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Ramping Procurement and Bid-Cost Recovery in Real-Time Market
Deep penetration of renewable energy resources has increased net-load uncertainty and the need for both upward and downward ramping capability in the real-time market. This challenge is particularly pronounced in renewable-rich regions such as those operated by the California Independent System Operator (CAISO) and Midcontinent Independent System Operator (MISO). The substantially increased ramping needs and inadequate ramping support has resulted in the excess use of system reserves and increasing renewable curtailments Clyde24CAISOramp, MISO:Schedule29. Since 2016, both MISO and CAISO have developed Flexible Ramp Product or Ramp Capability Product (herein referred to as FRP), aimed at positioning generators to provide ramping supports via energy and ramp-capacity co-optimizations cavicchi18ramp.
Significant challenges remain, however. First, because existing FRP implementations do not use bid-based ramp procurement and the ramp demand curves are set administratively, ramp prices are often too low under current Locational Marginal Pricing (LMP)-based energy and ramp pricing cavicchi18ramp,FERC15upliftE-2_14. Thus, LMP and ramp capacity shadow price may not be sufficiently high to incentivize generation and storage resources to provide ramping support in real-time dispatch.
Second, to procure ramping capacity to meet anticipated ramping needs in upcoming hours, generators may need to forgo profitable real-time energy opportunities. As a result, energy LMPs may fall below generators' bid-in costs CAISO:25BCRMIO, leading to under-compensation that must be addressed through out-of-market uplift payments.
One of the common out-of-market uplift settlements is the make-whole payment (MWP), also known as real-time bid cost recovery (BCR), which compensates underpaid generators for their losses.\footnote{BCR provides uplifts when market revenues fail to cover a resource’s start-up, minimum-load, or energy-bid costs over the course of a day. We focus on real-time BCR/MWP and energy-bid costs.} However, even though LMP is non-discriminative, the uplift-adjusted energy price is discriminative and the underlying payment mechanism non-transparent. More significantly, perhaps, is that the out-of-market settlement incentivizes strategic bidding behavior that withholds capacity to gain higher profit from out-of-market BCR CAISO:22BCR.
This paper establishes a framework for analyzing ramping procurement in the real-time electricity market, identifies the causes of out-of-market BCR uplifts, and proposes new pricing mechanisms to incentivize generators to provide ramping capabilities. Prior work by Cavicchi and Harvey cavicchi18ramp provides a comprehensive overview of industry practices for ramp capability dispatch and pricing. Building on their work and multiple industry reports in Clyde24CAISOramp, MISO:Schedule29, CAISO:23FRP, we develop concrete models for energy-ramping co-optimization and present both analytical and empirical studies that examine price volatility, out-of-market uplift payments, generator profits, demand payments, and operational efficiency.
Ramping capability procurement serves two purposes cavicchi18ramp: (i) accommodating intertemporal net-load changes and (ii) reserving ramping headroom to hedge against forecast errors. The former motivates {\em multi-interval rolling-window dispatch}, while the latter motivates FRPs. Together, FRP and rolling-window dispatch position resources in advance for anticipated ramping events under real-time operational uncertainty.
For FRP design, prior research has focused on the setting of ramping requirements and on the relationship between FRP and multi-interval dispatch. It has been recognized that FRP are generally not equivalent to multi-interval rolling-window dispatch. In Zhang26Ramp, the authors establish equivalence between FRP and multi-interval dispatch in a one-shot optimization with perfect foresight, and prove that such equivalence does not hold under rolling-window dispatch with imperfect forecasts and limited look-ahead horizons. Other studies focus one day ahead operation with FRP GhaljeheiKhorsand22ADAFRP, YurdakulEla25FRPDA and analyze the impact of ramping requirement settings on system performance wuhug15TPSfrp. In practice, recent enhancements to FRP address network congestion and the choice of look-ahead horizons CAISO:23FRP, cavicchi18ramp.
For {\em multi-interval rolling-window dispatch} in real-time market, the literature progresses along two main directions. The first focuses on operational efficiency under uncertainty using model predictive control (MPC)-style formulations. Studies compare single-interval and multi-interval dispatch and examine the impact of forecast uncertainty on real-time operations Biggar:22EJ, ela15TPSrtFRP. To further enhance performance under uncertainty, stochastic optimization methods have been proposed Cho:23OR, Werner23CDCpricing. These works demonstrate efficiency gains for generators with intertemporal ramping constraints ela15TPSrtFRP and storage with intertemporal state-of-charge constraints Zhao&Zheng&Litvinov:19TPS.
The second line of research focuses out-of-market uplifts and on out-of-merit dispatch. Out-of-merit dispatch refers to resources being dispatched at prices below their bid-in costs CAISO:25BCRMIO, which results in under-compensation Hogan:20, Mays:24EE. Such out-of-merit dispatch arises from multiple sources, including unit commitment for fast-start resources with ramping requirements FERC15upliftE-2_14, wanghobbs15TPSfrp, as well as limited look-ahead horizons and forecast uncertainty in rolling-window dispatch ZhangKory19rampuplift, Cho:23OR. To compensate generators, system operators rely on out-of-market uplifts—such as lost opportunity cost (LOC) and MWP/ BCR.\footnote{LOC compensates generators for their individual ex-post optimal profit—a stronger form of uplifts than MWP/BCR, which only cover energy-bid costs.} While these mechanisms correct ex-post profit shortfalls, they raise concerns regarding transparency and strategic behavior CAISO:25BCRMIO. Consequently, substantial efforts have been devoted to developing alternative real-time pricing methods that reduce uplifts Hogan:20, Zhao&Zheng&Litvinov:19TPS. Prior work shows that LOC uplifts are unavoidable under uniform in-market pricing Guo&Chen&Tong:21TPS, Chen&Guo&Tong:20TPS, whereas MWP uplifts can be eliminated under certain uniform pricing mechanisms ChenTong25PESGM.
