ( i ) . Incorp orating this zero constant into (8), L ρ ( x, p, λ ∗ ) = X a,t,i c t a ( i ) x t a ( i ) + X a,t λ t, ∗ a  r t a − r t, ∗ a  + ρ X a,t   r t, ∗ a − r t, ∗ a   . (12) Applying the triangle inequalit y to the RHS of (12): L ρ ( x, p, λ ∗ ) ≤ X a,t,i c t a ( i ) x t a ( i ) + X a,t λ t, ∗ a ( p t a − p t, ∗ a ) + X a,t λ t, ∗ a X i ∈O t a  q t a ( i ) x t, ∗ a ( i ) − q t a ( i ) x t a ( i )  + ρ X a,t   | p t a − p t, ∗ a | + X i ∈O t a | q t a ( i ) x t a ( i ) − q t a ( i ) x t, ∗ a ( i ) |   . (13) The RHS of Eq. (13) has tw o additional terms as follo ws, compared to Eq. (11): X a,t λ t, ∗ a ( p t a − p t, ∗ a ) + ρ X a,t | p t a − p t, ∗ a | . (14) Recall the deﬁnition of the linear set P in Section 3, ∀ p t a ∈ P , it satisﬁes P a,t p t a = 0 . By adding the term P a,t p t a to Eq. (14), and considering the constrain t p t a ∈ P , the net injection subproblem is form ulated as min p ∈ P Ψ( p ) := min p ∈ P X a,t p t a + X a,t λ t, ∗ a ( p t a − p t, ∗ a ) + ρ X a,t | p t a − p t, ∗ a | ! . (15) Replace ab o v e in Eq. (13), we ha v e the following b ound: L ρ ( x, p, λ ∗ ) ≤ Φ ag gr e ( x ) + Ψ( p ) − X a,t p t a . (16) 13 F or the optimal dispatch p ∗ ∈ P , Ψ( p ∗ ) = 0 , and therefore, the optimal ob jective function v alue of (15) is alw ays non-p ositiv e, and thus z M I P = min x ∈ X,p ∈ P L ρ ( x, p, λ ∗ ) ≤ min x ∈ X Φ ag gr e ( x ) + min p ∈ P Ψ( p ) − min p ∈ P X a,t p t a ≤ min x ∈ X Φ ag gr e ( x ) . (17) Step 4: Contr adiction and Optimality. Assume an order i can improv e its welfare by deviating to x a ( i )  = x ∗ a ( i ) . This implies f a,i ( x a ( i )) < f a,i ( x ∗ a ( i )) , and consequen tly min x ∈ X Φ ag gr e ( x ) < Φ ag gr e ( x ∗ ) = z M I P . Ho w ever, this contradicts the lo wer b ound z M I P established in (17). Therefore, { x t, ∗ a ( i ) } t ∈T is the optimal solution for eac h decen tralized problem, thus the ALD mec hanism is incentiv e-compatible. F ollowing the pro of of Theorem 2, the following prop osition characterizes the reven ue for transmis- sion system op erators. Prop osition 1 (Aggregate Congestion Rent) . The summation of the ﬁnal term in (9) over al l or- ders, given by − P a,t,i λ t, ∗ a q t a ( i ) x t, ∗ a ( i ) , e quals the total c ongestion r evenue al lo c ate d to the T r ansmission System Op er ator. Pr o of. First, recall the optimal net injection p t, ∗ a = P i ∈O t a q t a ( i ) x t, ∗ a ( i ) (e.g., 1b). The total reven ue term can b e expressed in a vector form as − X a,t,i λ t, ∗ a q t a ( i ) x t, ∗ a ( i ) = − X t ∈T X a ∈A λ t, ∗ a p t, ∗ a = − X t ∈T λ t, ∗⊤ p t, ∗ . (18) F ollowing the no dal p ow er balance p ∗ t = B ⊤ f ∗ t (W o o d et al., 2013), where B ∈ R | L |×|A| is the line-no de incidence matrix and f ∗ t is the optimal ﬂo w vector, we obtain − X t ∈T λ t, ∗⊤ ( B ⊤ f ∗ t ) = X t ∈T ( B λ t, ∗ ) ⊤ ( − f ∗ t ) . (19) The term B λ t, ∗ yields a v ector of dual price diﬀerentials across each interconnection (e.g., λ t, ∗ a ′ − λ t, ∗ a ′′ for a line from a ′ to a ′′ ). Therefore, ( B λ t, ∗ ) ⊤ ( − f ∗ t ) is the aggregate congestion rent during [ t − 1 , t ] . This concludes the pro of. F rom Prop osition 1, we can deduce that the individual optimization ob jectives deﬁned in Def. 2 eﬀectiv ely internalize transmission rents through the ALD congestion prices established in Def. 1. Prop osition 2 (Reven ue Adequacy for TSO of Radial Net works) . The aggr e gate daily c ongestion r ent of a r adial network under the ALD pricing me chanism is p ositive 1 . Pr o of. The pro of of Prop osition 2 follows from that the energy alwa ys ﬂows from a low-price area to a high-price area (i.e., λ t, ∗ a ′ < λ t, ∗ a ′′ ) for a radial netw ork structure, whic h is the same as the con v ex lo cational marginal pricing. 1 Note that the rev enue adequacy property may not hold for the TSO of a general net work structure. 14 3.2.2. Individual r evenue ade quacy In this section, w e examine whether the ALD pricing mechanism inheren tly guarantees individual rev enue adequacy—a prop ert y often lacking in existing pricing models such as the IP pricing mecha- nism. While some mechanisms, such as the completely p ositiv e programming (CPP) based approach (Guo et al., 2025) and the primal-dual pricing framework (Ruiz et al., 2012), enforce reven ue adequacy via explicit constraints (i.e., surplus ≥ 0 ), suc h a “forced” form ulation can inadverten tly compromise incen tive compatibility (Guo et al., 2025). In contrast, the ALD mec hanism ensures non-negative sur- plus for supply orders as an in trinsic structural outcome. Although this guarantee do es not universally extend to the demand orders (see Corollary 1). The follo wing theorem establishes this prop ert y for all supply orders. Theorem 3 (Non-negativit y of Supply-Order Surplus) . Assuming zer o deviation for a supply or der i (i.e., x t a ( i ) = x t, ∗ a ( i ) , ∀ t ∈ T ), its daily surplus is non-ne gative under the ALD pricing me chanism deﬁne d in Def. 1. Pr o of. Let π t a = LM P t a + ρ denote the eﬀectiv e settlement price under the ALD pricing mechanism. F or a supply order i in bidding area a , the daily surplus is deﬁned as S a,i = X t ∈T  π t a q t a ( i ) − c t a ( i )  x t a ( i ) , (20) where c t a ( i ) = p t a ( i ) q t a ( i ) . Recall the individual optimization problem with ob jectiv e function f a,i in (7). F or the supply order i , we ha v e q t a ( i ) > 0 . By comparing the optimal dispatch x ∗ a ( i ) with the null dispatc h 0 ∈ X i , w e observe: 1. f a,i (0) = 0 : In that case, the nonconv ex rew ard of the supply order i is precisely settled due to the deviation. 2. f a,i ( x ∗ a ( i )) = − S a,i : When ev aluated at the optimal solution x ∗ a ( i ) (with zero deviation), the minimization ob jective in (7) reduces to the additive in v erse of the surplus. Since x ∗ a ( i ) is the optimal solution of (7), (see the pro of of Theorem 2), it follows that f a,i ( x ∗ a ( i )) ≤ f a,i (0) . Consequently , − S a,i ≤ 0 , which implies S a,i ≥ 0 . Thus, every supply order is guaranteed to be rev enue adequate. Theorem 3 considers the surplus of supply orders, which is critical for eac h pricing mec hanism. In the follo wing corollary , the surplus of demand orders will also b e inv estigated. Corollary 1 (Demand-Side P AO-F ree Condition) . Consider a demand or der i ( q t a ( i ) < 0 ), the ALD pricing me chanism do es not unc onditional ly guar ante e the r evenue ade quacy (i.e., non-ne gative sur- plus). However, r evenue ade quacy holds for demand or der i if its bidding pric e p t a ( i ) exc e e ds the oﬀer pric e of any supply or ders. (A we aker yet suﬃcient c ondition for r evenue ade quacy of demand or ders is that the bid pric e p t a ( i ) r emains higher than the oﬀer pric e of any cle ar e d supply or der i (i.e., x t, ∗ a ( i ) > 0 ) for al l t ∈ T . ) 15 Pr o of. F or the ﬁrst claim, an illustrativ e example is provided (see Example 1 of the Supplemen t), and w e fo cus on the second claim. By replacing the ob jective function in (7) by f t a,i ( x t a ( i )) , the optimization problem b ecomes min X t ∈T f t a,i ( x t a ( i )) , s.t. x t a ( i ) ∈ X i , ∀ a ∈ A , t ∈ T , λ t, ∗ a b y Def. 1 , ρ ∈ [ ρ ∗ , ρ ] . (21) The same argument as the pro of of Theorem 2 can b e used to prov e that ∀ t ∈ T , x t, ∗ a ( i ) is the optimal solution of the single-p eriod problem: min f t a,i ( x t a ( i )) s.t. x t a ( i ) ∈ X i , ∀ a ∈ A , ∀ t ∈ T , λ t, ∗ a b y Def. 1 , ρ ∈ [ ρ ∗ , ρ ] (22) Therefore, ∀ t ∈ T , if x t, ∗ a ( i ) > 0 , then the eﬀectiv e clearing price inequality LM P t a + ρ ≥ p t a ( i ) , holds for a supply order i by Theorem 3. This implies that during eac h time perio d ( t − 1 , t ) , the bidding prices of those cleared supply orders are cov ered b y LM P t a + ρ . Recall the c hoice of the dual optimal v ariable in Eq. (5), for eac h demand order i , we will hav e p t a ( i ) > LM P t a + ρ, ∀ t ∈ T . The daily surplus of demand order i calculated b y S a,i = X t ∈T | q t a ( i ) |  p t a ( i ) − ( LM P t a + ρ )  x t, ∗ a ( i ) , (23) is th us non-negative. Therefore, the bidding price condition is a suﬃcient condition for the reven ue adequacy of demand orders under the ALD pricing mec hanism. F rom Corollary 1, it is clear that, unlik e the LP mo del, the MILP form ulation exhibits unique prop erties. In the LP mo del, all