diff --git a/opm/simulators/flow/BlackoilModel.hpp b/opm/simulators/flow/BlackoilModel.hpp
index 4182958b0..2ccf4a093 100644
--- a/opm/simulators/flow/BlackoilModel.hpp
+++ b/opm/simulators/flow/BlackoilModel.hpp
@@ -1009,6 +1009,7 @@ namespace Opm {
 
             const auto tol_cnv = use_relaxed_cnv ? param_.tolerance_cnv_relaxed_ : param_.tolerance_cnv_;
             const auto tol_mb  = use_relaxed_mb ? param_.tolerance_mb_relaxed_ : param_.tolerance_mb_;
+            const auto tol_cnv_energy = use_relaxed_cnv ? param_.tolerance_cnv_energy_relaxed_ : param_.tolerance_cnv_energy_;
 
             // Finish computation
             std::vector<Scalar> CNV(numComp);
@@ -1031,7 +1032,9 @@ namespace Opm {
                     CR::ReservoirFailure::Type::Cnv,
                 };
 
-                const Scalar tol[2] = { tol_mb, tol_cnv, };
+                Scalar tol[2] = { tol_mb, tol_cnv, };
+                if (has_energy_ && compIdx == contiEnergyEqIdx)
+                    tol[1] = tol_cnv_energy;
 
                 for (int ii : {0, 1}) {
                     if (std::isnan(res[ii])) {
diff --git a/opm/simulators/flow/BlackoilModelParameters.cpp b/opm/simulators/flow/BlackoilModelParameters.cpp
index 23eae8231..b2abb3890 100644
--- a/opm/simulators/flow/BlackoilModelParameters.cpp
+++ b/opm/simulators/flow/BlackoilModelParameters.cpp
@@ -40,6 +40,8 @@ BlackoilModelParameters<Scalar>::BlackoilModelParameters()
     tolerance_mb_relaxed_ = std::max(tolerance_mb_, Parameters::Get<Parameters::ToleranceMbRelaxed<Scalar>>());
     tolerance_cnv_ = Parameters::Get<Parameters::ToleranceCnv<Scalar>>();
     tolerance_cnv_relaxed_ = std::max(tolerance_cnv_, Parameters::Get<Parameters::ToleranceCnvRelaxed<Scalar>>());
+    tolerance_cnv_energy_ = Parameters::Get<Parameters::ToleranceCnvEnergy<Scalar>>();
+    tolerance_cnv_energy_relaxed_ = std::max(tolerance_cnv_energy_, Parameters::Get<Parameters::ToleranceCnvEnergyRelaxed<Scalar>>());
     tolerance_wells_ = Parameters::Get<Parameters::ToleranceWells<Scalar>>();
     tolerance_well_control_ = Parameters::Get<Parameters::ToleranceWellControl<Scalar>>();
     max_welleq_iter_ = Parameters::Get<Parameters::MaxWelleqIter>();
@@ -115,6 +117,11 @@ void BlackoilModelParameters<Scalar>::registerParameters()
     Parameters::Register<Parameters::ToleranceCnvRelaxed<Scalar>>
         ("Relaxed local convergence tolerance that applies for iterations "
          "after the iterations with the strict tolerance");
+    Parameters::Register<Parameters::ToleranceCnvEnergy<Scalar>>
+        ("Local energy convergence tolerance (Maximum of local energy errors)");
+    Parameters::Register<Parameters::ToleranceCnvEnergyRelaxed<Scalar>>
+        ("Relaxed local energy convergence tolerance that applies for iterations "
+         "after the iterations with the strict tolerance");
     Parameters::Register<Parameters::ToleranceWells<Scalar>>
         ("Well convergence tolerance");
     Parameters::Register<Parameters::ToleranceWellControl<Scalar>>
diff --git a/opm/simulators/flow/BlackoilModelParameters.hpp b/opm/simulators/flow/BlackoilModelParameters.hpp
index 01df8c35c..ff61fd9af 100644
--- a/opm/simulators/flow/BlackoilModelParameters.hpp
+++ b/opm/simulators/flow/BlackoilModelParameters.hpp
@@ -52,6 +52,12 @@ struct ToleranceCnv { static constexpr Scalar value = 1e-2; };
 template<class Scalar>
 struct ToleranceCnvRelaxed { static constexpr Scalar value = 1.0; };
 
+template<class Scalar>
+struct ToleranceCnvEnergy { static constexpr Scalar value = 1e-4; };
+
+template<class Scalar>
+struct ToleranceCnvEnergyRelaxed { static constexpr Scalar value = 2.0e-4; };
+
 template<class Scalar>
 struct ToleranceWells { static constexpr Scalar value = 1e-4; };
 
@@ -151,6 +157,10 @@ public:
     Scalar tolerance_cnv_;
     /// Relaxed local convergence tolerance (can be used when iter >= min_strict_cnv_iter_ && cnvViolatedPV < relaxed_max_pv_fraction_).
     Scalar tolerance_cnv_relaxed_;
+    /// Local energy convergence tolerance (max of local energy errors).
+    Scalar tolerance_cnv_energy_;
+    /// Relaxed local energy convergence tolerance (can be used when iter >= min_strict_cnv_iter_ && cnvViolatedPV < relaxed_max_pv_fraction_).
+    Scalar tolerance_cnv_energy_relaxed_;
     /// Well convergence tolerance.
     Scalar tolerance_wells_;
     /// Tolerance for the well control equations