Despite extensive research on FRP and {\em multi-interval rolling-window dispatch}, few studies jointly examine operational efficiency, out-of-merit dispatch, and out-of-market compensation in energy-ramping co-optimization. FRPs are typically compensated through opportunity-cost-based pricing, under the assumption that generators awarded ramping opportunity costs can recover foregone energy revenues YurdakulEla25FRPDA. While this assumption holds under perfect foresight, it breaks down in rolling-window real-time market with forecast errors and limited look-ahead horizons. This paper fills this gap by demonstrating that existing schemes for energy and FRPs can still lead to out-of-merit dispatch and out-of-market uplifts, even when FRP payments are applied. We further propose pricing alternatives and analyze operational efficiency across FRP implementations in single- and multi-interval dispatch.
This paper provides both analytical and empirical evaluations of real-time market with ramping procurement through energy–ramping co-optimization. We examine the mechanisms leading to out-of-market uplifts and investigate uniform pricing for procuring ramping capacity without BCR/MWP.
Analytical contributions: First, we develop an energy-ramping co-optimization framework that captures the essential features of practical FRPs, from which we characterize fundamental properties of FRP dispatch and pricing. Second, while it is intuitive that sufficiently high energy prices can eliminate MWPs, the least-cost uniform pricing mechanism that achieves this objective remains unknown. We derive two uniform pricing—maximum dispatch cost pricing (MDCP) and maximum temporal locational marginal pricing (MTLMP)—and theoretically show that MDCP eliminates MWP (Proposition (ref)) and MTLMP does so in most cases (Proposition (ref)). By eliminating MWPs, these pricing schemes provide stronger incentives for ramping support. Moreover, the zero-MWP property of MDCP enables us to establish truthful bidding incentives for price-taking generators (Theorem (ref)).
Empirical contributions: First, we find that the relative efficiency of single- versus multi-interval dispatch depends fundamentally on the system's ramping capability and forecast error. Multi-interval dispatch incurs higher upfront costs to secure ramping capacity but reduces penalties in scenarios with tight ramping requirements, whereas single-interval dispatch is more cost-effective under high forecast uncertainty due to avoiding over-procurement. Second, we observe that LMPs in rolling-window dispatch are highly volatile and frequently negative during constrained periods, causing under-compensation and weak ramping incentives. Multi-interval dispatch generally yields higher prices and stronger ramping incentives by reflecting opportunity costs associated with future intervals. Third, relative to LMP, the proposed uniform pricing mechanisms (MDCP and MTLMP) eliminate out-of-market MWPs, increasing generator profits and ramping incentives while raising total consumer payments. Last, CAISO dataset exhibits higher net load volatility and more extreme ramping events than ERCOT, leading to larger differences in cost and pricing outcomes between dispatch and pricing methods, whereas ERCOT’s smoother net-demand profile results in smaller gaps and more stable pricing.
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In real-time operations, system operators forecast demand and issue dispatch instructions for energy and ramping products at five- or fifteen-minute intervals. Real-time operations vary significantly regarding look-ahead horizons. For instance, MISO employs a single-interval dispatch based on a short-term forecast MISO:Schedule29, whereas the CAISO utilizes a multi-interval dispatch framework that optimizes trajectories over several future intervals CAISO:23FRP. In this section, we review industry practices for energy and ramping procurement, detail the modeling assumptions adopted in this study, and present a general formulation for energy–ramping co-optimization that encompasses both single- and multi-interval dispatch.
Existing operators, such as CAISO CAISO:11FRP and MISO MISO:Schedule29, utilize co-optimization engines to jointly clear energy, ramping, and reserves, yet they differ in their temporal focus and pricing mechanisms. CAISO procures its FRP primarily during the 15-minute real-time unit commitment to ensure sufficient maneuverability for the 5-minute dispatch, utilizing a multi-settlement rule that allows for financial buy-backs based on real-time price fluctuations. Conversely, MISO integrates its upward/downward ramp capability directly into the 5-minute real-time energy and operating reserve market using a ramp capability demand curve. While CAISO’s model emphasizes managing the uncertainty between the 15-minute and 5-minute intervals, MISO’s approach focuses on the marginal opportunity cost of energy to ensure the dispatch does not exhaust the ramping capacity needed for upcoming intervals.
In this paper, we abstract away unit commitment decisions, assuming commitment statuses are fixed. This exclusion avoids the pricing non-convexities associated with binary variables, thereby allowing us to isolate the uplift components specifically attributable to rolling-window dispatch and ramping constraints within the real-time economic dispatch.
The primary objective of this work is to analyze single- and multi-interval ramping procurement efficiency and MWP/BCR rather than to replicate full-scale real-time operations. To maintain analytical tractability, we adopt the following simplifications. (i) Transmission congestion and losses are ignored; as shown in the appendix and Chen&Guo&Tong:20TPS, the framework can be extended to incorporate the DC optimal power flow formulations used in real-time markets. (ii) Generation costs are assumed linear, however, all theoretical results naturally extend to convex piecewise-affine cost functions widely used in practice ChenTong25PESGM. (iii) Ramp capability demand curves are omitted. (iv) Cost-causation–based settlements for ramping CAISO:11FRP are abstracted to focus on system effects of real-time ramping procurement and pricing. We adopt a simplified settlement framework in which generators receive energy and ramping payments directly from demand. This abstraction preserves system operator revenue neutrality while enabling transparent analysis and straightforward extension to more detailed settlement designs.
While these assumptions simplify the operational setting and abstract from certain implementation details in industry practice, the proposed framework preserves the fundamental coupling between energy and ramping decisions. This enables a rigorous analysis of ramping and pricing effectiveness.