bidding orders can participate in the mark et, and the market-clearing price LMP t a can ensure that there are no parado xically accepted orders. In the MILP mo del, because of the p ow er balance equation, the market op erator will ﬁrst address the mark et-clearing issue. An exp ensiv e supply order ma y sell electricity to a demand order with a low bidding price, which makes the demand order’s surplus negativ e, but that negative surplus can b e absorb ed by another demand order that has a high bidding price, which makes the ov erall welfare larger than without letting the exp ensiv e supply order pro duce electricit y . Hence, the o verall welfare after the selection by the condition of Corollary 1 can b e less than without the selection. If the market op erator wan ts to put all bidding orders into the market without selection, we prop ose eliminating paradoxically accepted orders (P AOs) from demand orders as follo ws: this w ould put additional money in to the market. In the individual optimization problem (7), the term ρ P t q t a ( i ) x t, ∗ a ( i ) serv es as a non-conv ex reward for the supply order ( q > 0 ); ho w ever, it imp oses a non-conv ex cost on the demand order ( q < 0 ). T o eliminate negativ e surplus among demand orders, w e prop ose a remedy approac h based on targeted comp ensation. Let O nwb denote the set of “not-well-behav ed” demand orders whose daily surplus is negative under the standard ALD price mec hanism. By injecting additional rev en ue: a transfer pa yment of Ω i = − 2 ρ P t q t a ( i ) x t, ∗ a ( i ) for each i ∈ O nwb , the ﬁnal term in the reform ulated individual 16 ob jective of (7) becomes − ρ P t | q t a ( i ) x t, ∗ a ( i ) | . Consequently , for an y demand order i ∈ O nwb , the optimization yields f a,i ( x ∗ a ( i )) ≤ f a,i (0) = 0 , ensuring the daily surplus non-negative. This remedy approac h, as shown by an illustrativ e example (Example 1 of the Supplemen t), can eﬀectively restore rev enue adequacy across all demand orders. W e now proceed with paradoxically rejected orders (PR Os), in tro duced b y the following theorem. Theorem 4 (PRO-F ree Prop erty of ALD) . Under the ALD pricing me chanism deﬁne d in Def. 1, the market-cle aring solution for mo del (1) is fr e e of p ar adoxic al ly r eje cte d or ders (PR Os). Pr o of. The pro of follo ws from the optimality of the ALD-based market-clearing. Supp ose, for the sak e of contradiction, that an order i is paradoxically rejected, i.e., x ∗ i = 0 despite the order b eing " in-the- money " based on the market price. By the incentiv e compatibility prop ert y established in Theorem 2, any deviation from the optimal dispatch x ∗ i = 0 to x i > 0 would result in a low er individual surplus for that order. Sp eciﬁcally , for a demand order, the marginal rev enue at the given ALD price would b e insuﬃcient to co ver its supply cost plus the exact p enalt y term. Similarly , for a demand order, the utilit y gained from the purchase w ould b e outw eighed by the total pa yment required, including the p enalt y . In conclusion, since the ALD mechanism ensures that the optimal dispatch maximizes eac h order’s surplus relative to the p enalty-adjusted prices, no order has a p ositive surplus to b e accepted at the clearing price, th us eliminating PROs. Note that if the individual optimization problem (7) has only the feasible solution 0 , we w ould not call it a PRO, ev en if the market-clearing price is higher than its cost or lo wer than its bidding price; see the second last example of the Supplement. In addition, the last example of the Supplemen t will displa y how the ALD pricing mechanism eliminates the PROs and P AOs. 4. The mo diﬁed SA VLR metho d The ALD pricing framew ork, based on MILP strong dualit y , w as detailed in Section 3.2. Since MILP is NP-hard, the computational burden of large-scale market clearing increases signiﬁcantly with prob- lem size. Although commercial solv ers utilize adv anced metho ds, such as branch-and-cut to solv e the primal problem, computing the exact dual prices remains a clear challenge. Consequently , the primary ob jective of this section is to presen t a computationally tractable solution methodology sp eciﬁcally designed for the dual problem to ensure a zero dualit y gap. 4.1. The r elate d work Recall Lemma 1, the dual problem (3) is concav e and sub-diﬀeren tiable. T o compute the optimal dual v ariable, an iteration pro cedure is needed. How ev er, computing the exact subgradient at eac h iteration step is computationally exp ensive, as eac h step requires solving a relaxed version of the original large-scale MILP . T o address this, v arious decomp osition techniques ha ve b een dev elop ed to partition 17 the problem into smaller, easily handled sub-problems. Consequently and also in the relev ant literature, a surrogate subgradient is emplo yed in place of the exact subgradien t to alleviate the computational burden. The surrogat e subgradien t metho d for MILP dual optimization was pioneered in (Zhao et al., 1999), with its conv ergence prop erties established in (Bragin et al., 2015). Subsequent adv ancements in tro duced adaptiv e step sizes and re-initialization techniques (Bragin et al., 2016) to accelerate con ver- gence. T o further enhance numerical accuracy and stabilit y , the Surrogate Absolute-V alue Lagrangian Relaxation (SA VLR) w as dev elop ed (Bragin et al., 2018) b y incorp orating an L 1 -norm p enalt y . Building up on the SA VLR framework, this pap er prop oses a modiﬁed SA VLR metho d to solv e problem (3). Our approach is sp eciﬁcally designed to eliminate the dualit y gap b y determining the optimal multiplier λ ∗ and exact p enalty co eﬃcient ρ ∈ [ ρ ∗ , ρ ] . While this strengthening procedure en tails a higher computational cost, it ensures global optimality for a non-con v ex mark et clearing. Details will b e explained in the following sections. 4.2. The de c omp ose d primal up date In the ALD formulation (3), the global constraint (1b) is relaxed but subsequently p enalized by the L 1 norm. Since this p enalt y term maintains the coupling b etw een v ariables, w e temp orarily exclude it to obtain the decomp osed subproblems. Deﬁnition 4 (MILP Decomp osition) . W e c an de c omp ose the fe asible set X as X i by exploiting the sep ar able structur e in (1) , and the set of net inje ctions P is not sep ar able. The de c omp ose d fe asible sets ar e: 1. X t E ,d,a,i := { x t E ,d,a ( i ) | x t E ,d,a ( i ) ∈ X t E ,d,a,i ⊂ R 1 } , the fe asible set for a demand elementary or der i at time t and ar e a a . 2. X E ,s,a,i := { x E ,s k ,a ( i ) | x E ,s,a ( i ) ∈ X E ,s,a,i ⊂ R |T | } , the fe asible set of a supply elementary or der i at ar e a a for al l time steps, (se e the c oupling c onstr aint: lo ad gr adient c ondition (1h) ). 3. X B O,a,i := { x B O,a ( i ) | x B O,a ( i ) ∈ X B O,a,i ⊂ R 1 } , the fe asible set of a blo ck or der i at the ar e a a which is not c ouple d with others. 4. X B O,a,g − c := { x g − c B O,a | x g − c B O,a ∈ X B O,a,g − c ⊂ R n g − c } , the fe asible set of blo ck or ders which ar e c ouple d to gether at the ar e a a , (se e the linke d c onstr aint or exclusive gr oup c onstr aint (1f) ). 5. X F H B ,a,i := { x F H B ,a ( i ) | x F H B ,a ( i ) ∈ X F H B ,a,i ⊂ R |T | } , the ﬂexible hourly or der i at the ar e a a , (se e the c oupling c onstr aint (1g) ). 6. P := { p ∈ P ⊂ R |A|×|T | } , the fe asible set of net inje ction variables deﬁne d in Se ction 3 explaine d b elow: - Set the last no de as the r efer enc e no de, i.e., p t |A| = p t r ef , p t = h p t r e d p t r ef i . Then by (1c) , p t r ef + 1 ⊤ |A|− 1 p t r e d = 0 , ∀ t ∈ T . 