Feasible Operating Region (FOR) Overview

• Feasible Operating Region (FOR) is the constrained envelope of admissible system states defined by physical, operational, or combinatorial limits.
• Methodologies such as directional optimization, inner approximations, and data-driven sampling are used to compute and approximate FOR boundaries.
• FOR insights support system safety and robust trajectory planning by reducing high-dimensional constraints to actionable lower-dimensional representations.

Searching arXiv for recent and relevant papers on "Feasible Operating Region" and closely related "feasible region" usages across domains. Feasible Operating Region (FOR) denotes, across several research literatures, the set of admissible operating states or performance vectors of a system under its governing physical, operational, or combinatorial constraints. In immune recognition, the term refers to the biologically useful subset of the speed–energy–accuracy landscape in which a proofreading circuit achieves high accuracy, short mean first-passage time (MFPT), moderate power dissipation, and a sufficiently large output signal (Cui et al., 2017). In power systems, it usually denotes the set of feasible active and reactive power exchange points at a point of common coupling (PCC) or other interconnection interface, defined by power-flow equalities and operational inequalities and used as a safety envelope for online control and coordination (Klein-Helmkamp et al., 2024). Related literatures use closely allied expressions such as feasible region, feasible operation region, flexibility region, robust feasible region, and feasible planning region, but the unifying idea is the same: a FOR is an admissible boundary in a reduced coordinate space that preserves the relevant constraints of a higher-dimensional system (Riaz et al., 2019).

1. Cross-domain meaning and scope

The term is domain-dependent in its coordinates, but not in its logic. It always identifies a constrained envelope of realizable macroscopic states. In distribution-grid and TSO/DSO work, the coordinates are typically active and reactive interconnection flows in the $PQ$-plane; in T-cell proofreading they are speed, dissipation, and error; in internal combustion engine modeling they are engine speed and torque/load; in legged locomotion they are horizontal center-of-mass coordinates; and in permutation theory they are limiting vectors of local pattern frequencies (Böttcher et al., 2023).

Domain	Coordinates	Meaning of the region
T-cell recognition	error, MFPT, dissipation, output	fast and accurate decisions at intermediate energy cost
Power systems	$P,Q$ at PCC or interface	all valid interconnection power flows or feasible PCC power-flow points
Internal combustion engines	$\omega_{eng}, \mathrm{T}_{eng}$	full speed-load map where steady-state exergy balance is computable
Legged locomotion	$c_{xy}$	CoM projections satisfying static equilibrium, friction, and actuation limits
Consecutive permutation patterns	$(v_\pi)_{\pi\in S_k}$	realizable limiting consecutive-pattern frequency vectors

This diversity is explicit in the literature. Distribution-grid studies describe the FOR as containing “all valid interconnection power flows” and as defining “safe operating conditions for the distribution grid” (Böttcher et al., 2023). Virtual-power-plant studies describe it as the set of all feasible dispatch power points, while distinguishing it from the subset that can be reached within a specified response time (Riaz et al., 2019). In early T-cell recognition, the feasible operating regime is narrower: it is not the entire high-accuracy region, but the subset with “highest accuracy and a high output signal,” visible as a narrow band in numerical phase diagrams (Cui et al., 2017).

A plausible implication is that FOR functions as a reduction operator: it compresses a full constrained model into a lower-dimensional object that can be communicated, optimized over, or interpreted.

2. Mathematical structure

A recurring formal pattern is existential projection. In online power-system operation, the FOR is introduced as

$\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$

where $s_{\text{PCC}}$ is the complex load flow at the PCC and $\mathbf{x}$ the full system state. The region is therefore the feasible slice of the high-dimensional AC-feasible set projected onto the PCC load-flow variables (Klein-Helmkamp et al., 2024). The same projection logic appears in multi-interconnection TSO–DSO coordination, where the FOR is the projection of a DSO feasible set $\mathcal{F}\subset\mathbb{R}^n$ onto interface variables $u$, i.e.

$P,Q$0

with internal variables eliminated (Bandeira et al., 27 Oct 2025).

In the T-cell proofreading literature, the coordinates are not physical interface flows but performance observables. Accuracy is quantified by

$P,Q$1

speed by the MFPT to produce the final foreign product $P,Q$2, and energy consumption by steady-state power dissipation

$P,Q$3

For irreversible KPR ($P,Q$4), the minimum error is the Hopfield limit

$P,Q$5

For the standard example $P,Q$6, $P,Q$7 s, $P,Q$8 s, this gives $P,Q$9 (Cui et al., 2017). The feasible operating regime is the subset of this speed–energy–accuracy plane that also yields a sufficiently strong output signal.