Let the complete dispatch horizon be $[T]:=\{1,..., T\}$. Denote ${\cal T}_{t'} := \{t', t' + 1, \dots, t' + W - 1\}$ as a rolling-window look-ahead horizon with $t'$ as the binding interval and the rest are advisory intervals (Fig. (ref) right). This means that, in each rolling-window, the operator optimizes over a look-ahead horizon of $W$ periods. $W = 1$ represents single-interval dispatch ${\cal T}_{t'} = \{t'\}$; $W > 1$ represents multi-interval dispatch.
Let $[N] := \{1, \dots, N\}$ denote the set of all generation units. The operator solves the optimization ${\cal G}_{t'}$ in (ref) to determine the generation $\mathbf{g}_{t'}:=(g_{it})_{ i \in [N],t \in {\cal T}_{t'}}$, ramping up capacity $\mathbf{\overline{r}}_{t'}:=(\overline{r}_{it})_{ i \in [N],t \in {\cal T}_{t'}}$, and the ramping down capacity $\mathbf{\underline{r}}_{t'}:=(\underline{r}_{it})_{ i \in [N],t \in {\cal T}_{t'}}$.
The objective (ref) is to minimize the system operating cost. $c_{it}$ is the marginal generation cost from generator $i$'s offer at time $t$, $l_t $ represents renewable curtailments, and $s_t$, $\overline{s}_t$, and $\underline{s}_t$ are slack variables associate with energy balance and ramping requirements. These slack variables signal scarcity events; when they become nonzero, scarcity penalties are triggered in the objective through administratively determined penalty prices, including load shedding penalty $(p_t)$, ramp down and up shortfall penalties $(\overline{p}_t, \underline{p}_t)$, and renewable curtailment penalty $m_{t}$. These penalty prices are typically much higher than the highest generation cost $c_{it}$. The penalty function in the objective is $J(\lbf_{t'}, \sbf_{t'}, \underline{\sbf}_{t'},\overline{\sbf}_{t'}):=\sum_{t\in {\cal T}_{t'}}(p_{t}s_t+ \overline{p}_{t} \overline{s}_t+ \underline{p}_{t} \underline{s}_t+ m_{t} l_t) $.
Constraints (ref) ensure supply–demand balance, while (ref)(ref) procures sufficient ramp-up and ramp-down capacities. (ref) is the ramping constraint for look ahead intervals, and (ref) specifies the initial ramping constraint based on unit's ramping capability $r^{{\mbox{\rm\tiny U}}}_{i}$ and $r^{{\mbox{\rm\tiny D}}}_{i}$. Constraint (ref) defines the feasible dispatch range for energy, which is smaller than the physical generation capacity $\overline{g}_i$ because upper and lower headrooms are reserved for ramping procurement (Fig. (ref), top right). The upper headroom represents reserved capability to increase generation, limited by both the unit’s maximum capacity $\overline{g}_i$ and its upward ramp rate. Symmetrically, the lower headroom ensures the generator can safely decrease its output without violating its minimum operating limits or downward ramp rates. Constraint (ref) restricts procured ramping capacities according to the physical ramping limits $r^{{\mbox{\rm\tiny U}}}_i$ and $r^{{\mbox{\rm\tiny D}}}_{i}$. The dual variables are denoted by $\lambda_t, \overline{\eta}_t$, and $\underline{\eta}_t$.
The operator is given the following forecasts and system parameters (Fig. (ref) left): (i) net demand forecast $\{\hat{d}_t\}_{ t \in {\cal T}_{t'}}$ (defined as load minus variable renewable generation and assumed to be nonnegative), (ii) ramp up requirements $\{\overline{\omega}_t\}_{ t \in {\cal T}_{t'}}$ and ramp down requirements $\{\underline{\omega}_t\}_{ t \in {\cal T}_{t'}}$, and (iii) the initial generation level $\{g_{i(t'-1)}\}_{i \in [N]}$. To forecast net demand and ramping up and ramp down requirements, we assume a Gaussian distribution for the historical and future net demand. Based on this assumption, we utilize a rolling horizon Minimum Mean Square Error (MMSE) estimator to dynamically forecast the expected net demand trajectory $\{\hat{d}_t\}_{ t \in {\cal T}_{t'}}$ and the probability distribution of the net demand. The system-wide upward and downward flexible ramping requirements $\{\overline{\omega}_t\}_{ t \in {\cal T}_{t'}}$ and $\{\underline{\omega}_t\}_{ t \in {\cal T}_{t'}}$ are then analytically derived to ensure sufficient capacity is reserved to cover the 95% confidence interval of the conditional forecast error (Fig. (ref) bottom left). Specifically, following the forecast model, the ramping requirements are designed to cover both the expected intertemporal change in net demand and the uncertainty margins of the forecast error—denoted by the upward $\overline{U}_{t}$ and downward $\underline{U}_{t}$ margins—such that $\overline{\omega}_{t'}\geq \Delta \hat{d}_{t'}:= \hat{d}_{t'+1}-\hat{d}_{t'}$. (Fig. (ref) top left). Detailed math formulations for these dynamic load forecasts and ramping requirements are in the appendix.