18 - The c onstr aints (1i) , (1j) c an b e formulate d as p ∈ P ramp , and the c onstr aints (1k) , (1l) c an b e written as f ∈ F ramp . R e c al l the line-no de incidenc e matrix B = h B 1 b i , wher e b c orr esp onds to the r efer enc e no de, then f = B 1 ( B ⊤ 1 B 1 ) − 1 p t r e d ∈ P ramp . - The c onstr aints ab ove c an b e c omp actly r epr esente d as p ∈ P . Note that Def. 4 presents the interdependencies of the feasible region of each individual problem in Def. 2. F or the ease of notation, w e still use X i to represen t an order feasible set. During the iteration pro cess, given λ k +1 ∈ R |A|×|T | , in order to av oid directly solving (3), according to the decomp osition in Def. 4, one can solve the following subproblems instead. • F or the subproblem sub jected to an order feasible set X i : x k +1 i : = arg min x i ∈ X i ˜ L ρ ( x i , p k , λ k +1 ) , ˜ L ρ ( x i , p k , λ k +1 ) := ( c i − q ⊤ i λ k +1 ) ⊤ x i + ρ ∥ p k − ⟨ q i , x i ⟩ − ⟨ q − i , x k − i ⟩∥ 1 , (24) where q − i is the quan tity vector excluding the part of subproblem i . • F or the subproblem sub jected to P : p k +1 : = arg min p ∈ P ˜ L ρ ( x k , p, λ k +1 ) , ˜ L ρ ( x k , p, λ k +1 ) := λ k +1 ⊤ p + ρ ∥ p − X i ⟨ q i , x k i ⟩∥ 1 . (25) Among the subproblems, those with feasible sets deﬁned in 1, 2, and 6 are Linear Programming (LP), while those with feasible sets deﬁned in 3, 4, and 5 are small-scale MILPs. Sp eciﬁcally , the subproblem with feasible set 4 represents the largest MILP . Giv en that most decomp osed problems in volv e only a single v ariable, computational eﬃciency can b e signiﬁcantly improv ed. Deﬁnition 5 (The Surrogate Optimality Conditions) . F or the subpr oblem (24) , the surr o gate op- timality c ondition is deﬁne d as ˜ L ρ ( x k +1 i , p k , λ k +1 ) < ˜ L ρ ( x k i , p k , λ k +1 ) , and for the subpr oblem (25) , the c ondition is ˜ L ρ ( x k , p k +1 , λ k +1 ) < ˜ L ρ ( x k , p k , λ k +1 ) . The decomp osition in Def. 4 and the surrogate optimality conditions in Def. 5 lead to the approach of up dating primal v ariables in Algorithm 1. F or a large p enalt y co eﬃcien t ρ , the classical SA VLR metho d may conv erge to a feasible but not optimal p oint λ ′ ∈ R |A|×|T | , see (Bragin et al., 2018, Prop. 1). Therefore, the authors of (Bragin et al., 2018) reduce the p enalt y co eﬃcien t per iteration, b y ρ k +1 = ρ k /β , β > 1 , aiming at satisfying the surrogate optimality condition [Def. 5]. In the worst case ρ = 0 , the SA VLR metho d decreases to the Surrogate Lagrangian Relaxation metho d in (Bragin et al., 2015), where there are theorems ensuring the conv ergence to the dual optimal p oin t. How ever, if the p enalt y co eﬃcien t is decreased, the duality gap can b e enlarged, see Fig. 2(a), whic h will destro y the prop osed pricing mechanism. Therefore, we apply the steps introduced b elow instead. 19 Algorithm 1 The Primal Up date Approac h Input: λ k +1 , ρ j +1 > 0 ▷ A dual v ariable v alue, and p enalty co eﬃcient Output: x k +1 , p k +1 ▷ Numerical v alue of order acceptance and net injection 1: for x i ∈ X i in Def. 4 do 2: x k +1 i ← Eq. (24) 3: if Def. 5 is satisﬁed then ▷ Chec k surrogate optimality condition 4: x k +1 ← [ x k +1 i , x k − i ] , p k +1 ← p k ▷ Primal blo c k up date 5: break 6: else 7: p k +1 ← Eq. (25) ▷ Go to net injection subproblem 8: if Def. 5 is satisﬁed then 9: x k +1 ← x k , p k +1 ← p k +1 10: else 11: Com bine X i sets, reformulating (24), and return to Step 1 12: if Step 11 cannot b e further adv anced then 13: [ x k +1 , p k +1 ] ← arg min L ρ j +1 ( x, p, λ k +1 ) ▷ MILP (3) 14: end if 15: end if 16: end if 17: end for Remark 1 (Mo diﬁcation of SA VLR) . The steps 11, 12, 13 of Algorithm 1 ar e the critic al steps for the mo diﬁc ation of the classic al SA VLR metho d, which ensur es that the outer level appr o ach to b e intr o duc e d in A lgorithm 2 c omputes the exact p enalty c o eﬃcient ρ ∈ [ ρ ∗ , ρ ] . By the com bination technique in tro duced in Remark 1, the distance of the diﬀerence betw een the surrogate subgradient g s = p k +1 − P i ⟨ q i , x k +1 i ⟩ , and the real subgradient g r = p k +1 − ⟨ q , x k +1 ⟩ : ∥ g s − g r ∥ decreases. Then, it is more likely that the surrogate optimality condition will b e satisﬁed. The w orst case is that all the subproblems need to b e combined, i.e., solv e MILP (3) to compute the real subgradient; in that case, the decomp osition loses its v alidity , which constitutes the primary limitation of our mo diﬁed approach. 4.3. The dual up date A t iteration k , with λ k +1 ∈ R |A|×|T | , ρ k +1 > 0 , and primal solution ( x k +1 , p k +1 ) up dated b y Algorithm 1, the dual v ariable up date is as follows: s k +1 = 1 − 1 ( k + 1) r k , M k ≥ 1 , r k ∈ [0 , 1] , σ k +1 = (1 − 1 M k ( k + 1) s k +1 ) σ k ∥ p k − P i ⟨ q i , x k i ⟩∥ 1 ∥ p k +1 − P i ⟨ q i , x k +1 i ⟩∥ 1 , (26) d k +1 = p k +1 − X i ⟨ q i , x k +1 i ⟩ , λ k +2 = λ k +1 + σ k +1 d k +1 , (27) 20 Algorithm 2 The mo diﬁed SA VLR Metho d Input: λ 1 , ρ 1 > 0 , β > 1 , and optimal primal solution ( x ∗ , p ∗ ) of (2). Output: λ j +1 , x j +1 , p j +1 1: Initialization: 2: ( x 1 , p 1 ) ← arg min x,p L ρ 1 ( x, p, λ 1 ) , ˆ z 1 ← z M I P 3: σ 1 ← ˆ z 1 − L ρ 1 ( λ 1 ,x 1 ,p 1 ) ∥⟨ q ,x 1 ⟩− p 1 ∥ 2 1 , d 1 ← p 1 − ⟨ q, x 1 ⟩ , λ 2 = λ 1 + σ 1 d 1 j ← 1 4: while ∥ L ρ j ( x j +1 , p j +1 , λ j +1 ) − z M I P ∥ 1 / ∥ z M I P ∥ 1 > ϵ b do 5: k ← 1 , ρ j +1 ← β ρ j ▷ P enalty co eﬃcient up date 6: while ∥ λ k +1 − λ k ∥ > ϵ d do 7: x k +1 , p k +1 ← Algorithm 1 with input λ k +1 , ρ j +1 . 8: Up date σ k +1 as (26) 9: if k > N then 10: if σ k +1 ≪ z M I P − z ALD ρ ( λ k ) ∥ p k +1 −⟨ q ,x k +1 ⟩∥ 2 1 or σ k +1 > z M I P − z ALD ρ ( λ k ) ∥ p k +1 −⟨ q ,x k +1 ⟩∥ 2 1 then 11: σ k +1 = z M I P − z ALD ρ ( λ k ) ∥ p k +1 −⟨ q ,x k +1 ⟩∥ 2 1 ▷ z ALD ρ ( λ k ) ← MILP (3) 12: end if 13: end if 14: Up date d k +1 as (27) 15: λ k +2 ← λ k +1 + σ k +1 d k +1 , 16: k ← k + 1 ▷ Dual up date 17: end while 18: λ j +1 ← λ k +1 19: ( x j +1 , p j +1 ) ← arg min L ρ j +1 ( x, p, λ j +1 ) ▷ Outer level 20: end while where in (26), the sequences { M k } and { r k } monotonically decrease to M ≥ 1 and r ∈ [0 , 1] , re- sp ectiv ely , and the generated step length σ k +1 is called an adaptive step length (Bragin et al., 2016, Sections 3.2.1). An example of the parameter sequences can b e M k = 2000 / √ k + 1 , r k = 1 / √ k + 0 . 01 . Before in tro ducing the con vergence theorem, we introduce the following remark. Remark 2. The te chnique use d at steps 10 and 11 of Algorithm 2 is c al le d Stepsize R e-initialization, (Br agin et al., 2016, Se ction 3.2.2), which is use d to avoid oscil lation. This te chnique involves solving the c entr alize d MILP (3) and is applie d only after suﬃcient iter ations of the surr o gate sub gr adient. The con vergence theorem for the sequence of generated dual v ariable v alues is then in tro duced. Theorem 5 (Dual Con vergence) . With the surr o gate sub gr adient d k +1 := p k +1 − P i ⟨ q i , x k +1 i ⟩ up date d fr om Algorithm 1, and the se quenc e { λ 1 , λ 2 , · · · } gener ate d by A lgorithm 2 c onver ges to a unique ﬁxe d p oint λ ∗ , wher e λ ∗ = arg max λ ∈ R |A|×|T | z ALD ρ ( λ ) ,(se e Br agin et al. (2015, The or em 2.1) and Br agin et al. (2018, The or em 1)). The preceding discussion leads to the tw o-lev el Algorithm 2. A similar approac h for computing the exact p enalt y co eﬃcien t can b e found in (Chen and Chen, 2010, Section 4.1). F urthermore, as 21 sho wn in (Lefeb vre and Sc hmidt, 2024, Theorem 14), ∀ λ ∈ R |A|×|T | the exact penalty co eﬃcient ρ ( λ ) can b e computed in p olynomial time. Consequently , while the cen tralized MILP is o ccasionally solv ed, the computational complexity of Algorithm 2 is primarily determined b y the primal MILP sub- problems. By exploiting the computational eﬃciency of high-p erformance commercial MILP solvers, the prop osed algorithm demonstrates strong scalabilit y on large-scale mark et-clearing instances. W e therefore conclude that the ALD pricing mec hanism in tro duced in Def. 1 is b oth theoretically rigorous and computationally tractable. 5. Illustrativ e examples This section compares the ALD pricing mechanism (prop osed in Section 3.2) with existing literature b enc hmarks. W e utilize a represen tativ e example from (Sc hiro et al., 2016) which, despite mo deling a unit commitment problem, eﬀectively illustrates the prop erties of diﬀerent pricing sc hemes. W e com- pare ALD with In teger Programming (IP) pricing (O’Neill et al., 2005), Semi-Lagrangian Relaxation (SLR) pricing (Araoz and Jörnsten, 2011), Primal-Dual (PD) pricing (Ruiz et al., 2012), and Con vex Hull pricing (CHP) (Sc hiro et al., 2016). T o comprehensively ev aluate these mec hanisms, w e analyze the Lost Opp ortunity Cost (LOC), Individual Surplus, T otal Consumer Pa yment, and Uplift Pa ymen t. These metrics directly corresp ond to the critical market properties of incen tiv e compatibilit y , individual rev en ue adequacy , economic eﬃciency , and pricing transparency , resp ectiv ely . 