In legged locomotion, the feasible region is likewise a projected set: $\omega_{eng}, \mathrm{T}_{eng}$0 where $\omega_{eng}, \mathrm{T}_{eng}$1 enforces static equilibrium and friction, and $\omega_{eng}, \mathrm{T}_{eng}$2 enforces joint-torque feasibility. The region is a 2D horizontal convex set in the CoM plane, but it is only locally valid because the contact Jacobians are configuration-dependent (Orsolino et al., 2019).

Combinatorial usage is more rigidly geometric. For consecutive permutation patterns, the feasible region $\omega_{eng}, \mathrm{T}_{eng}$3 is the set of all limiting consecutive-pattern frequency vectors and is exactly the cycle polytope of the overlap graph: $\omega_{eng}, \mathrm{T}_{eng}$4 Its dimension is

$\omega_{eng}, \mathrm{T}_{eng}$5

its vertices are normalized incidence vectors of simple cycles, and its defining equations are conservation-of-flow constraints (Borga et al., 2019).

Process-systems flexibility analysis offers another analytical form. If the primitive constraints are $\omega_{eng}, \mathrm{T}_{eng}$6, then with $\omega_{eng}, \mathrm{T}_{eng}$7, an R-function representation defines the feasible region by

$\omega_{eng}, \mathrm{T}_{eng}$8

Here the FOR is the superlevel set of a single implicit function rather than a sampled polygon or optimization-derived boundary (Kucherenko et al., 7 Mar 2025).

3. Construction and approximation methods

FORs are computed by several distinct methodological families. One common approach is directional optimization. In AC power systems, the FOR boundary can be generated by maximizing feasible loading in different directions of the $\omega_{eng}, \mathrm{T}_{eng}$9-$c_{xy}$0 plane through an angle-based procedure: $c_{xy}$1 subject to

$c_{xy}$2

with $c_{xy}$3. This yields the boundary of achievable PCC points in all directions and a visual feasibility map in the 2D $c_{xy}$4-plane (Klein-Helmkamp et al., 2024). Related work approximates the FOR by a polygon with 36 vertices at $c_{xy}$5 resolution and uses those vertices as controller targets (Klein-Helmkamp et al., 16 Jun 2025).

A second family constructs inner approximations. For radial distribution networks, a polyhedral restriction $c_{xy}$6 is built around a known feasible operating point $c_{xy}$7. The $c_{xy}$8-stable set

$c_{xy}$9

is combined with a fixed-point map

$(v_\pi)_{\pi\in S_k}$0

If $(v_\pi)_{\pi\in S_k}$1, Brouwer’s fixed-point theorem guarantees at least one AC-feasible solution in $(v_\pi)_{\pi\in S_k}$2 (Christianen et al., 2023). This yields a certified FOR in the sense that every point in the restriction is guaranteed feasible.

A third family relies on linearization and projection. Fast mapping at TSO–DSO interfaces uses a linear power-flow approximation, expresses DER, current, and voltage constraints as a high-dimensional polytope $(v_\pi)_{\pi\in S_k}$3, and then applies Fourier-Motzkin Elimination to project onto $(v_\pi)_{\pi\in S_k}$4. On a modified IEEE 33-bus test system, the proposed FME-FOR took 0.79 s, compared with 7.5 s for the benchmark optimization method (Patig et al., 2022). Related works use piecewise linearization, mixed-integer formulations, convexification, or DistFlow SOCP relaxations to accelerate FOR construction for ancillary-service studies and operational management (Nerowski et al., 2024).

Sampling-based methods remain important. For virtual power plants, Monte Carlo feasibility estimation samples resource operating points, solves AC power flow, and retains only samples satisfying bus-voltage and thermal constraints. The resulting FOR is “basically the aggregated PQ capability curve of a VPP,” and the convex hull of sampled points is used to approximate the boundary in the figures (Riaz et al., 2019). Particle-swarm-based sampling has also been used to trace non-convex FOR edges and attach cost metadata to them in hierarchical flexibility markets (Sarstedt et al., 2021).