Real-time dispatch is a sequential process with rolling-windows (Fig. (ref) bottom right). The power system operator solves problem ${\cal G}_{t'}$ in (ref), and implements only the following decisions for the {\em binding} interval: \beq g^{{\rm\tiny R}}_{it'} := g^*_{it'}, r^{{\rm\tiny R}}_{it'} := r^*_{it'}, \overline{r}^{{\rm\tiny R}}_{it'} := \overline{r}^*_{it'}, \eeq where superscript * denotes the optimal solution for energy dispatch and ramping procurement. For the next rolling window to obtain the dispatch at the binding interval $t'+1$, the operator update the known parameter $g_{it'}=g^{{\mbox{\rm\tiny R}}}_{it'}$ for the new initial generation based on the solution (ref) from the last rolling window. The operator updates the forecast $\{\hat{d}_{t}\}_{t \in {\cal T}_{t'+1}}$, ramp up requirements $\{\overline{\omega}_t\}_{ t \in {\cal T}_{t'+1}}$, and ramp down requirements and $\{\underline{\omega}_t\}_{ t \in {\cal T}_{t'+1}}$ based on the real time information. Then, with updated parameters, operator resolves ${\cal G}_{t'+1}$ via (ref).
For the {\em single interval dispatch} with $W =1$, (ref) represents the single-interval optimal dispatch decision in (ref). For the {\em multi-interval dispatch} with $W >1$, although (ref) produces optimal solutions over the entire horizon ${\cal T}_{t'}$, only the first {\em binding} interval results at $t'$ are implemented, while subsequent rolling-window decisions serve as {\em advisory} signals.
To ensure generators follow real-time dispatch signals, the power system operators provide in-market and out-of-market payments. In-market payments are determined by real-time market prices, such as locational marginal pricing (LMP). Out-of-market settlement is made outside the real-time dispatch optimization (e.g. typically at the end of the day when all the dispatches are realized.). It aims to compensate generators when in-market payments are insufficient to cover their offered marginal costs (i.e., when a generator is dispatched at a price below its bid). Such cases arise in real-time markets with uncertainty and intertemporal coupling CAISO:25BCRMIO. Together, these two payments ensure generators are not under-compensated and maintain dispatch following incentives. However, out-of-market payments distort the transparency of uniform LMP and incentivize strategic bidding behaviors. To address these issues, we propose two uniform pricing methods---max dispatch cost pricing (MDCP) and max temporal locational marginal pricing (MTLMP)---that eliminate out-of-market payments. Both MDCP and MTLMP coincide with LMP when ramping constraints are non-binding.
Real-time uniform pricing signals determine in-market payments for both energy and FRP. Following the marginal pricing principle, the real-time energy price LMP at the time interval $t$ derived from envelope theorem is given by \beq \pi^{\tiny LMP}_{t} := \frac{\partial F}{\partial \hat{d}_{t}} = \lambda^*_{t}, \eeq where $F$ denotes the optimal objective value in (ref) and $ \lambda^*_{t}$ is the optimal dual associated with the balance constraint (ref).
The up and down FRP prices $(\pi^{{\mbox{\rm\tiny U}}}_{t},\pi^{{\mbox{\rm\tiny D}}}_{t})$ are similarly obtained by envelope theorem, i.e., \beq \pi^{{\rm\tiny U}}_{t}:= \frac{\partial F}{\partial \overline{\omega}_{t}}=\overline{\eta}^*_t, \pi^{{\rm\tiny D}}_{t}:= \frac{\partial F}{\partial \omega_{t}}=\eta^*_t, \eeq following principles analogous to reserve capacity pricing CAISO:11FRP, WuPapalexopoulos04TPSreservepricing. $\overline{\eta}^*_t$ and $\underline{\eta}^*_t$ are the optimal dual variables of the upward and downward ramping constraints (ref), respectively.
The total in-market payment to generator $i$ at the time interval $t$ consists of an energy payment ${\cal P}^{\text{\tiny LMP}}_{it} $ and a ramp capability payment ${\cal R}_{it}$, \ie \beq {\cal P}^{\tiny LMP}_{it} := \pi^{\tiny LMP}_{t} g_{it}^{{\rm\tiny R}}, \eeq \beq {\cal R}_{it} := \pi^{{\rm\tiny U}}_{t} \overline{r}^{{\rm\tiny R}}_{it} + \pi^{{\rm\tiny D}}_{t} \underline{r}^{{\mbox{\rm\tiny R}}}_{it} , \eeq where $g_{it}^{{\mbox{\rm\tiny R}}}$, $\overline{r}^{{\mbox{\rm\tiny R}}}_{it} $, and $\underline{r}^{{\mbox{\rm\tiny R}}}_{it} $ denote the {\em binding} real-time dispatch of energy, upward ramping, and downward ramping, respectively, as determined by (ref).
Due to uncertainty and temporal coupling, LMP alone may fail to provide sufficient dispatch-following incentives for all generators. Out-of-market payments—namely make-whole payments (MWP) and lost opportunity costs (LOC)—address this issue. MWP compensates generators for underpayment. LOC compensates for foregone profits from alternative optimal dispatches. LOC provides stronger incentive alignment, while MWP serves as a baseline measure ensuring immediate compensation. In practice, MWP is used more frequently by assuming generators with immediate compensation are willing to follow the real-time dispatch signal.
Denote ${\cal M}_{it}(\cdot)$ as the interval-based MWP\footnote{In practice, daily MWPs are often adopted for real-time BCR uplifts, \ie $\max\{0, \sum_{t \in {\cal H}} (c_{it} g^{{\mbox{\rm\tiny R}}}_{it})- \pi_t g^{{\mbox{\rm\tiny R}}}_{it}\}$ CAISO:24BCR. We compute both interval-based MWPs and daily MWPs in the simulation and appendix sections. } for generator $i$ at time $t$, which can be computed by \beq {\cal M}_{it}(\pi_t, g^{{\rm\tiny R}}_{it}) := \max\{0, c_{i} g^{{\rm\tiny R}}_{it}- \pi_t g^{{\rm\tiny R}}_{it}\}, \eeq where $\pi_t$ is the market price. MWP is positive only when the price is less than it's bids ($\pi_t < c_{i}$). Thus, by paying MWPs, the operator eliminates under-compensation. Note that out of market payment MWP is discriminative over different generators, although LMP for in market payment is uniform.