5.1. A demonstr ative example (Schir o et al., 2016) The 1-ar e a c ase . The optimization problem is form ulated as follows: min 2500 x 1 + 500 x 2 s.t. 50 x 1 + 50 x 2 = 35 , x 1 ∈ [0 . 2 , 1] , x 2 ∈ { 0 , 1 } . (28) It is ob vious that the optimal solution of (28) is x 1 = 0 . 7 , x 2 = 0 , and the corresp onding optimal ob jective function v alue is 1750 . The 2-ar e a c ase . The optimization problem is as follo ws: min 2500 x 1 + 500 x 2 s.t.   p 1 − p 1   =   50 x 1 − 35 50 x 2   , p 1 = f , − 10 ≤ f ≤ 10 , x 1 ∈ [0 . 2 , 1] , x 2 ∈ { 0 , 1 } . (29) Eviden tly , the optimal solution of (29) is x 1 = 0 . 7 , x 2 = 0 , p 1 = 0 , f = 0 , and the corresp onding optimal ob jective function v alue is 1750 . 22 Benchmark pricing to the 1-ar e a c ase . Given that the pricing mechanisms under comparison are well established and the illustrativ e example is straightforw ard, detailed deriv ations are omitted for brevity . It is noteworth y that the implementation complexit y of the Primal-Dual (PD) pricing mec hanism primarily stems from the linearization of the reven ue adequacy constraint via the Big-M metho d and the asso ciated discretization pro cedures. In this instance, the LP relaxation yields a price of 10 EUR/MWh. Notably , the PD price exhibits a signiﬁcant departure from this LP relaxation b enc hmark (see (Ruiz et al., 2012, abstract)), while closely aligning with the results of Semi-Lagrangian Relaxation (SLR). The p erformance metrics for eac h mec hanism are summarized in T able 2, with the follo wing notations: LMP (Lo cational Marginal Price, EUR/MWh), UP (Uplift Pa ymen t, EUR), LOC (Lost Opp ortunit y Cost, EUR), and S (Surplus, EUR). LMP UP( G 1 ) UP( G 2 ) LOC( G 1 ) LOC( G 2 ) IP 50.00 0.00 2000.00 0.00 2000.00 SLR 50.00 0.00 0.00 0.00 2000.00 PD 50.42 0.00 0.00 0.00 2000.00 CHP 10.00 1000.00 0.00 1000.00 0.00 ALD 10.00 0.00 0.00 0.00 0.00 S( G 1 ) S( G 2 ) T otal paymen t 0.00 2000.00 3750.00 0.00 0.00 1750.00 0.00 0.00 1750.00 -400.00 0.00 1350.00 0.00 0.00 1750.00 T able 2: The results of each pricing mechanism of the 1-area case. Benchmark pricing to the 2-ar e a c ase . The n umerical results for each pricing mec hanism in the 2-area scenario are summarized in T able 3. T o accommo date the multi-area setting, the T ransmission System Op erator (TSO) is incorp orated as a critical mark et entit y . W e ev aluate the TSO’s ﬁnancial p erformance by analyzing its surplus, lost opp ortunit y cost, and uplift paymen t, denoted resp ectively as S ( TSO ) , LOC ( TSO ) , and UP ( TSO ) (all in EUR). Notably , the LMPs derived from the LP relaxation are LMP 1 = 50 EUR/MWh and LMP 2 = 10 EUR/MWh. These prices result in rev enue inadequacy for the TSO, as the price diﬀeren tial incen tivizes cross-area transmission while the physical ﬂo w remains zero. As illustrated in T able 3, the primal-dual pricing results in this 2-area case are close to the LP relaxation b enc hmarks, consisten t with Ruiz et al. (2012, abstract). Comp ar ative analysis . The results in T able 2 clearly demonstrate that the ALD pricing mechanism ac hieves the highest economic eﬃciency , as evidenced by its total pa yment of 1750 EUR, the low est among all listed pricing mechanisms. In particular, the ALD mechanism op erates without any up- lift pa yments (UP ( G 1 ) = UP ( G 2 ) = 0 EUR), ensuring that price signals remain transparent to all 23 mark et orders. F urthermore, the v anishing lost opp ortunity costs (LOC ( G 1 ) = LOC ( G 2 ) = 0 EUR) n umerically v alidate the incentiv e compatibilit y prop ert y established in Theorem 2. Complemen ting the comparative analysis in T able 2, T able 3 pro vides the TSO’s computational results. Since the general performance of each pricing mechanism remains consistent across b oth tables, in T able 3 w e fo cus sp eciﬁcally on the TSO’s ﬁnancial p osition. Under PD pricing, the TSO incurs a lost opp ortunit y cost (LOC) of 400 EUR, reveal ing p oten tial incentiv e issues. Similarly , CHP pricing yields a 400 EUR LOC, which must b e cov ered by uplift pa ymen ts. In con trast, the ALD pricing mec hanism eliminates both the LOC and the need for uplift paymen ts, thereb y ensuring incen tive compatibilit y and an uplift-free settlement for the TSO in this scenario. LMP 1 LMP 2 UP( G 1 ) UP( G 2 ) LOC( G 1 ) LOC( G 2 ) IP 50.00 50.00 0.00 2000.00 0.00 2000.00 SLR 50.00 50.00 0.00 0.00 0.00 2000.00 PD 50.42 10.00 0.00 0.00 0.00 0.00 CHP 50.00 10.00 0.00 0.00 0.00 0.00 ALD 25.00 25.00 0.00 0.00 0.00 0.00 S( G 1 ) S( G 2 ) UP(TSO) S(TSO) LOC(TSO) T otal paymen t 0.00 2000.00 0.00 0.00 0.00 3750.00 0.00 0.00 0.00 0.00 0.00 1750.00 0.00 0.00 0.00 0.00 400.00 1750.00 0.00 0.00 400.00 400.00 400.00 2150.00 0.00 0.00 0.00 0.00 0.00 1750.00 T able 3: The results of each pricing mechanism of the 2-area case. It is noteworth y that the results of each pricing mechanism of this academic example m utually conﬁrm their prop erties presented in T able 1. 5.2. R esults of the ALD pricing me chanism This section illustrates the p erformance of the ALD pricing mec hanism introduced in Deﬁnition 1 using the b enc hmark case from Section 5.1. The 1-ar e a c ase . Applying the ALD approach to (28), and reformulating the absolute v alue op erator yields the follo wing optimization problems: min x 1 ,x 2 2500 x 1 + 500 x 2 + λ (35 − 50 x 1 − 50 x 2 ) + ρ | 35 − 50 x 1 − 50 x 2 | s.t. x 1 ∈ [0 . 2 , 1] , x 2 ∈ { 0 , 1 } . ↓ Reformulation min x 1 ,x 2 ,q 2500 x 1 + 500 x 2 + λ (35 − 50 x 1 − 50 x 2 ) + ρq s.t. q ≥ 35 − 50 x 1 − 50 x 2 , q ≥ − (35 − 50 x 1 − 50 x 2 ) q ≥ 0 , x 1 ∈ [0 . 2 , 1] , x 2 ∈ { 0 , 1 } . (30) 24 Eq. (30) inv olv es a bilinear expression λ (35 − 50 x 1 − 50 x 2 ) . Fixing λ = λ ∗ ∈ R , the mo del transforms in to a simple MILP . This allows for the c haracterization of the dual function under diﬀeren t p enalt y co eﬃcien ts ρ , with the results depicted in Fig. 2(a). 20 0 20 40 60 D u a l V a r i a b l e 0 500 1000 1500 Dual F unction V alue = 1 0 = 2 0 = 3 0 = 4 0 = 5 0 = 5 5 (a) Cen tralized ALD 0.2 0.4 0.6 0.8 1.0 L o c a l V a r i a b l e x 1 0 250 500 750 1000 Objective V alue (b) ALD for generator 1 0.00 0.25 0.50 0.75 1.00 L o c a l V a r i a b l e x 2 0 500 1000 1500 2000 Objective V alue (c) ALD for generator 2 Figure 2: The ALD approach to 1-area case Fig. 2(a) clearly iden tiﬁes ρ ∗ = 40 as the threshold that closes the duality gap. F or ρ = ρ ∗ , the unique optimal dual v ariable is λ ∗ = 10 , whereas for ρ = 45 , the optimal dual set expands to Λ = [5 , 15] . Based on the ALD pricing rules in Deﬁnition 1, b oth (40 , 10) and (45 , 5) constitute v alid ALD price signals. The resulting individual optimization problems, as deﬁned in Def. 2, are provided b elo w: F or generator 1: min 2500 x 1 + λ ∗ ( − 50 x 1 ) + ρ | 35 − 50 x 1 | − ρ × 50 × x ∗ 1 , s.t. x 1 ∈ [0 . 2 , 1] , x ∗ 1 = 0 . 7; (31) F or generator 2: min 500 x 2 + λ ∗ ( − 50 x 2 ) + ρ | − 50 x 2 | − ρ × 50 × x ∗ 2 , s.t. x 2 ∈ { 0 , 1 } , x ∗ 2 = 0 . (32) The ob jective functions of problems (31) and (32) are illustrated in Fig. 2(b) and Fig. 2(c), resp ectiv ely . These ﬁgures depict the case with ρ = 40 and λ ∗ = 10 ; notably , the results for the case where ρ = 50 and λ ∗ = 0 exhibit similar characteristics. Note that the optimization problems are formulated as minimization problems; a p ositive ob jective v alue corresp onds to a ﬁnancial loss. It is clear from the observ ation that x ∗ 1 = 0 . 