Data-driven approaches increasingly appear when full models or parameters are unavailable. In coal-mine virtual power plants, a learning-based feasible region assessment (LFRA) method uses inverse optimization and historical dispatch data to estimate unknown parameters $(v_\pi)_{\pi\in S_k}$5 and reconstruct a surrogate feasible region $(v_\pi)_{\pi\in S_k}$6. The case study reports many learned belt-conveyor parameter values with relative errors below 1%, boundary error under 0.3% for minimum belt-conveyor power, below 1% for maximum belt-conveyor power, and about 3% average error for power-exchange bounds (Huang et al., 2 Mar 2025). In nonlinear MPC, low-discrepancy sampling and kernel SVM classification are used to learn inner and outer approximations of the feasible-region boundary from feasibility labels (Zhou et al., 2020).

Direct analytical representations are rarer but notable. R-functions permit a closed-form implicit description of feasibility regions from primitive inequalities, including nonconvex and disjoint regions (Kucherenko et al., 7 Mar 2025). In legged robotics, the Iterative Projection algorithm directly computes the 2D CoM feasible region without first constructing higher-dimensional wrench polytopes (Orsolino et al., 2019).

4. Safety, robustness, and trajectory semantics

In control-oriented power-system work, the FOR is not merely a static capability diagram. It is explicitly the reference set of safe operating states against which trajectories are evaluated. If $(v_\pi)_{\pi\in S_k}$7 denotes the projection of the full system state at time $(v_\pi)_{\pi\in S_k}$8, the key feasibility assumption is

$(v_\pi)_{\pi\in S_k}$9

A safe trajectory set satisfies

$\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$0

a conditionally safe trajectory set requires only the final converged state to lie in $\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$1, and an unsafe set is one for which at least one final state lies outside the FOR (Klein-Helmkamp et al., 2024). This set-inclusion language turns the FOR into a certificate of closed-loop admissibility.

The same interpretation appears in online feedback optimization (OFO). The controller is safe if the reachable trajectory set remains inside the FOR: $\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$2 The OFO controller uses projected gradient descent in closed loop with the physical grid layer. Its behavior depends strongly on the gain $\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$3, the sensitivity matrix $\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$4, and the weighting matrix $\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$5. In a meshed high-voltage sub-transmission case, $\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$6 produced stable convergence for all vertices of $\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$7, whereas $\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$8 led to oscillatory behavior and transient infeasibility; about 52% of $\mathcal{F} = \left\{ s_{\text{PCC} \mid g(s_{\text{PCC}, \mathbf{x}) = 0,\; h(s_{\text{PCC}, \mathbf{x}) \le 0 \right\},$9 was reachable within the first two iterations, but for $s_{\text{PCC}}$0 the trajectory-set analysis showed $s_{\text{PCC}}$1 (Klein-Helmkamp et al., 16 Jun 2025).

Robustness under uncertainty can be expressed as an even stricter inclusion. In the same online-control setting, robustness requires

$s_{\text{PCC}}$2

The disturbance model includes Gaussian load uncertainty, uniform measurement noise, and bounded uncertainty in the sensitivity map (Klein-Helmkamp et al., 2024). In unbalanced distribution networks, the analogous object is the robust feasible region (RFR),

$s_{\text{PCC}}$3

the subset of the feasible region that remains valid for all admissible impedance realizations (Liu et al., 2022).

The T-cell literature supplies a different kind of robustness argument. The existence of a feasible operating regime is argued to be broadly generic for KPR-based biochemical networks because, with a small but nonzero bypass rate, very slow speeds or very low dissipation eventually allow bypass events to dominate. Accuracy therefore ceases to improve monotonically with time or energy, and a finite intermediate regime emerges where speed, accuracy, and energy are jointly acceptable (Cui et al., 2017).

5. Representative realizations

In early T-cell recognition, the FOR is a narrow intermediate-energy band in the speed–energy–accuracy plane. The paper’s core KPR figure divides parameter space into regions A–D, with region A as the high-accuracy discrimination regime and the feasible operating region highlighted within it where the output $s_{\text{PCC}}$4 is also strong enough to matter biologically. For $s_{\text{PCC}}$5, KPR can reach error $s_{\text{PCC}}$6, with power consumption about $s_{\text{PCC}}$7 ATP/s, far below the cell’s estimated $s_{\text{PCC}}$8 ATP/s budget, while remaining consistent with the observed 1–5 minute decision window. Adaptive sorting maintains absolute discrimination but is much slower than KPR and struggles to achieve low error on the experimentally relevant $s_{\text{PCC}}$9 s timescale (Cui et al., 2017).