Eliminating out-of-market payments improves market transparency, as such payments often distort real-time uniform price signals and incentivize strategic bidding CAISO:25BCRMIO. As shown in Guo&Chen&Tong:21TPS, no uniform price can remove LOC for all generators. Nonetheless, uniform prices are preferred for transparency, and we next propose two such mechanisms—MDCP and MTLMP—that achieve zero MWP.
Define the real-time MDCP at time $t$ as \beq \pi^{{\rm\tiny MDCP}}_{t} := {\rm max} \{c_{1t}\mathbbm{1}_{[g^{{\rm\tiny R}}_{1t}>0]}, ..., c_{Nt}\mathbbm{1}_{[g^{{\rm\tiny R}}_{Nt}>0]}, p_{t}\mathbbm{1}_{[s^{{\rm\tiny R}}_{t}>0]}\}, \eeq where $\mathbbm{1}_{[g^{{\mbox{\rm\tiny R}}}_{it}>0]}$ indicates the dispatch status of generator $i$. The indicator $\mathbbm{1}_{\cal X}$ equals 1 if ${\cal X}$ is true, otherwise it's zero. In normal operation, MDCP equals the maximum marginal cost among dispatched generators; in scarcity events with nonzero slack variable $s^{{\mbox{\rm\tiny R}}}_{t}$ for load shedding, MDCP equals to the load shedding penalty prices $p_t$. The superscript $^{{\mbox{\rm\tiny R}}}$, consistent with (ref), denotes the dispatch outcome realized in the binding interval of the rolling-window dispatch. Assume realized demand $d_t \in \mathbb{R}^+$, under MDCP, the MWP computed by (ref) are eliminated for all generators. Moreover, the total demand payment is minimized, provided that the price respects the scarcity condition that penalizes load shedding. This result is formally stated below.
The proof is in the appendix. Here, the real-time dispatch $g^{{\mbox{\rm\tiny R}}}_{it}$ is computed by (ref). Under MDCP, generators $i$ at time $t$ receive in-market energy payment computed by \beq {\cal P}^{\tiny MDCP}_{it} := \pi^{\tiny MDCP}_{t} g_{it}^{{\rm\tiny R}}, \eeq ramping payment (ref), and zero out-of-market MWP uplifts.
Define the real-time MTLMP at time $t$ as \beq
\eeq where temporal locational marginal pricing\footnote{TLMP definition here is different but consistent with Guo&Chen&Tong:21TPS, which computes the derivative at the optimal solution.} (TLMP) is given by $\pi^{{\mbox{\rm\tiny TLMP}}}_{it}:= - \frac{\partial}{\partial g_{it}} V_{it}(g^*_{it}, \rbf^*)$ with $\mathbf{r}^*:=\{\mathbf{\overline{r}}^*_t, \mathbf{\underline{r}}^*_t\}_{t\in {\cal T}_{t'}}$ and $V_{it}(g^*_{it}, \rbf^*):=F(g^*_{it}, \rbf^*)-c_{it}g^*_{it}$. $\mathbf{r}^*$ represents the optimal ramping procurement. $V_{it}(g^*_{it}, \rbf^*)$ represents the optimal total generation cost in (ref) excluding the contribution from generator $i$ in interval $t$. When deriving the energy price MTLMP, we fix all ramping products at the optimal solution to exclude the influence of ramping products. That way, in normal cases without load shedding, TLMP evaluates the marginal energy cost of losing generation for generator $i$ at time $t$. Thus, MTLMP represents the system’s worst marginal cost of losing generation. In scarcity cases with nonzero slack variable $s^{{\mbox{\rm\tiny R}}}_{t}$ for load shedding, we set MTLMP equals the load shedding penalty $p_t$. Through the envelope theorem, MTLMP can be computed by the optimal dual values of (ref). Detailed derivations of MTLMP are provided in the appendix.
MTLMP have several advantages: (i) it rewards generators with higher ramp capabilities, as generators without binding ramping constraints usually receive payments above its bid-in costs; (ii) it reflects the system’s marginal generation loss; (iii) it extends naturally to network-constrained systems; and (iv) it generally eliminates MWP (Proposition (ref)).
The proof is in the appendix. Although MTLMP may not guarantee zero MWP in every edge case, such scenarios (pure ramp-down scheduling) are rare in practice. Under MTLMP, generators receive in-market energy payment \beq {\cal P}^{\tiny MTLMP}_{it} := \pi^{\tiny MTLMP}_{t} g_{it}^{{\rm\tiny R}}, \eeq and ramping payment computed by (ref). From Proposition (ref), most resources won't receive out-of-market MWP uplifts.
Because MDCP is defined by the bid-in-costs, there is a valid concern that such a pricing mechanism may incentivize generators to bid untruthfully, \eg by inflating their reported costs to increase profit. The same question applies to LMP under the classical single-interval dispatch without ramping constraints, for which the answer is that bidding truthfully at marginal cost is optimal for price-taking generators. However, under LMP in multi-interval rolling-window dispatch, a price-taking generator does have an incentive to deviate from truthful bidding to profit from out-of-market payments Guo&Chen&Tong:21TPS.
Theorem (ref) below generalizes the truthful-bidding results for price-taking generators under MDCP, demonstrating that truthful bidding is a local Nash-equilibrium strategy. Here, we make a simplifying assumption by considering the one-shot dispatch of (ref) within a rolling window $\mathcal{T}_{t'}$, where the operator has the demand forecast for the entire horizon and solves the dispatch in a single step. The obvious discrepancies between the one-shot and rolling-window dispatches require some justification. Specifically, given any realization of forecasts over the entire horizon, the profit realized by the one-shot dispatch is a tight upper bound of the profit from the rolling window dispatch. In particular, there exist forecasts for which the rolling-window dispatches match those from the one-shot dispatch. This means that the optimal bidding strategy cannot be improved uniformly across all demand forecasts unless we have additional information about the forecast properties. For a price-taking generator that submits its bid before delivery without foresight of demand, Theorem (ref) implies it should bid truthfully as a local optimal strategy.