7 and x ∗ 2 = 0 are the resp ective optimal solutions. Consequently , neither generator has an incen tive to deviate, as an y departure from these v alues w ould decrease their surplus. This observ ation corrob orates the incen tive compatibility property in troduced in Theorem 2. The observ ation that the optimal ob jectiv e v alues are zero implies that the generators’ costs are fully reco vered by their reven ues, thereby v alidating Theorem 3 regarding individual reven ue adequacy . The 2-ar e a c ase . W e extend the ALD framework to the 2-area conﬁguration (29). The dual functions for v arious p enalty co eﬃcien ts are depicted in Fig. 3, identifying the critical threshold ρ ∗ = 25 required to eliminate the duality gap. At this threshold, the optimal dual set is Λ = { λ 1 ∈ [25 , 75] , λ 2 = 25 } . Increasing the p enalt y to ρ = 30 expands the set to Λ = { λ 1 ∈ [20 , 80] , λ 2 ∈ [20 , 30] } . F or market clearing, the op erator ma y select ( ρ, λ ∗ 1 , λ ∗ 2 ) = (25 , 25 , 25) as price signals. The decentralized opti- mization for generators follo ws the structure of Eq. (31)–(32), with the resulting ob jective landscap es 25 20 40 60 80 1 15 20 25 30 35 2 1100 1200 1300 1400 1500 1600 1700 (a) The dual function with ρ = ρ ∗ = 25 20 40 60 80 1 15 20 25 30 35 2 1450 1500 1550 1600 1650 1700 1750 (b) The dual function with ρ = 30 > ρ ∗ Figure 3: The ALD approach to the 2-area case aligning with Fig. 2(b) and Fig. 2(c). This conﬁrms that incentiv e compatibility and reven ue adequacy are preserv ed in m ulti-area conﬁgurations. F urthermore, in the absence of congestion, the no dal price diﬀeren tial remains zero ( λ ∗ 1 − λ ∗ 2 = 0 ) regardless of whether (25 , 25 , 25) or (30 , 20 , 20) is c hosen. Under these signals, the TSO has no ﬁnancial incentiv e to initiate p o wer transfers, as the zero price yields no congestion rev enue, while an y deviation would incur a p enalt y . Consequently , the TSO also ac hieves rev enue adequacy in this scenario. The comprehensive results for the ALD approac h are presen ted in T able 2 and T able 3, with the comparativ e analysis provided ab o ve. 6. Numerical exp eriments In this section, we present n umerical examples that apply Algorithm 1 and Algorithm 2 to demon- strate the ALD pricing mechanism. The example we emplo y is the Nordic day-ahead electricity mark et. The bidding areas and transmission lines are sho wn in Fig. 4. The supply oﬀer prices, and the demand bidding prices are pro duced such that each demand bidding price exceeds the largest supply oﬀer price at eac h t ∈ T , i.e., satisfying the suﬃcien t condition in Corollary 1. T o make the prices more realis- tic, w e set the supply oﬀer prices to gradually reach their p eak v alues during the time p erio ds [7 , 11] and [17 , 21] . The capacities are conﬁgured suc h that the total p otential supply capacity exceeds the maxim um aggregate demand to ensure that the primal problem (1) is feasible. The n umber of orders in each bidding area is generated by random n umber generators, and T able 4 shows the n umber of diﬀeren t t yp es of orders in the bidding area 10 . The ramping co eﬃcien ts of netw ork constraints are generated in accordance with the ratios of realistic energy sources, e.g., hydro p ow er, gas, n uclear, etc., in each Nordic country . A dditionally , we use the Fixed Binary-V ariable (FBV) pricing method pre- 26 sen ted in (Chatzigiannis et al., 2016, Section 5) as a comparison. Distinct from the traditional Integer Programming (IP) pricing (O’Neill et al., 2005), our approac h follows the logic of (Chatzigiannis et al., 2016, Section 5). W e refer to this as Fixed Binary-V ariable Pricing, in which the binary commitment status is ﬁxed at its optimal v alue in the ﬁnal price-setting stage. This ensures that the resulting LMPs reﬂect the marginal cost without including uplift pa yments. It is worth noting that the fundamental properties of ma jor pricing mechanisms ha ve b een rigor- ously compared in Section 5 using an academic case. In this stylized study , further comparison with other mec hanisms would primarily entail complex implementation, suc h as characterizing the conv ex h ulls of feasible regions under exclusiv e-group, linked-block, and ﬂexible hourly constraints, without yielding additional theoretical insights. Consequently , Fixed Binary-V ariable pricing is selected as the represen tative b enc hmark for this large-scale computation. Moreo ver, all n umerical simulations are ex- ecuted in the Pyomo mo deling en vironment using the Gurobi solver. All computations are p erformed on a mac hine equipp ed with an Apple M4 Pro c hip and 24 GB of RAM. NO4(6) NO3(5) SE2(9) SE1(8) FI(2) NO5(7) NO1(3) SE3(10) NO2(4) SE4(11) DK2(1) DK1(0) l 11 l 6 l 12 l 13 l 7 l 5 l 15 l 4 l 14 l 8 l 10 l 9 l 16 l 1 l 3 l 2 l 0 Figure 4: The Nordic day-ahead electricity market top ology . n E ,d n E ,s n P B ,d n P B ,s n RB ,d n RB ,s n F H B ,d n F H B ,s 23 12 6 6 6 8 5 3 T able 4: The num ber of each order in bidding area 10 (SE3). 6.1. A stylize d c omputation of the Nor dic ele ctricity market The netw ork top ology , shown in Fig. 4, consists of |A| = 12 no des and | L | = 17 lines ov er |T | = 24 time steps. The resulting mathematical mo del is substan tial, containing 9,640 v ariables, including 2,248 binary decision v ariables p er instance. Despite this complexity , sensitivity analysis using Algorithm 2 27 with diﬀerent initial settings λ and ρ demonstrated robust p erformance, with a maximum computation time of 40 min utes. In addition, under the suﬃcient condition in Corollary 1, demand orders cannot b e PROs, as their willingness to pay is suﬃciently high. Any rejection of suc h orders would stem from structural constraints, suc h as link ed-blo c k constraint, etc. Consequen tly , potential PROs are limited to the supply side. Or der Surplus . T able 5 presents order surpluses for p oten tial PROs under b oth pricing mechanisms. The results indicate that the ALD mechanism eﬀectively eliminates all PROs by rendering deviations unproﬁtable; for instance, deviations that previously app eared attractive (e.g., surpluses of 23,628.62 EUR and 55,188.95 EUR) result in signiﬁcant losses under ALD (-320,933.42 EUR and -288,976.80 EUR, resp ectively). Similarly , as sho wn in T able 6, the ALD approac h resolves all Parado xically A ccepted Orders (P AOs) inherent to the Fixed Binary-V ariable (FBV) metho d. Speciﬁcally , a supply regular blo ck order in Area 1 transits from a deﬁcit of -241,980.01 EUR to a p ositiv e surplus of 14,553.13 EUR, while a ﬂexible hourly order in Area 5 impro ves from -65,626.68 EUR to 45,515.78 EUR. Congestion R ent . T able 7 details the congestion reven ue of each transmission line. The aggregate TSO reven ue under ALD pricing reac hes 71,460.28 EUR, larger than the 3,120.43 EUR generated by the Fixed Binary-V ariable metho d. This p ositiv e total reven ue observ ation implies that for a general net work structure, it can be rev enue adequate for the TSO under the ALD pricing mec hanism. It is notew orthy that ALD do es not guarantee non-negativ e congestion ren ts on ev ery individual line; for instance, Lines 3 and 5 exhibit negativ e ren ts of -83,937.70 EUR and -35,143.99 EUR, resp ectiv ely . Finally , a structural analysis of T ables 5 and 6 reveals that under Fixed Binary-V ariable pricing, PR Os consist exclusively of supply proﬁle blo c k orders, whereas P AOs are comp osed en tirely of supply regular blo c k orders and ﬂexible hourly orders. LMP and Congestion Pric e (CP) Analysis . Fig. 5 and Fig. 6 display LMPs of four represen ta- tiv e bidding areas and the congestion prices of four transmission lines, resp ectiv ely . Under the ALD mec hanism, the eﬀectiv e settlement LMPs are calculated as λ t, ∗ a + ρ , with the p enalty parameter set to ρ = 30 . 58 (where ρ ∈ [ ρ ∗ , ρ ] ). Congestion prices are deﬁned based on the assumed ﬂow directions: DK2 → SE4, NO1 → NO5, NO4 → SE1, and SE3 → FI. Both ﬁgures reveal a temp oral pattern, with p eak LMPs and congestion prices app earing in the time in terv als [7, 11] and [17, 21]. As illustrated in Fig. 5, ALD-deriv ed LMPs are generally higher than those from the Fixed Binary-V ariable (FBV) pricing. How ev er, as evidenced in Fig. 6, no consistent dominance relationship exists b et w een the tw o mec hanisms regarding congestion prices. W elfar e A nalysis . Finally , the aggregate net surplus across all orders is calculated as 111,539,574.81 EUR for the FBV metho d and 111,471,234.95 EUR for the ALD metho d. The T otal Social W elfare, 28 deﬁned as the sum of the total order surplus and the TSO’s congestion reven ue, is Fixed Binary-V ariable: 111 , 539 , 574 . 81 + 3 , 120 . 43 = 111 , 542 , 695 . 24 ( EUR ) ALD: 111 , 471 , 234 . 95 + 71 , 460 . 28 = 111 , 542 , 695 . 24 ( EUR ) The iden tical so cial w elfare demonstrates no w elfare loss, attributing to the suﬃcien t condition in Corollary 1. Signiﬁcan tly , these computational results v alidate Prop osition 1, conﬁrming that the summation of individual surplus functions (7) c haracterizes the reven ue allo cated to the TSO. 0 10 20 DK1 25 50 75 The LMP 0 10 20 FI 50 75 0 10 20 NO4 0 50 The LMP 0 10 20 SE3 25 50 75 Fixed Binary- V ariable ALD Figure 5: LMPs of DK1 FI NO4, and SE3. 