In internal combustion engines, the FOR is the full speed-load map over which a mean-value exergy model is well defined. The state variables are primarily $\mathbf{x}$0 and $\mathbf{x}$1, and each feasible point supports a steady-state exergy balance

$\mathbf{x}$2

with exergy inflows, outputs, and destructions computed from map data and a crank-angle-resolved model. For a turbocharged 6.4 L V8 diesel with 20% EGR, the static maps show that $\mathbf{x}$3 remains greater than 1 everywhere, mechanical work fraction reaches about 38%, heat-transfer exergy drops to about 2.5% at high speed/load, friction losses depend mainly on speed, and the residual $\mathbf{x}$4 stays modest, around a few percent on average (Pozzato et al., 2021).

In legged locomotion, the feasible region replaces the classical support polygon by requiring not only tipover and slip avoidance but also existence of joint torques within limits. The modified Iterative Projection method computes this actuation-aware region at least 20× faster for 3 point contacts and about 50× faster for 4 point contacts than routes based on higher-dimensional wrench polytopes. Solve times are below 10 ms in 4-stance and below 7.5 ms in 3-stance in 99.5% of computations, enabling operation around 100–133 Hz in quadruped stance configurations. Experiments on the HyQ robot demonstrated rough-terrain locomotion while carrying an extra 10 kg payload (Orsolino et al., 2019).

In permutation theory, the feasible region is completely explicit rather than approximate. For fixed pattern size $\mathbf{x}$5, $\mathbf{x}$6 is the cycle polytope of the overlap graph, has dimension $\mathbf{x}$7, vertices given by simple cycles, and faces corresponding to non-empty full subgraphs. The paper further shows that classical pattern limits and consecutive pattern limits are independent, implying that the scaling limit of a sequence of permutations induces no constraints on the local limit, and vice versa (Borga et al., 2019).

A plausible implication is that the phrase “operating region” ranges from biologically viable nonequilibrium performance windows to algebraically exact convex-geometric objects, but in each case the region identifies what is not merely conceivable, but realizable.

6. Related concepts and recurrent distinctions

A central distinction in power-system flexibility work is between feasibility and flexibility. For virtual power plants, the FOR represents what can physically and electrically be sustained at a given instant, whereas the Flexibility Operating Region (FXOR) is the subset reachable from the current dispatch point within a specified response time $\mathbf{x}$8: $\mathbf{x}$9 Thus a point may lie in the FOR but be excluded from the FXOR because of activation-time or ramp-rate constraints (Riaz et al., 2019).

A second distinction is between a static capability envelope and a time-varying subset. In cross-voltage-level power-flow control, FORs describe theoretically achievable operational states of assets or network areas, while Flexibility Regions (FRs) are time-specific subsets reduced by exogenous factors such as solar irradiance, wind availability, and load demand. Operationally, the FOR is computed for a fixed network topology and resource pool, whereas FRs are recomputed periodically (Nerowski et al., 2024).

A third distinction is between operation and planning. In distribution-grid planning, the FOR is the $\mathcal{F}\subset\mathbb{R}^n$0 capability region of a given grid stage, while the Feasible Planning Region (FPR) is a set of FORs aligned along a cost axis $\mathcal{F}\subset\mathbb{R}^n$1. Lower-cost stages have smaller FORs and higher-cost stages have larger ones; the FPR therefore represents the admissible trade-off between interconnection power flow and distribution-grid expansion cost (Böttcher et al., 2023).

Other neighboring notions include the Operating Envelope (OE), geometrically linked to feasible regions for DERs, and the Robust Feasible Region (RFR), the subset of a feasible region that remains valid under uncertainty (Liu et al., 2022). In process-systems and nonlinear-MPC literature, the dominant terms are “feasibility region,” “operability region,” and the feasible region of the controller; these are not always labeled FOR, but they instantiate the same set-theoretic idea of admissible conditions under constraints (Kucherenko et al., 7 Mar 2025).

Across these usages, the most persistent misconception is to equate a FOR with a maximal boundary independent of time, signal strength, controllability, or internal realizability. The surveyed work repeatedly rejects that simplification. A FOR may exclude high-accuracy but biologically useless T-cell states, power-system setpoints that are feasible but not safely reachable, CoM positions inside a support polygon that violate torque limits, or dispatch points that are feasible in principle but not within a required response time (Cui et al., 2017).