The proof of Theorem (ref) is provided in the Appendix. The key step is to establish that the price markup under MDCP over the marginal cost is the same for all bids in $\Cc_i$. By Topkis' monotonicity theorem, increasing bid monotonically decreases the market-cleared generation quantity, therefore, monotonically decreasing the profit. $c_{it}^\dagger$ denotes the true marginal cost of generator $i$ at time $t$ and $\boldsymbol{\pi}^{{\mbox{\rm\tiny MDCP}}}_{-i}$ denotes the MDCP when generator $i$ is excluded from the dispatch problem (ref). The restriction $\mathbf{c}_i \ge \mathbf{c}_i^\dagger$ is without loss of generality, since bidding below marginal cost leads to weakly lower profits under MDCP, which is guaranteed to be no less than the bid-in cost for dispatched units. Theorem (ref) supports that truthful bidding of every generator is a local Nash equilibrium under MDCP, \ie generator $i$ has no profitable deviation locally within the ${\cal C}_i:=\{\cbf: \cbf^\dagger_{i} \le \cbf \le \overline{\cbf}_i\}$.
The main weakness of Theorem (ref) is the price-taking assumption of all participants, and the Nash-equilibrium-like statement implies that, assuming all other generators bid truthfully, generator $i$ should also bid truthfully. Another weakness is that truthful bidding is held locally around the true marginal cost. Both weaknesses also apply to the classical single-interval dispatch under LMP. Note also that the region ${\cal C}_i$ for which truthful bidding holds cannot be enlarged in general because, when generator $i$ is an MDCP setter, increasing its bid slightly above its marginal cost does not change the dispatch quantity but strictly increases its profit.
We compare single interval and rolling-window multi-interval dispatch frameworks under LMP, MDCP, and MTLMP. The performance is evaluated using metrics including out of market settlement (MWP), demand payment, and operation efficiency (dispatch cost). In the appendix, we provide additional results and examples to provide more insights of ramping procurement and MWP.
Our realtime market simulation is grounded in consecutive 100-day rolling windows of realized net-load trajectories derived from re-scaled CAISO and ERCOT 15-minute data.\footnote{Specifically, we re-scaled the data through an affine transformation such that both datasets share the exact same minimum and maximum values throughout the entire 100-day period.} In each rolling window ${\cal T}_{t'}$, the $W$-interval look-ahead flexible ramp dispatch ((ref)) is implemented to produce the dispatch and the up/down ramp reserve capacity of the binding interval $t'$. The forecasted net demand in ((ref)) is produced based on the MMSE predictor using past $96$ samples, and the ramp reserve requirements are calculated based on a Gaussian approximation of the conditional probability distributions within the look-ahead window (see details in Appendix (ref)).
The top panels of Fig. (ref) illustrate the complete sets of realized net-demand trajectories over the 100-day simulation period for the CAISO and ERCOT systems, respectively. In these plots, each distinct color corresponds to a different day within the sequence, illustrating the daily chronological variations in the net load. Across all days, critical periods for flexibility assessment consistently emerge during the steep morning ramp-down and the afternoon/sunset ramp-up, which are highly pronounced in CAISO data. In comparison, ERCOT net-demand trajectories remain noticeably flatter, and the variability comes from the wind uncertainty during the day.
For the single-interval dispatch, each time interval represents 15 minutes MISO:Schedule29.\footnote{In practice, MISO solves (ref) every 10 minutes, while its ramping uncertainty window spans 10--30 minutes. CAISO solves (ref) every 15 minutes. In simulation, the parameter settings do not necessarily replicate exact industry practices, but they are chosen to be comparable for analytical purposes.} Dispatch decisions were computed from (ref) with $W=1$, representing a greedy dispatch without consideration of future demand forecasts. Here, we procured ramping capacity of the current 15-minute interval. {Across all dispatch models, the real-time prices (LMP, MDCP, and MTLMP) are computed from (ref), (ref), and (ref), respectively.
For the multi-interval rolling-window dispatch, the look-ahead window was set to $W=4$, \ie one hour, with each interval again representing 15 minutes CAISO:23FRP. Ramp capacities were procured for each future interval. For the system operating cost comparison, we evaluated two variations of the multi-interval model: the fully dynamic uncertainty margin approach (denoted as M), where uncertainty margins naturally expand over the look-ahead horizon using the formulas detailed in Appendix (ref); and the “15-Minute” uncertainty approach (denoted as 15-m), which statically uses the ramping requirement calculated for the binding interval across all future advisory intervals (see Section (ref) for further details).
We considered a single-bus system with three generators, each with different bid-in marginal costs. This simplified setup allowed us to focus on ramping influence ignoring the network congestion. Extensions to networked systems can be implemented following the approach in the appendix and also in Chen&Guo&Tong:20TPS. The bid-in marginal generation costs $\cbf$ for G1, G2, and G3 were 25, 30, and 50 \$/MWh, respectively. Maximum generation capacities $\overline{g}_i$ were all 500 MW. The marginal penalty costs for reserves and curtailments were set at 80 \$/MWh CAISO:20FRPRefinements. As shown in the bottom of Fig. (ref), generator ramping capabilities were varied across ten configurations, labeled S1--S10. Case S1 corresponded to the lowest ramping capability, while S10 represented the highest.