0 10 20 l 2 50 0 The CP 0 10 20 l 8 20 0 20 0 10 20 l 1 1 0 50 The CP 0 10 20 l 1 4 0 20 Fixed Binary- V ariable ALD Figure 6: Congestion prices of l 2 l 8 l 11 , and l 14 . 7. Conclusion and the future w ork This paper prop oses a no v el pricing mec hanism, termed the Augmen ted Lagrangian and Dual- it y (ALD) pricing mechanism, deriv ed from the strong dualit y framework of Mixed-In teger Linear Programming (MILP). W e demonstrate that once the primal so cial w elfare maximization problem is solv ed to optimality , the ALD pricing mechanism eﬀectively eliminates all Parado xically Rejected Orders (PROs) and Parado xically Accepted Orders (P AOs) among supply orders. F urthermore, we establish that P AOs among demand orders can b e eliminated pro vided that demand bid prices exceed the maximal supply oﬀer prices. These prop erties are supp orted by rigorous theoretical pro ofs and are b enc hmarked with ma jor pricing mechanisms in the literature to highlight the adv antages of the ALD pricing approac h. Finally , we hav e mo diﬁed the SA VLR algorithm (Bragin et al., 2018) to com- pute the exact p enalty co eﬃcients required to close the duality gap and the optimal dual v ariables. This mo diﬁcation ensures that the prop osed ALD pricing mechanism is computationally tractable, as v alidated by the computational results. T o further extend this research, sev eral promising tra jectories are identiﬁed: 1) Mo del General- ization: While this study formulates the mark et-clearing problem as an MILP , incorp orating more complex features could lead to Mixed-Integer Quadratic Programming (MIQP) (Gu et al., 2020) or 29 T able 5: Surplus of Poten tial PROs under Deviation across the T w o Pricing Mec hanisms (v alues in EUR). Area T ype ALD Fixed Binary-V ariable 1 PBs -320933.42 23628.62 1 PBs -311859.08 20116.34 1 PBs -330400.36 29700.51 1 PBs -311803.45 21623.49 1 PBs -338078.87 31933.73 1 PBs -281835.01 29642.45 1 PBs -274894.84 34316.63 2 PBs -293594.67 46974.28 2 PBs -304967.75 66581.78 2 PBs -296806.04 61679.44 2 PBs -291870.20 59921.81 2 PBs -288976.80 55188.95 6 PBs -305562.97 9211.71 Area T ype ALD Fixed Binary-V ariable 6 PBs -292515.60 5584.97 7 PBs -341547.04 1461.39 10 PBs -178153.35 200758.35 10 PBs -194856.67 185135.18 10 PBs -187285.31 207090.16 10 PBs -195764.04 197509.82 10 PBs -201818.50 190176.82 10 PBs -177305.86 161165.21 11 PBs -303860.11 24955.95 11 PBs -332687.96 39930.19 11 PBs -307721.71 30519.63 11 PBs -286929.02 39001.60 11 PBs -290551.67 33186.83 T able 6: Surplus of P AOs under the T wo Pricing Mechanisms (v alues in EUR). Area T ype ALD Fixed Binary-V ariable 0 RBs 14553.13 -241980.01 0 RBs 46802.42 -186208.88 0 RBs 37922.19 -217987.58 1 RBs 69378.67 -103914.42 1 RBs 103046.33 -63429.62 1 RBs 71680.03 -92404.38 2 FHBs 21672.06 -16827.11 3 PBs 82489.51 -18668.97 3 RBs 214432.84 -143719.35 3 RBs 205945.53 -103015.72 3 RBs 192840.15 -131127.68 3 RBs 249548.58 -67867.13 3 FHBs 39548.60 -8649.98 4 RBs 94217.03 -134868.43 4 RBs 153001.69 -91859.72 Area T ype ALD Fixed Binary-V ariable 4 FHBs 58413.02 -11775.41 5 RBs 311282.16 -78148.65 5 FHBs 45515.78 -65626.68 6 RBs 176777.06 -65749.15 6 FHBs 45271.51 -6579.98 7 PBs 72123.80 -4861.81 7 RBs 155260.26 -92365.57 7 RBs 173642.32 -86077.40 8 RBs 93790.23 -210338.61 8 RBs 51920.53 -224680.03 9 RBs 162331.97 -75942.15 9 FHBs 35304.13 -8964.91 11 RBs 103812.06 -31966.78 11 RBs 140318.61 -7339.24 11 FHBs 25994.79 -21321.93 30 T able 7: Congestion Rents across Lines under the T wo Pricing Mechanisms (v alues in EUR). Line ALD Fixed Binary-V ariable 0 8166.97 20368.84 1 -152489.56 -92750.42 2 74401.55 74936.62 3 -83937.70 -83308.50 4 124440.82 110795.22 5 -35143.99 -39330.08 6 22205.46 31606.41 7 -3698.31 2307.88 Line ALD Fixed Binary-V ariable 8 21910.85 3972.68 9 3663.32 -5400.03 10 8829.46 15065.89 11 -79154.66 -84060.20 12 1391.44 -2152.80 13 4217.57 24297.86 14 62495.10 12103.53 15 -31221.42 6754.30 16 57043.53 76253.08 ev en Mixed-Integer Nonlinear Programming (MINLP) formulations (Lefebvre and Sc hmidt, 2024). Ap- plying the prop osed ALD framework to these more sophisticated mo dels presen ts foreseeable c hallenges and a promising area for future exploration. 2) Algorithm Accele ration for 15-Minute Dispatc h: In resp onse to the Europ ean market’s transition to wards a 15-minute time in terv al, the n um b er of in teger v ariables will increase signiﬁcantly . T o main tain strict mark et-clearing deadlines under this resolution, future w ork will fo cus on augmenting the prop osed mo diﬁed SA VLR algorithm with adv anced primal heuristics and parallel computing architectures to further accelerate the con vergence of the primal com binatorial problem. A c kno wledgments The authors thank the Sw edish Energy Agency for the ﬁnancial supp ort of this researc h work (grant n umber: P2022-00738), the advisors from the Nord P o ol, and the PhD studen ts at KTH, energy mark et researc h group, for their useful comments. During the preparation of this work, Zhen W ang used Go ogle Gemini in order to edit the lan- guage. After using this to ol, the co-authors reviewed and edited the conten t as needed. W e take full resp onsibilit y for the con tent of the published article. F or repro ducibilit y and further researc h, the ALD pricing implementation, mo diﬁed SA VLR algo- rithm, and Nordic test cases are av ailable on GitHub under the MIT license: https://github.com/ Zhen- Wang- Sudo/nordic- electricity- ald- codes . References Ah unba y , M.Ş., Bichler, M., Dob os, T., Knörr, J., 2024. Solving large-scale electricity mark et pricing problems in p olynomial time. European Journal of Op erational Research 318, 605–617. doi: https: //doi.org/10.1016/j.ejor.2024.05.020 . 31 Ah unba y , M.Ş., Bichler, M., Knörr, J., 2025. Pricing optimal outcomes in coupled and non- con vex mark ets: Theory and applications to electricit y markets. Operations Researc h 73, 178–193. doi: https://doi.org/10.1287/opre.2023.0401 . Andrianesis, P ., Bertsimas, D., Caramanis, M.C., Hogan, W.W., 2021. Computation of con vex hull prices in electricit y markets with non-conv exities using dan tzig-wolfe decomp osition. IEEE T rans- actions on P ow er Systems 37, 2578–2589. doi: 10.1109/TPWRS.2021.3122000 . Araoz, V., Jörnsten, K., 2011. Semi-Lagrangian approach for price discov ery in mark ets with non- con vexities. Europ ean Journal of Op erational Research 214, 411–417. doi: https://doi.org/10. 1016/j.ejor.2011.05.009 . Azizan, N., Su, Y., Dvijotham, K., Wierman, A., 2020. Optimal pricing in markets with nonconv ex costs. Operations Research 68, 480–496. doi: https://doi.org/10.1287/opre.2019.1900 . Bjørndal, M., Jörnsten, K., 2008. Equilibrium prices supp orted b y dual price functions in mark ets with non-conv exities. Europ ean Journal of Op erational Researc h 190, 768–789. doi: https://doi. org/10.1016/j.ejor.2007.06.050 . Bragin, M.A., Luh, P .B., Y an, B., Sun, X., 2018. A scalable solution metho dology for mixed-integer linear programming problems arising in automation. IEEE T ransactions on Automation Science and Engineering 16, 531–541. doi: 10.1109/TASE.2018.2835298 . Bragin, M.A., Luh, P .B., Y an, J.H., Stern, G.A., 2016. An eﬃcient approach for solving mixed-integer programming problems under the monotonic condition. Journal of Con trol and Decision 3, 44–67. doi: https://doi.org/10.1080/23307706.2015.1129916 . Bragin, M.A., Luh, P .B., Y an, J.H., Y u, N., Stern, G.A., 2015. Con vergence of the surrogate lagrangian relaxation metho d. Journal of Optimization Theory and applications 164, 173–201. doi: https: //doi.org/10.1007/s10957- 014- 0561- 3 . By ers, C., Hug, G., 2022. Economic impacts of near-optimal solutions with non-con vex pricing. Electric P ow er Systems Research 211, 108287. doi: https://doi.org/10.1016/j.epsr.2022.108287 . By ers, C., Hug, G., 2023. Long-run optimal pricing in electricit y markets with non-con v ex costs. Europ ean Journal of Op erational Researc h 307, 351–363. doi: https://doi.org/10.1016/j.ejor. 2022.07.052 . Chatzigiannis, D.I., Dourb ois, G.A., Bisk as, P .N., Bakirtzis, A.G., 2016. Europ ean day-ahead electricit y market clearing mo del. Electric P o w er Systems Research 140, 225–239. doi: https: //doi.org/10.1016/j.epsr.2016.06.019 . Chen, Y., Chen, M., 2010. Extended duality for nonlinear programming. Computational optimization and applications 47, 33–59. doi: https://doi.org/10.1007/s10589- 008- 9208- 3 . 