Fig. (ref) compares hourly LMP, MDCP, and MTLMP averaged over 100 days' empirical datasets at CAISO and ERCOT. To distinguish between the dispatch models, pricing schemes under single-interval dispatch are denoted as S-LMP, S-MDCP, and S-MTLMP, while those under multi-interval dispatch are denoted as M-LMP, M-MDCP, and M-MTLMP.
The economic relationships and pricing behaviors observed in S3 were consistent across all ramping scenarios (S1--S10). As generators’ physical ramping capabilities increased beyond S3, ramping constraints became less binding, leading to more stable pricing signals and convergence across the different pricing mechanisms. Here we summarize key observations associated with S3 from Fig. (ref).
As a primary observation, the ERCOT dataset exhibited significantly lower price variability than CAISO. In the CAISO data, LMP showed substantially higher variations than both MDCP and MTLMP under single-interval and multi-interval dispatch. Specifically, the CAISO LMP signal is characterized by severe fluctuations; it dips sharply into negative values during the morning ramp-down, whereas MDCP and MTLMP effectively removed these deep price drops and appropriately increased prices during the steep afternoon ramp-up. Notably, negative LMP in CAISO reflected the effect of the curtailment of generation in the pricing signal and led to under-compensation and higher MWPs (see Sec. (ref)).
To compare the pricing benchmarks, LMP was generally lower than MDCP and MTLMP. For instance, during morning periods, procuring sufficient downward ramping capability required more generators to be dispatched. This increased the highest marginal cost among dispatched units, directly driving up the MDCP. The LMP, however, did not increase, as the extra marginal energy can still be supplied by cheaper generators that possess available upper headroom. Furthermore, the price profiles for MDCP and MTLMP were nearly identical across all single and multi-interval dispatch models. Interestingly, for the ERCOT dataset with lower net-demand volatility, the prices across all three methods (LMP, MDCP, and MTLMP) remained roughly the same.
When comparing each optimization approach, the relationship between multi-interval and single-interval prices depended on the timing of future ramping needs. Multi-interval prices were higher in the morning because a marginal increase in demand during the current binding interval exacerbated the difficulty of ramping down in subsequent intervals. Because multi-interval pricing explicitly considered both present and future operational costs, this anticipated future ramping-down scarcity was reflected as a higher price in the binding interval. In the sunset period, multi-interval prices can actually fall below single-interval prices. During periods of steep upward ramping, a marginal increase in generation in the current interval effectively pre-positioned the system, alleviating future ramping constraints and making future operations cheaper. Consequently, this anticipated future cost reduction translated into a lower overall marginal price for the multi-interval model compared to a myopic single-interval approach.
It is important to note that while Fig. (ref) reports the daily average prices—where MDCP and MTLMP generally appear to exceed or bound LMP—there is no strict mathematical ordering among the three pricing schemes. Depending on the specific binding constraints, generator bids, and system conditions within any individual dispatch interval, all possible ordinal relationships between LMP, MDCP, and MTLMP can materialize in practice.
We summarize the results for average daily generator profits (Table (ref)), MWP (Table (ref)), and demand payments (Table (ref)).
As the physical system became more constrained (moving from scenario S8 towards S1), we found that the magnitude of profits, MWP fractions, and payments noticeably increased, reflecting the higher operational difficulty and tighter constraints in both CAISO and ERCOT cases. On the other hand, as the system became less constrained (moving towards S8), the total profits and payments under the different pricing schemes converged with each other, mirroring the stabilization and convergence observed in the pricing signals themselves.
Generator profit trends (Table (ref)), including profits from energy, FRP, and MWP, highlight the impact of the different pricing mechanisms. Profits with LMP dispatch were consistently lower than those under MDCP and MTLMP. While MDCP generally resulted in the highest generator profits across most scenarios, we noted a specific exception in the ERCOT case under scenario S3, where M-MTLMP yielded the highest profit.
The differences in generator profitability are closely tied to MWPs, which are shown in Table (ref). MWPs were zero under MDCP and very close to zero under MTLMP across all ramping and forecast settings, validating Propositions (ref)--(ref). Although MTLMP may yield nonzero MWPs when $g^*_{it} = \underline{r}^*_{it}$, this case happened with low frequency and only when the ramping down price was positive. In contrast, MWPs under LMP were significantly higher. We present interval-based MWP in the main text and daily MWP in Appendix Sec. (ref). Both types of MWPs were comparable to or even larger than generator profits when ramping capabilities were highly constrained, like in Case S1 with frequent negative LMPs. To explicitly highlight the severity of out-of-market uplifts, the percentages shown in parentheses in Table II represent the MWP divided by the pure energy revenue (calculated without considering the MWP). Comparing the datasets, the severe variations in price in the CAISO data caused a massive portion of the total energy revenue to come from MWPs (reaching up to 41.1% in S1). This highlights the high frequency and severity of negative or excessively low LMPs in that case. In contrast, the relative MWP fraction of the energy revenue in the ERCOT case was significantly smaller.
Table (ref) illustrates the demand payments (including MWPs), showing that LMP consistently yields the lowest demand payments. In contrast, MDCP and MTLMP have higher demand payments because they internalize the out-of-market MWPs into the uniform in-market price. With limited ramping capacities in Case S1, MDCP added approximately 10% more to the demand payment compared with MTLMP, while the change of MWP was small. This showed MDCP paid a high demand price to completely remove all MWPs in Propositions (ref). Comparatively, MTLMP in Propositions (ref) can mostly remove MWP while yielding a lower demand payment. Although this increased the upfront cost to consumers, incorporating these costs directly into the uniform market clearing price greatly improves market transparency. It provided a true, cost-reflective signal for flexibility and eliminates the out-of-market uplifts that distort generator bidding incentives.
This subsection provides an empirical comparison of operational efficiency (total cost) between single and multiple interval flexible ramp dispatch models, independently of the pricing method. Based on the binding-interval objective values of (ref), Fig. (ref) shows the system operating cost including generation cost, ramping reserve costs (for ramp up and ramp down shortfalls), and penalties for energy imbalance (load shedding and renewable curtailment). Lower system operating costs represent higher efficiencies. Variations in total operating cost directly reflected differences in the quantities of energy imbalance, ramping shortfall, and reserve utilization procured in real time, because the energy imbalance penalty price and ramping reserve prices were exogenously fixed in simulation.
Current FRP implementations often rely on short-term uncertainty margins $\overline{U}_{t}$ and $\underline{U}_{t}$ (top left of Fig. (ref)) that do not fully capture the increasing variance of net load over longer horizons. As noted by CAISO, the “15-minute” uncertainty used in current FRP designs is substantially smaller than the actual uncertainty that materializes over a multi-interval horizon CAISO:25FRP_DMM. To evaluate the impact of uncertainty margin modeling and isolate the economic value of accurately capturing the growth of uncertainty over time, as opposed to merely incorporating a multi-interval horizon, we evaluated three frameworks: the myopic Single-interval (S) approach, a 15-minute (15-m) uncertainty multi-interval approach, and the fully dynamic actual uncertainty Multi-interval (M) approach. In M approach, flexible ramping requirements expand across later advisory intervals as forecast variance increases (details in Appendix (ref)). In contrast, the 15-m approach determines the uncertainty margin based solely on the immediate next interval and applies this same margin uniformly across all subsequent look-ahead intervals (see Section (ref) for further details). The observations are summarized as follows.
First, the majority of operating costs arose from baseline generation rather than penalty terms across all the scenarios. However, the efficiency gap between multi- and single-interval dispatch was primarily driven by penalties associated with energy imbalances and ramping shortfalls, particularly in tightly constrained scenarios (S1–S2). Moreover, evaluating the differences between the datasets, CAISO experienced more severe ramping events, resulting in significantly higher average operating costs. In contrast, ERCOT exhibited lower total costs due to the less physically demanding net-load profile and less challenging intra-day variability.
Second, when the system possessed sufficient physical flexibility to absorb net-load volatility (S5-S10), single- and multi-interval dispatch exhibited nearly identical performance. In these scenarios, the advanced preparation provided by the multi-interval look-ahead offered little to no additional value, and a single-interval dispatch was sufficient to maintain feasibility without triggering severe shortages.
Third, focusing on the differences between the forecasting frameworks, under highly constrained conditions, the performance of the 15-minute (15-m) uncertainty approach is notable. By incorporating a look-ahead horizon, it avoided the short-sightedness of single-interval dispatch and was consistently more cost-effective than the single-interval (S) approach. However, because it relied on a static representation of uncertainty, it failed to capture the growth in forecast variance across future intervals and thus remained more expensive than the fully actual uncertainty (M) approach. Overall, multi-interval dispatch with actual uncertainty (M) achieved the lowest cost. However, in the CAISO case, the system was so heavily challenged by steep physical ramping events that the primary economic benefit came simply from having any look-ahead horizon. As a result, the cost difference between (15-m) and (M) was relatively small.
Overall, when ramping requirements were properly specified, multi-interval dispatch consistently reduced total operating costs and improved system efficiency relative to single-interval dispatch. This finding is consistent with insights from stochastic control and with cavicchi18ramp: maintaining ramp capability requires higher production costs in the current binding interval, which is economically justified only when the expected future value of that ramp capability (e.g., avoiding a penalty) exceeds the immediate cost.
High renewable penetration has intensified uncertainty and ramping requirements in the real-time electricity market, challenging existing dispatch and pricing mechanisms. This paper studies single- and multi-interval energy–ramping co-optimized real-time market with a focus on bid cost recovery (BCR), truthful bidding, and operational efficiency. We propose two uniform pricing mechanisms—max dispatch cost pricing (MDCP) and max temporal locational marginal pricing (MTLMP). We theoretically show that MDCP eliminates BCR out-of-market uplifts and preserve truthful bidding incentives for price-taking generators. We compare single- and multi-interval dispatch efficiency with ramping procurement under various forecast settings.
Empirically, we evaluate single- and multi-interval dispatch with LMP, MDCP, and MTLMP. Our results show that: (i) LMP is highly volatile and often negative in severe ramping scenarios, whereas MDCP and MTLMP eliminate negative prices and out of market BCR, yielding higher generator profits but higher demand payments; (ii) multi-interval dispatch internalizes future opportunity costs, leading to higher energy prices during constrained ramp-down periods to prevent future scarcity, while potentially lowering prices during ramp-up by anticipating increased supply availability; (iii) removing BCR uplifts improves pricing transparency but increases demand payments; and (iv) the relative efficiency of dispatch models depends critically on system flexibility and forecast accuracy: although dynamic multi-interval dispatch generally minimizes total operating costs in tightly constrained grids by pre-positioning resources, simpler industry practices—such as single-interval or fixed-margin multi-interval dispatch—are highly competitive alternatives that perform just as efficiently in systems with high ramping flexibility or less challenging net demand profiles.
Several avenues for future research remain. First, this study focused on intertemporal ramping constraints within a copper-plate model. While MTLMP naturally extends to network-constrained settings, MDCP requires the specific modifications outlined in the appendix to address congestion. Second, we do not model differing time granularity between real-time dispatch and real-time unit commitment for energy and ramping procurement. Third, our current formulation couples generator minimum output limits with ramp-down requirements; alternative formulations that decouple these constraints could mitigate potential under-compensation issues. Finally, our empirical study is conducted on a small-scale system with positive net load; we anticipate that the qualitative insights will generalize to larger networks with various load scenarios, though further validation is needed.