32 F eizollahi, M.J., Ahmed, S., Sun, A., 2017. Exact augmented lagrangian duality for mixed integer linear programming. Mathematical Programming 161, 365–387. doi: https://doi.org/10.1007/ s10107- 016- 1012- 8 . Gribik, P .R., Hogan, W.W., P ope, S.L., et al., 2007. Market-clearing electricit y prices and energy uplift. Cam bridge, MA , 1–46. Gu, X., Ahmed, S., Dey , S.S., 2020. Exact augmented lagrangian dualit y for mixed integer quadratic programming. SIAM Journal on Optimization 30, 781–797. doi: https://doi.org/10.1137/ 19M127169 . Guo, C., Bo dur, M., T a ylor, J.A., 2025. Copositive dualit y for discrete energy markets. Managemen t Science doi: https://doi.org/10.1287/mnsc.2023.00906 . Herrero, I., Ro dilla, P ., Batlle, C., 2015. Electricit y mark et-clearing prices and in vestmen t incen tives: The role of pricing rules. Energy Economics 47, 42–51. doi: https://doi.org/10.1016/j.eneco. 2014.10.024 . Hua, B., Baldick, R., 2016. A conv ex primal formulation for conv ex hull pricing. IEEE T ransactions on P ow er Systems 32, 3814–3823. doi: 10.1109/TPWRS.2016.2637718 . Ilea, V., Bov o, C., et al., 2018. Europ ean da y-ahead electricit y market coupling: Discussion, mo deling, and case study . Electric Po w er Systems Research 155, 80–92. doi: https://doi.org/10.1016/j. epsr.2017.10.003 . Kn ueven, B., Ostrowski, J., Castillo, A., W atson, J.P ., 2022. A computationally eﬃcient algorithm for computing con v ex h ull prices. Computers & Industrial Engineering 163, 107806. doi: https: //doi.org/10.1016/j.cie.2021.107806 . Kuang, X., Lamadrid, A.J., Zuluaga, L.F., 2019. Pricing in non-conv ex markets with quadratic deliv- erabilit y costs. Energy Economics 80, 123–131. doi: https://doi.org/10.1016/j.eneco.2018.12. 022 . Lefeb vre, H., Sc hmidt, M., 2024. Exact augmented lagrangian dualit y for noncon vex mixed-integer nonlinear optimization. optimization-online.org . Lib eropoulos, G., Andrianesis, P ., 2016. Critical review of pricing sc hemes in markets with non-con vex costs. Operations Research 64, 17–31. doi: https://doi.org/10.1287/opre.2015.1451 . Madani, M., Ruiz, C., Siddiqui, S., V an V yve, M., 2018. Conv ex hull, IP and Europ ean electricit y pricing in a Europ ean p o w er exc hanges setting with eﬃcient computation of con vex hull prices. arXiv preprin t arXiv:1804.00048 doi: https://doi.org/10.48550/arXiv.1804.00048 . 33 Madani, M., V an V yv e, M., 2015. Computationally eﬃcien t MIP form ulation and algorithms for Europ ean day-ahead electricit y mark et auctions. Europ ean Journal of Op erational Research 242, 580–593. doi: https://doi.org/10.1016/j.ejor.2014.09.060 . Madani, M., V an V yve, M., 2017. A MIP framework for non-con vex uniform price day-ahead electricit y auctions. EURO Journal on Computational Optimization 5, 263–284. doi: https://doi.org/10. 1007/s13675- 015- 0047- 6 . Madani, M., V an V yv e, M., 2018. Revisiting minimum proﬁt conditions in uniform price da y-ahead electricit y auctions. Europ ean Journal of Op erational Researc h 266, 1072–1085. doi: https://doi. org/10.1016/j.ejor.2017.10.024 . Martin, A., Müller, J.C., Pokutta, S., 2014. Strict linear prices in non-con vex Europ ean da y-ahead electricit y markets. Optimization Metho ds and Softw are 29, 189–221. doi: https://doi.org/10. 1080/10556788.2013.823544 . Ma ys, J., Morton, D.P ., O’Neill, R.P ., 2021. In v estment eﬀects of pricing schemes for non-conv ex mark ets. Europ ean Journal of Op erational Researc h 289, 712–726. doi: https://doi.org/10.1016/ j.ejor.2020.07.026 . Milgrom, P ., W att, M., 2022. Linear pricing mec hanisms for markets without conv exit y , in: Pro ceedings of the 23rd A CM Conference on Economics and Computation, pp. 300–300. doi: https://doi.org/ 10.1145/3490486.3538310 . O’Neill, R.P ., Sotkiewicz, P .M., Hobbs, B.F., Rothk opf, M.H., Stewart Jr, W.R., 2005. Eﬃcien t mark et-clearing prices in markets with noncon vexities. Europ ean Journal of Op erational Research 164, 269–285. doi: https://doi.org/10.1016/j.ejor.2003.12.011 . Ruiz, C., Conejo, A.J., Gabriel, S.A., 2012. Pricing non-con vexities in an electricity po ol. IEEE T ransactions on Po w er Systems 27, 1334–1342. doi: 10.1109/TPWRS.2012.2184562 . Sc hiro, D.A., Zheng, T., Zhao, F., Litvinov, E., 2016. Conv ex h ull pricing in electricity mark ets: F ormulation, analysis, and implementation challenges. IEEE T ransactions on Po wer Systems 31, 4068–4075. doi: 10.1109/TPWRS.2015.2486380 . Sleisz, Á., Divén yi, D., Raisz, D., 2019. New formulation of p o w er plan ts’ general complex orders on Europ ean electricity mark ets. Electric P ow er Systems Research 169, 229–240. doi: https://doi. org/10.1016/j.epsr.2018.12.028 . Stev ens, N., P apav asiliou, A., Smeers, Y., 2024. On some adv an tages of conv ex hull pricing for the Europ ean electricity auction. Energy Economics 134, 107542. doi: https://doi.org/10.1016/j. eneco.2024.107542 . 34 T o czyłowski, E., Zoltowsk a, I., 2009. A new pricing scheme for a m ulti-p erio d p o ol-based electricity auction. Europ ean Journal of Operational Researc h 197, 1051–1062. doi: https://doi.org/10. 1016/j.ejor.2007.12.048 . Vlac hos, A., Dourb ois, G., Bisk as, P ., 2016. Comparison of tw o mathematical programming mo dels for the solution of a conv ex p ortfolio-based Europ ean da y-ahead electricit y mark et. Electric Po w er Systems Researc h 141, 313–324. doi: https://doi.org/10.1016/j.epsr.2016.08.007 . W ang, C., Peng, T., Luh, P .B., Gribik, P ., Zhang, L., 2013. The subgradient simplex cutting plane metho d for extended lo cational marginal prices. IEEE T ransactions on Po w er Systems 28, 2758–2767. doi: 10.1109/TPWRS.2013.2243173 . W o o d, A.J., W ollen b erg, B.F., Sheblé, G.B., 2013. Po w er generation, op eration, and control. John wiley & sons. Y ang, Z., Zheng, T., Y u, J., Xie, K., 2019. A uniﬁed approach to pricing under noncon v exity . IEEE T ransactions on Po w er Systems 34, 3417–3427. doi: 10.1109/TPWRS.2019.2911419 . Y u, Y., Guan, Y., Chen, Y., 2020. An extended integral unit commitment form ulation and an iterative algorithm for conv ex h ull pricing. IEEE T ransactions on Po w er Systems 35, 4335–4346. doi: 10. 1109/TPWRS.2020.2993027 . Zhao, X., Luh, P .B., W ang, J., 1999. Surrogate gradien t algorithm for lagrangian relaxation. Jour- nal of optimization Theory and Applications 100, 699–712. doi: https://doi.org/10.1023/A: 1022646725208 . 35 Supplemen tary Material In tro duction This do cument pro vides supplemen tary pro ofs and illustrativ e examples to further explain the lemmas and theorems presen ted in Section 3 of the pap er. Pro of of [Lemma 1, Section 3.1] Pr o of. F or the v ariable λ ∈ R |A|×|T | , the inner-lev el optimization problem is a minimization problem of a set of linear functions with respect to λ . Therefore, the ob jective function z ALD ρ ( λ ) of the dual problem is piecewise linear, and due to that, given a vector λ 1 ∈ R |A|×|T | , the gradients of z ALD ρ ( λ 1 ) can b e diﬀerent from diﬀerent directions. F or a one-dimensional case, the deriv ative of z ALD ρ ( λ 1 ) from the left hand side can b e diﬀeren t from that of the right hand side. Therefore, z ALD ρ ( λ ) is non-diﬀerentiable. The dual v ariable λ ∈ R |A|×|T | , and therefore the feasible domain is conv ex. Given λ 1 ∈ R |A|×|T | , λ 2 ∈ R |A|×|T | , α ∈ [0 , 1] , w e hav e z ALD ρ ( αλ 1 + (1 − α ) λ 2 ) = min x ∈ X,p ∈ P L ρ ( x, p, α λ 1 + (1 − α ) λ 2 ) ≥ α min x ∈ X,p ∈ P L ρ ( x, p, λ 1 ) + (1 − α ) min x ∈ X,p ∈ P L ρ ( x, p, λ 2 ) = αz ALD ρ ( λ 1 ) + (1 − α ) z ALD ρ ( λ 2 ) . Hence, the dual optimization problem is concav e. Since the ob jectiv e function z ALD ρ ( λ ) is concav e and piecewise linear, it is sub-diﬀeren tiable. Addi tionally , the subgradient of the dual ob jective function z ALD ρ ( λ ) at a v ector λ ∗ ∈ R |A|×|T | denoted as ∂ z ALD ρ ( λ ∗ ) satisﬁes, z ALD ρ ( λ ) ≤ z ALD ρ ( λ ∗ ) + ∂ z ALD ρ ( λ ∗ ) ⊤ ( λ − λ ∗ ) . Example 1 The examples in this section complemen t [Corollary 1, Theorem 4, Section 3.2.2]. They serv e to demonstrate the p ossibility of negativ e surplus of demand orders under the ALD pricing mechanism, v alidate the prop osed remedy , and illustrate the resulting congestion price signals. All prices are expressed in EUR/MWh. The cost, reven ue, surplus, etc are in EUR. 36 1) Consider the follo wing optimization problem. min − 23 y 1 − 10 y 2 − 16 y 3 + 14 x 1 + 20 x 2 , or equiv alently max 23 y 1 + 10 y 2 + 16 y 3 − 14 x 1 − 20 x 2 , s.t. p 1 = x 1 − y 3 , p 2 = x 2 − y 1 − y 2 , f = p 1 , p 2 = − p 1 , f ∈ [ − 100 , 100] , y 1 ∈ [0 , 100] , y 2 ∈ [0 , 20] , y 3 ∈ { 0 , 50 } , x 1 ∈ { 0 , 120 } , x 2 ∈ { 0 , 50 } . (33) In Problem 33, we consider a t wo-area mark et consisting of tw o supply block orders ( x 1 , x 2 ), one demand blo ck order ( y 3 ), and t wo demand elementary orders ( y 1 , y 2 ). The cost minimization problem yields an optimal ob jective v alue of − 730 (corresponding to a welfare of 730 EUR), with the optimal solution: x ∗ 1 = 120 , x ∗ 2 = 0 , y ∗ 1 = 70 , y ∗ 2 = 0 , y ∗ 3 = 50 , and a cross-zonal ﬂo w f ∗ = 70 . Under the ALD approac h, the dual solution is characterized by λ 1 ∈ [14 , 14 . 5] and λ 2 ∈ [14 , 20] under the p enalt y co eﬃcien t threshold ρ ∗ = 9 required to close the duality gap. Applying the ALD pricing mec hanism [Def. 1, Section 3.2], the resulting price signals are LM P 1 = 14 (EUR/MWh) and LM P 2 = 14 (EUR/MWh). Consequently , the demand order y 3 has a commo dit y cost of 14 × 50 (EUR) and a non-conv ex charge of 9 × 50 (EUR), leading to a surplus of 16 × 50 − (14 × 50 + 9 × 50) = − 350 (EUR). This negative surplus identiﬁes y 3 as a Parado xically A ccepted Order (P A O), which is due to the fact that certain demand bid prices are lo wer than the supply oﬀer prices. Remedial Approach : F or the demand order y 3 , if a transfer pa ymen t of 2 × 9 × 50 = 900 (EUR) is applied, the adjusted surplus b ecomes (16 + 9 − 14) × 50 = 550 (EUR). This conﬁrms that the prop osed remedy restores the individual reven ue adequacy of the paradoxically accepted orders among demand orders. Congestion Price Signals : F urthermore, this example reveals that the raw ALD pricing mecha- nism ma y fail to yield a congestion price signal despite the presence of net work congestion. Note that the cross-zonal price diﬀerence by ALD pricing is LM P 1 − LM P 2 = 14 − 14 = 0 (EUR/MWh). F or comparison, the LP relaxation provides optimal dual v ariables λ 1 = 16 and λ 2 ∈ [16 , 20] . By selecting LM P 1 = 16 (EUR/MWh) and LM P 2 = 17 (EUR/MWh), the resulting congestion price for the TSO is 1 (EUR/MWh). Note that the price signals provided by LP relaxation also eﬀectively eliminate b oth P AOs and PROs. This case highlights a critical limitation: raw ALD dual v ariables ma y fail to internalize net work congestion even when ﬂo w limits are activ e. T o address this, the introduction of the parameter η (as prop osed in [Deﬁnition 1, Section 3.2]) pro vides the necessary ﬂexibility to restore congestion price signals. This ensures reven ue adequacy for the TSO, particularly in scenarios of netw ork congestion, while maintaining the theoretical adv an tages of the ALD framew ork. Note that by adding η = 1 , the ALD pricing mechanism gives LM P 1 = 14 (EUR/MWh) and LM P 2 = 15 (EUR/MWh), implying a congestion price of 1 EUR/MWh for the TSO. 37 2) Scenario with excluded non-comp etitiv e bids: When demand orders with bid prices b elo w supply oﬀers are excluded, the mark et-clearing problem is formulated as follows: max 23 y 1 + 20 y 2 + 16 y 3 − 14 x 1 − 10 x 2 s.t. p 1 = x 1 − y 3 , p 2 = x 2 − y 1 − y 2 f = p 1 , p 2 = − p 1 , f ∈ [ − 100 , 100] y 1 ∈ [0 , 100] , y 2 ∈ [0 , 20] , y 3 ∈ { 0 , 50 } x 1 ∈ { 0 , 120 } , x 2 ∈ { 0 , 50 } (34) The optimal solution is x ∗ 1 = 120 , x ∗ 2 = 50 , y ∗ 1 = 100 , y ∗ 2 = 20 , y ∗ 3 = 50 , indicating full acceptance of all orders with a cross-zonal ﬂo w f ∗ = 70 . Under the ALD framew ork, the solution is characterized by λ 1 = 13 and λ 2 ∈ [13 , 15] . Setting ρ = 1 (where ρ > ρ ∗ = 0 ) and applying the mec hanism in [Def. 1, Section 3.2], the prices are determined as LM P 1 = 13 (EUR/MWh) and LM P 2 = 14 (EUR/MWh). This yields η = 1 (EUR/MWh), representing the congestion price for the TSO. F urthermore, all orders satisfy the reven ue adequacy requiremen t. In contrast, while LP relaxation-based prices (e.g., LM P 1 = LM P 2 = 14 or 15 (EUR/MWh)) also eliminate P A Os and PROs, they do not generate a congestion price for the TSO and th us fail to represent netw ork congestion. Case Study: Dep endency and Dispatch Restrictions This example illustrates a scenario where the mark et-clearing price exceeds a sp eciﬁc supply order’s oﬀer price, yet the order’s dispatch remains at zero due to feasibility constrain ts. Consider the follo wing problem: min − 50 y 1 − 35 y 2 + 40 x 1 + 30 x 2 + 32 x 3 s.t. x 1 + x 2 = y 1 + y 2 , ( λ ) x 3 ≤ x 1 , ( ξ ) y 1 ∈ [0 , 50] , y 2 ∈ [0 , 50] , x 1 ∈ [0 , 30] x 2 ∈ [0 , 50] , x 3 ∈ [0 , 30] (35) Ob viously , Problem (35) is an LP , where there are tw o demand orders ( y 1 , y 2 ) and three supply orders ( x 1 , x 2 , x 3 ). The optimal solution is y ∗ 1 = 50 , y ∗ 2 = 0 , x ∗ 1 = 0 , x ∗ 2 = 50 , x ∗ 3 = 0 , yielding the optimal ob jective v alue of − 1000 . An y λ ∗ ∈ [35 , 40] constitutes an optimal dual solution. Although λ ∗ > 32 (the oﬀer price of x 3 ), the supply order x 3 is not classiﬁed as a Parado xically Rejected Order (PRO). This is b ecause its feasibility is dep endent on the dispatc h of its paren t order ( x 1 ). F ollowing [Deﬁnition 2, Section 3.2], w e ﬁrst ev aluate the individual optimization of the parent order x 1 : max ( λ ∗ − 40) x 1 , s.t. x 1 ∈ [0 , 30] (36) Since λ ∗ ≤ 40 , x ∗ 1 = 0 is an optimal solution. Subsequently , the individual optimization for the child order x 3 is form ulated as: max ( λ ∗ − 32) x 3 , s.t. 0 ≤ x 3 ≤ x ∗ 1 (37) 38 Giv en x ∗ 1 = 0 , the only feasible solution for (37) is x ∗ 3 = 0 . Consequen tly , x 3 is not a PR O as its zero-dispatc h is a requirement of feasibility rather than a rejection of a proﬁtable bid. Case Study: Elimination of PROs and P A Os This example demonstrates how the ALD pricing mec hanism eliminates P arado xically Rejected Orders (PROs) and P aradoxically A ccepted Orders (P AOs) in a Unit Commitment (UC) framework. Consider the follo wing optimization problem (Pure Integer Programming): min 100 x 1 + 315 x 2 + 45 x 3 s.t. 4 x 1 + 9 x 2 + x 3 = 10 , x i ∈ { 0 , 1 } , i = 1 , 2 , 3 . (38) Three supply blo c k orders pro duce electricity to meet a 10 MWh demand. The oﬀer prices are 25, 35, and 45 EUR/MWh, with capacities of 4, 9, and 1 MWh, respectively . The optimal solution is x ∗ 1 = 0 , x ∗ 2 = 1 , x ∗ 3 = 1 , with a total cost of 360 EUR. Under LP relaxation, the LMP is 35 EUR/MWh. A t this price, the supply order 1 ( x ∗ 1 = 0 ) b ecomes a PRO b ecause its oﬀer (25 EUR/MWh) is b elo w the price, while the supply order 3 ( x ∗ 3 = 1 ) b ecomes a P AO b ecause its oﬀer (45 EUR/MWh) exceeds the price. Applying the ALD solution ( λ ∗ = 31 . 7 , ρ = 13 . 37 ), we ev aluate the p erformance of the prop osed ALD pricing mechanism as follo ws: • Supply order 1 ( x ∗ 1 = 0 ): By follo wing the cen tralized dispatc h, its surplus is zero. If it w ere to deviate to x 1 = 1 , it would receiv e a commo dity rev enue of 31 . 7 × 4 EUR, but incur a cost of 25 × 4 EUR, and a non-con vex p enalt y of 13 . 37 × 4 EUR. The resulting surplus, (31 . 7 − 25 − 13 . 37) × 4 = − 26 . 68 EUR, whic h is negativ e. Thus, the supply order 1 has no incen tive to deviate, eliminating the PRO status. • Supply order 3 ( x ∗ 3 = 1 ): By follo wing the dispatch, it receives a total paymen t (commo dit y rev enue plus non-conv ex rew ard) of (31 . 7 + 13 . 37) × 1 = 45 . 07 EUR. Deducting the cost ( 45 × 1 ) EUR, the surplus of supply order 3 is 0 . 07 EUR. Since its surplus is non-negativ e, the P A O condition is resolv ed. F urthermore, it can b e veriﬁed that under the ALD pricing mechanism, supply order 2 has a non- negativ e surplus and has no incen tiv e to deviate from x ∗ 2 = 1 ; supply order 3 has no incen tive to deviate from x ∗ 3 = 1 . Consequently , the ALD mec hanism ensures individual rev enue adequacy for supply orders and incen tive compatibility for each order. 39

A Framework for Eliminating Paradoxical Orders in European Day-Ahead Electricity Markets through Mixed-Integer Linear Programming Strong Duality

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